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Generalized Serre–Tate Ordinary Theory

Adrian Vasiu∗

December 1, 2007

Abstract. We study a generalization of Serre–Tate theory of ordinary abelian varieties andtheir deformation spaces. This generalization deals with abelian varieties equipped withadditional structures. The additional structures can be not only an action of a semisimplealgebra and a polarization, but more generally the data given by some “crystalline Hodgecycles” (a p-adic version of a Hodge cycle in the sense of motives). Compared to Serre–Tateordinary theory, new phenomena appear in this generalized context.

We give an application of this theory to the existence of “good” integral models ofthose Shimura varieties whose adjoints are products of simple, adjoint Shimura varietiesof DH

l type with l ≥ 4.

Key words: p-divisible groups and objects, F -crystals, Newton polygons, ordinariness,canonical lifts, reductive group schemes, formal Lie groups, abelian varieties, Shimuravarieties, integral models, Hodge cycles, complex multiplication, and deformation spaces.

MSC 2000: 11G10, 11G15, 11G18, 14F20, 14F30, 14F40, 14G35, 14K10, 14K22, 14L05,14L15, and 17B45.

* Department of Mathematical Sciences, Binghamton University, Binghamton, NY13902-6000, U.S.A. e-mail: adrian@math.binghamton.edu

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1. Classical Serre–Tate ordinary theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2. Translation of the classical theory into our language . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3. Generalized language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4. Main results on Shimura p-divisible objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5. Standard Hodge situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6. Main results on standard Hodge situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7. On applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.8. Extra literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.9. Our main motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.10. More on organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1. Standard notations for the abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2. Descending slope filtration Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3. Elementary properties of reductive group schemes . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4. Natural Zp structures and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5. Lie F -isocrystals and parabolic subgroup schemes . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6. Ten group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.7. The opposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.8. Elementary crystalline complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Abstract theory, I: proofs of Theorems 1.4.1 and 1.4.2 . . . . . . . . . . . . . . . . . 50

3.1. On Hodge cocharacters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2. A general outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3. The essence of the proofs of Theorems 1.4.1 and 1.4.2 . . . . . . . . . . . . . . . . . . . . 56

3.4. The split, adjoint, cyclic context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5. The proof of the property 3.3 (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6. Formulas for group normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.7. The proof of the property 3.3 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.8. End of the proofs of Theorems 1.4.1 and 1.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.9. On Hasse–Witt invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Abstract theory, II: proofs of Theorems 1.4.3 and 1.4.4 . . . . . . . . . . . . . . . . 83

4.1. Proof of Theorem 1.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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4.2. Proofs of Theorem 1.4.4 and Corollary 1.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Abstract theory, III: formal Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1. Sign p-divisible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2. Minimal decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3. Formal Lie groups: the adjoint, cyclic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4. Formal Lie groups: the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Geometric theory: proofs of the Theorems 1.6.1 to 1.6.3 . . . . . . . . . . . . . . 108

6.1. Fontaine comparison theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2. Deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3. Proof of Theorem 1.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.4. The existence of geometric lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.5. Proofs of Theorems 1.6.2 and 1.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Examples and complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1. On the Newton polygon N0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.2. Modulo p interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3. Examples on p-ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4. On the Sh-ordinary locus O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.5. Example for U-ordinariness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8 Two special applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.1. Applications to a conjecture of Milne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.2. Applications to integral canonical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9 Generalization of the properties 1.1 (f) to (h) to the context of

standard Hodge situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.1. Generalization of the property 1.1 (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.2. Generalization of the property 1.1 (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9.3. Basic notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.4. Theorem (the existence of canonical formal Lie group structures for I0 ≤ 1) 165

9.5. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.6. Ramified valued points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

9.7. A functorial property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.8. Four operations for the case I0 ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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1 Introduction

In this monograph we develop a theory of ordinary abelian varieties and p-divisiblegroups with additional structures. We use it to give applications to the existence of “good”integral models of those Shimura varieties whose adjoints are products of simple, adjointShimura varieties of DH

l type with l ≥ 4.

The additional structures can be defined by an action of a semisimple algebra, by apolarization, or more generally by some “p-adic Hodge cycles”.

In this introduction we first begin by recalling the classical Serre–Tate ordinary theoryof abelian varieties and their deformation spaces. In our theory this will correspond to thecase of a general linear group, the more general case corresponding to a “p-adic Mumford–Tate group” (i.e., a p-adic reductive group scheme whose generic fibre fixes the Hodgetensors that define the additional structures). After having recalled the classical Serre–Tate ordinary theory, we will give the translation of part of it into our language. Then wewill explain the more general objects we consider and we will state our main results. Wewill end the introduction by mentioning applications and our main motivations.

For reader’s convenience, here is a list of standard notions and notations that pertainto (group) schemes. The reader ought to look into it only when needed.

Standard language and notations on (group) schemes. We recall that a groupscheme H over an affine scheme Spec(R) is called reductive, if it is smooth and affine andits fibres are connected and have trivial unipotent radicals (cf. [DG, Vol. III, Exp. XIX,Def. 2.7]). It is known that H is of finite presentation over Spec(R), cf. [DG, Vol. III, Exp.XIX, Sect. 2.1 or Rm. 2.9]. If Spec(R) is connected, then H is a semisimple group scheme(resp. is a torus) if and only if one fibre of H is a semisimple group (resp. is a torus) (cf.[DG, Vol. III, Exp. XIX, Cor. 2.6]). Let Hder, Z(H), Hab, and Had be the derived groupscheme of H, the center of H, the maximal commutative quotient of H, and the adjointgroup scheme of H (respectively). Thus we have Had = H/Z(H) and Hab = H/Hder,cf. [DG, Vol. III, Exp. XXII, Def. 4.3.6 and Thm. 6.2.1]. Moreover Hder and Had aresemisimple group schemes, Hab is a torus, and (cf. [DG, Vol. III, Exp. XXII, Cor. 4.1.7])Z(H) is a group scheme of multiplicative type. Let Z0(H) be the maximal torus of Z(H);the quotient group scheme Z(H)/Z0(H) is finite and of multiplicative type.

Let F be a closed subgroup scheme of H; if R is not a field, then we assume F issmooth. Let Lie(F ) be the Lie algebra over R of F . As R-modules, we identify Lie(F ) withKer(F (R[x]/x2) → F (R)), where the R-epimorphism R[x]/(x2) ։ R takes x to 0. The Liebracket on Lie(F ) is obtained by taking the total differential of the commutator morphismF ×Spec(R) F → F at identity sections. Often we say F and H are over R and we writeF ×R F instead of F ×Spec(R) F . For a finite, flat, affine morphism Spec(R) → Spec(R0),

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let ResR/R0F be the group scheme over Spec(R0) obtained from F through the Weil

restriction of scalars (see [BLR, Ch. 7, Sect. 7.6] and [Va5, Subsect. 2.3]).

By a parabolic subgroup scheme of H we mean a smooth, closed subgroup scheme PH

of H whose fibres are parabolic subgroup schemes over spectra of fields. The unipotentradical UH of PH is the maximal unipotent, smooth, normal, closed subgroup scheme ofPH ; it has connected fibres. The quotient group scheme PH/UH exists and is reductive, cf.[DG, Vol. III, Exp. XXVI, Prop. 1.6]. We identify UH with the closed subgroup schemeof Had which is the unipotent radical of the parabolic subgroup scheme Im(PH → Had) ofHad. By a Levi subgroup scheme LH of PH we mean a reductive, closed subgroup schemeof PH such that the natural homomorphism LH → PH/UH is an isomorphism.

For a free R-module M of finite rank, let GLM be the reductive group scheme overSpec(R) of linear automorphisms of M . Thus for a commutative R-algebra R1, GLM (R1)is the group of R1-linear automorphisms ofM ⊗RR1. If Lie(F ) is a free R-module of finiterank, then in expressions of the form GLLie(F ) we view Lie(F ) only as a free R-module;thus we view Lie(F ) as an F -module via the adjoint representation F → GLLie(F ). If f1,f2 ∈ EndZ(M), let f1f2 := f1 f2 ∈ EndZ(M). Let M∗ := HomR(M,R).

A bilinear form ψM : M ⊗R M → R on M is called perfect if it induces naturallyan R-linear isomorphism M→M∗. If ψM is perfect and alternating, let GSp(M,ψM) andSp(M,ψM ) be the reductive group schemes over R of symplectic similitude isomorphismsand of symplectic isomorphisms (respectively) of (M,ψM).

Let K be an algebraic closure of a field K. If T → S is a morphism of schemesand X (resp. XS or Xi with i as an index) is an S-scheme, let XT := X ×S T (resp.XT or Xi,T ) be the T -scheme which is the pull back of X (resp. of XS or Xi). IfT = Spec(R) → S = Spec(R0) is a morphism of affine schemes, we often denote XT andXi,T by XR and Xi,R (respectively). The same type of notations apply for morphismsof schemes and for (morphisms of) p-divisible groups. If R is a complete, local ring, weidentify the categories of p-divisible groups over Spf(R) and Spec(R) (cf. [dJ, Lem. 2.4.4]).

1.1. Classical Serre–Tate ordinary theory

Let p ∈ N be a prime. Let k be a perfect field of characteristic p. Let W (k) be thering of Witt vectors with coefficients in k. Let B(k) := W (k)[ 1p ] be the field of fractions

of W (k). Let σ := σk be the Frobenius automorphism of k, W (k), and B(k). Let r,d ∈ N ∪ 0 with r ≥ d. In late sixties Serre and Tate developed an ordinary theory forp-divisible groups and abelian varieties over k. It can be summarized as follows.

Ordinary p-divisible groups. A p-divisible group D over k of height r and dimension dis called ordinary if and only if one of the following five equivalent conditions holds for it:

(a) it is a direct sum of an etale p-divisible group and of a p-divisible group of multi-plicative type;

(b) its Newton polygon is below the Newton polygon of every other p-divisible groupover k of height r and dimension d;

(c) its Hasse–Witt invariant is equal to r − d;

(d) there exist isomorphisms between D[p]k and µµµdp ⊕ (Z/pZ)r−d;

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(e) there exist isomorphisms between Dk and µµµdp∞ ⊕ (Qp/Zp)

r−d.

Canonical lifts. If D is ordinary, then there exists a unique p-divisible group DW (k) overW (k) which lifts D and which is a direct sum as in (a). It is called the canonical lift of D.Via reduction modulo p we have an identity End(DW (k)) = End(D) of Zp-algebras.

Abelian varieties. An abelian variety A over k is called ordinary if its p-divisible groupis so. Ordinary abelian varieties are of particular importance as in the reduction modulop of the moduli spaces of principally polarized abelian varieties, they form open densesubspaces. If A is ordinary, then there exists a unique abelian scheme AW (k) over W (k)which lifts A and whose p-divisible group is a direct sum as in (a); it is called the canonicallift of A. If k is an algebraic closure F of the field Fp with p elements, then a theorem ofTate asserts that AW (k) has complex multiplication. If k is finite, then AW (k) is the uniquelift of A to W (k) which has the property that the Frobenius endomorphism of A lifts to it.

Deformation theory. Let Duniv be the universal p-divisible group over the deformationspace D(D) ofD (see [Il, Thm. 4.8]; based on [dJ, Lem. 2.4.4], we view D(D) as a scheme).If D is ordinary, then the main properties of Duniv are the following four:

(f) we have a short exact sequence 0 → D(1)univ → Duniv → D

(0)univ → 0 of p-divisible

groups over D(D), where D(1)univ is of multiplicative type and D

(0)univ is etale;

(g) if k = k, if V is a finite, discrete valuation ring extension of W (k), if K := V [ 1p ], if

DV is a p-divisible group over V that lifts D, if T ∗p (DK) is the dual of the Tate module of

DK , then the p-adic Galois representation Gal(K/K) → GLT∗p (DK)(Zp) factors through

the group of Zp-valued points of a connected, smooth, solvable, closed subgroup schemeof GLT∗

p (DK);

(h) if k = k, then the formal scheme defined by D(D) has a canonical structure of aformal torus over Spf(W (k)), the identity section corresponding to the canonical lift of D;

(i) if k = k, then there exist isomorphisms D(D)→Spec(W (k)[[x1, . . . , xd(r−d)]]) thatdefine canonical coordinates for D(D) which are unique up to a suitable action of the groupGLd(r−d)(Zp) of automorphisms of the d(r− d) dimensional formal torus over Spf(W (k)).

The above results on D, DW (k), and AW (k) were published as an appendix to [Me].The equivalence (a) ⇔ (b) is also a particular case of a theorem of Mazur (see [Ka2, 1.4.1]).All the above results on D(D) are contained in [Ka1], [Ka3], [Ka4], and [De4]. A variantfor principally quasi-polarized p-divisible groups can be deduced easily from the aboveresults and thus it is also part of the classical Serre–Tate ordinary theory. If k = k, thenthe formal torus structure of (h) is obtained naturally once one remarks that:

(h.a) the short exact sequence of (f) is the universal extension of D(0)univ→(Qp/Zp)

r−d

by D(1)univ→µµµd

p∞ (here the two isomorphisms are over D(D));

(h.b) the short exact sequence 0 → Zr−dp → Qr−d

p → (Qp/Zp)r−d → 0 gives birth

to a (push forward) coboundary morphism Hom(Zr−dp , µµµd

p∞) → Ext1((Qp/Zp)r−d, µµµd

p∞)which is an isomorphism between two formal tori over Spf(W (k)) of dimension d(r − d).

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We recall that a formal Lie group F over Spf(W (k)) is a contravariant functor fromthe category of local, artinian W (k)-schemes of residue field k into the category of groupswhich, when viewed as a functor into the category of sets, is representable by Spf(R),where R = W (k)[[x1, . . . , xm]] for some number m ∈ N ∪ 0 called the dimension of Fand for some independent variables x1, . . . , xm (see [Me, Ch. 2, 1.1.5] and [Fo1]).

1.2. Translation of the classical theory into our language

In Subsection 1.2.1 we recall few classical results. In Subsections 1.2.2 to 1.2.7 weinclude intrinsic crystalline interpretations of some parts of Section 1.1. Some interpreta-tions are new and they are meant: (i) to motivate the abstract notions to be introducedin Section 1.3, and (ii) to detail few of the new ideas we will use in this monograph.

Let D be a p-divisible group over k of height r and dimension d. Let (M,ϕ) bethe (contravariant) Dieudonne module of D. We recall that M is a free W (k)-moduleof rank r and ϕ : M → M is a σ-linear endomorphism such that we have pM ⊆ ϕ(M)and dimk(M/ϕ(M)) = d. Let ϑ : M → M be the σ−1-linear endomorphism that is theVerschiebung map of (M,ϕ); we have identities ϑϕ = ϕϑ = p1M . Let M := M/pM . Letϕ : M → M and ϑ : M → M be the reductions modulo p of ϕ and ϑ (respectively). Thek-vector space Ker(ϕ) = Im(ϑ) has dimension d. We recall that the classical Dieudonnetheory says that the category of p-divisible groups over k is antiequivalent to the categoryof (contravariant) Dieudonne modules over k (see [Fo1, Ch. III, Prop. 6.1 iii)], etc.).

1.2.1. Lifts. Each lift of D to a p-divisible group DW (k) over W (k) defines naturally adirect summand F 1 of M which modulo p is Ker(ϕ) (see [Me], [BBM], etc.); the directsummand F 1 is called the Hodge filtration of DW (k). We have ϕ−1(M) = M + 1

pF1. We

refer to the triple (M,F 1, ϕ) as a filtered Dieudonne module over k and to F 1 as a lift of(M,ϕ). It is well known that:

(a) if p ≥ 3 or if p = 2 and either D or its Cartier dual Dt is connected, then thecorrespondence DW (k) ↔ F 1 is a bijection.

The case p ≥ 3 is a consequence of the Grothendieck–Messing deformation theory (see[Me]). The general case is (for instance) a consequence of [Fo1, Ch. IV, Prop. 1.6]. In loc.cit. it is proved more generally that the following two properties hold:

(b) if p ≥ 3, then the category of p-divisible groups over W (k) is antiequivalent tothe category of filtered Dieudonne modules over k;

(c) if p = 2, then the category of p-divisible groups over W (k) that are connected(resp. that have connected Cartier duals) is antiequivalent to the category of filteredDieudonne modules over k that do not have Newton polygon slope 0 (resp. do not haveNewton polygon slope 1).

We emphasize that strictly speaking, [Fo1, Ch. IV, Prop. 1.6] is stated in terms ofHonda triples (M,ϕ( 1

pF 1), ϕ) and not in terms of filtered Dieudonne modules (M,F 1, ϕ).

The classical Hasse–Witt invariant HW (D) of D (or of (M,ϕ)) can be defined as

HW (D) := dimk(∩m∈Nϕm(M)).

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It is the multiplicity of the Newton polygon slope 0 of (M,ϕ).

1.2.2. End’s of Dieudonne modules. One can easily translate the theory of ordinaryp-divisible groups over k recalled in Section 1.1, in terms of the Dieudonne module (M,ϕ).We want to make such a translation intrinsic, expressed only in terms of the algebraicgroup GLM (or rather its Lie algebra EndW (k)(M)) and not in terms of the GLM -moduleM . Later on in our theory, we will replace GLM by a reductive, closed subgroup schemeof it. Next we will have a look back at Section 1.1 from such an intrinsic point of view.

The canonical identification EndB(k)(M [ 1p ]) = M [ 1p ] ⊗B(k) M∗[ 1p ] allows us to view

EndB(k)(M [ 1p ]) as the B(k)-vector space of an F -isocrystal over k. More precisely, we

will denote also by ϕ the σ-linear automorphism of EndB(k)(M [ 1p ]) that takes an element

x ∈ EndB(k)(M [ 1p ]) to ϕx ϕ−1 ∈ EndB(k)(M [ 1p ]). Let Ψ : EndW (k)(M) → EndW (k)(M)

be the σ-linear endomorphism defined by the rule: for x ∈ EndW (k)(M) we have

Ψ(x) := p(ϕ(x)) = pϕ x ϕ−1 = ϕ x ϑ ∈ EndW (k)(M).

The pair (EndW (k)(M),Ψ) is the tensor product of the Dieudonne modules of D and Dt.

Let Ψ : Endk(M) → Endk(M) be the reduction modulo p of Ψ. The integer

I(D) := dimk(∩m∈NΨm(Endk(M)))

computes the multiplicity of the Newton polygon slope 0 of Ψ and thus also the multiplicityof the Newton polygon slope −1 of (EndB(k)(M [ 1p ]), ϕ). Based on this and the last sentenceof the previous paragraph, we get the identity

I(D) = HW (D)HW (Dt).

In particular, if D has a quasi-polarization (i.e., it is isogeneous to Dt), then we haveHW (D) =

√I(D). Moreover, we easily get that

D is ordinary ⇐⇒ I(D) = d(r − d).

1.2.3. Characterisation of p-torsion subgroup schemes in terms of (Endk(M),Ψ).For g ∈ GLM (W (k)), let gad be its image in GLad

M (k). Let D1 be the p-divisible groupover k whose Dieudonne module is (M, gϕ). If k = k, then the σ-linear endomorphism Ψ“encompasses” both maps ϕ, ϑ :M →M . We exemplify this property modulo p.

Claim. If k = k, then there exists an isomorphism D[p]→D1[p] if and only if there existsan element h ∈ GLad

M (k) such that as σ-linear endomorphisms of Endk(M) we have anidentity hΨ = gadΨh (here we need k = k as Ψ can not “differentiate” between ϕ and itsGm(W (k))-multiples).

We check here the “if” part of the Claim. Let h ∈ GLM (W (k)) be an element such

that we have had

= h. As we have hΨ = gadΨh, the reduction modulo p of h normalizesKer(Ψ) = x ∈ Endk(M)|x(Im(ϑ)) ⊆ Ker(ϕ) and therefore it normalizes Ker(ϕ) = Im(ϑ).

8

This implies that (M,hϕh−1) is a Dieudonne module. Thus by replacing ϕ with hϕh−1,we can assume that h ∈ GLad

M (k) is the identity element; therefore Ψ = gadΨ. Wedefine M1 := ϕ( 1pF

1)/ϕ(F 1) and M0 := Im(ϕ). We have a direct sum decomposition

M =M1 ⊕M0 of k-vector spaces, cf. the identity ϕ−1(M) =M + 1pF

1. As Ψ = gadΨ, the

element gad fixes Im(Ψ). But Im(Ψ) is the direct summand Homk(M1,M0) of

Endk(M) = Endk(M1)⊕ Endk(M0)⊕ Homk(M1,M0)⊕ Homk(M0,M1)

and thus the centralizer of Im(Ψ) in Endk(M) is Im(Ψ)⊕ k1M . Therefore there exists an

element x ∈ EndW (k)(M) such that gad is the image inGLadM (k) of the element 1M+Ψ(x) ∈

GLM (W (k)). Let g1 := 1M + px ∈ GLM (W (k)). By replacing gϕ with

g1gϕg−11 = g1gϕg

−11 ϕ−1ϕ = g1g(1M +Ψ(x))−1ϕ,

g gets replaced by the element g1g(1M +Ψ(x))−1 whose image in GLadM (k) is the identity

element. Therefore we can assume that gad ∈ GLadM (k) is the identity element. Thus,

as D1[p] depends only on g modulo p, we can assume that g ∈ Z(GLM )(W (k)). In thislast case, as k = k there exists an isomorphism (M, gϕ)→(M,ϕ) defined by an element ofZ(GLM )(W (k)) and therefore in fact we have D→D1.

1.2.4. New p-divisible groups via Lie algebras. Until Section 1.3 we will work overan arbitray perfect field k of characteristic p and we take D to be ordinary and DW (k) tobe its canonical lift. We have ϕ(F 1) = pF 1. Let F 0 be the direct supplement of F 1 inM such that we have ϕ(F 0) = F 0. Let P+ and P− be the maximal parabolic subgroupschemes of GLM that normalize F 1 and F 0 (respectively). The map Ψ leaves invariantboth Lie(P+) and Lie(P−). For ∗ ∈ +,− let U∗ be the unipotent radical of P ∗ andlet S∗ be the solvable, closed subgroup scheme of GLM generated by U∗ and by theGm subgroup scheme of GLM that fixes F 0 and that acts as the inverse of the identicalcharacter of Gm on F 1. We remark that U∗ is constructed canonically in terms of theF -isocrystal (EndB(k)(M [ 1

p]), φ), as Lie(U∗)[ 1

p] is the Newton polygon slope ∗1 part of it.

Moreover S∗ is obtained from U∗ via a minuscule cocharacter µ : Gm → GLM , a p-adicHodge cocharacter analogue to the classical Hodge cocharacters.

We identify naturally Lie(U+) with HomW (k)(F0, F 1), Lie(U−) with HomW (k)(F

1, F 0),Lie(S+) with HomW (k)(F

0, F 1)⊕W (k)1F 1 , and Lie(S−) with HomW (k)(F1, F 0)⊕W (k)1F 1

(the last two identities are of W (k)-modules). The W (k)-module

E := EndW (k)(F0)⊕ HomW (k)(F

1, F 0)⊕ x ∈ EndW (k)(F1)|trace of x is 0

is left invariant by Ψ. If p does not divide d = rkW (k)(F1), then we have a direct

sum decomposition EndW (k)(M) = E ⊕ Lie(S+) left invariant by Ψ. In general (i.e.,regardless of the fact that p does or does not divide d), we can identify (Lie(S+),Ψ) with(EndW (k)(M)/E,Ψ) in such a way that HomW (k)(F

0, F 1) gets identified with its image inEndW (k)(M)/E and 1F 1 maps to the image in EndW (k)(M)/E of an arbitrary endomor-phism t ∈ EndW (k)(F

1) whose trace is 1.

9

Let D+W (k) be the p-divisible group of multiplicative type over W (k) whose filtered

Dieudonne module is (Lie(U+),Lie(U+), ϕ). We call it the positive p-divisible group overW (k) of D or of (M,ϕ). The p-divisible group D+(0) over k whose Dieudonne module is(Lie(S+), ϕ) is called the standard non-negative p-divisible group of D or of (M,ϕ); it isthe direct sum of Qp/Zp (over k) and of the special fibre D+ of D+

W (k).

Let D−W (k) be the etale p-divisible group over W (k) whose filtered Dieudonne module

is (Lie(U−), 0,Ψ). We call it the negative p-divisible group over W (k) of D or of (M,ϕ).The p-divisible group D−(0) over k whose Dieudonne module is (Lie(S−),Ψ) is called thestandard non-positive p-divisible group of D or of (M,ϕ); it is the direct sum of µµµp∞ (over

k) and of the special fibre D− of D−W (k).

1.2.5. Deformation theory revisited. Let R := W (k)[[x1, . . . , xd(r−d)]]. Let ΦR

be the Frobenius lift of R that is compatible with σ and that takes xi to xpi for all i ∈1, . . . , d(r−d). DeformingD is equivalent to deformingD+(0). The easiest way to see thisis to start with a versal deformation of D over Spec(R) and to show that the deformationof D+(0) we get naturally by identifying Lie(S+) with EndW (k)(M)/E is versal. At thelevel of filtered F -crystals over R/pR, this goes as follows.

Let u−univ : Spec(R) → U− be a formally etale morphism which modulo the idealI := (x1, . . . , xd(r−d)) of R defines the identity section of U−. We also identify u−univ withan R-linear automorphism of M ⊗W (k) R which modulo I is 1M . Faltings deformationtheory (see [Fa2, Thm. 10]) and [dJ, Lem. 2.4.4] assure us that there exists a p-divisiblegroup over Spec(R) such that: (i) its reduction modulo I is DW (k), and (ii) its filteredF -crystal over R/pR is the quadruple

C := (M ⊗W (k) R, F1 ⊗W (k) R, u

−univ(ϕ⊗ ΦR),∇),

where ∇ is an integrable, nilpotent modulo p connection on M ⊗W (k) R that is uniquely

determined by u−univ(ϕ⊗ ΦR). It is easy to see that the Kodaira–Spencer map of ∇ is anisomorphism. Thus we can naturally identify D(D) with Spec(R).

Similarly (via the identification Lie(S+) = EndW (k)(M)/E) we have a natural iden-

tification D(D) = Spec(R) under which the filtered F -crystal of the deformation of D+(0)

over D(D+(0)) is the following quadruple

C+(0) := ((EndW (k)(M)/E)⊗W (k) R,HomW (k)(F0, F 1)⊗W (k) R, u

−univ(ϕ⊗ ΦR),∇

+(0)),

where ∇+(0) is the connection on (EndW (k)(M)/E)⊗W (k) R induced naturally by ∇.

We explain why C+(0) defines indeed a versal deformation of D+(0). Let U−+ be theunipotent radical of the maximal parabolic subgroup scheme of GLEndW (k)(M)/E thatnormalizes the image of W (k)t in EndW (k)(M)/E (see Subsection 1.2.4 for t). Thegroup scheme U− acts naturally on EndW (k)(M)/E via inner conjugation and passageto quotients. The resulting homomorphism U− → GLEndW (k)(M)/E is a monomor-

phism that gives birth to an isomorphism q : U−→U−+. At the level of matrices thisis equivalent to the following statement: if C is a commutative W (k)-algebra and if

10

x ∈ HomW (k)(F1, F 0) ⊗W (k) C \ 0, then for nC := 1M⊗W (k)C + x =

(1 0x 1

)∈ U−(C)

there exists y =(0 y0 0

)∈ HomW (k)(F

0, F 1)⊗W (k) C such that the element

nCyn−1C − y =

(−yx 0

−xyx xy

)∈ EndW (k)(M)⊗W (k) C

has the property that the trace of −yx ∈ EndW (k)(F1)⊗W (k)C is non-zero. Here the 2×2

matrices are with respect to the direct sum decomposition M = F 1 ⊕ F 0 and thereforetheir entries are matrices themselves.

Let u−+univ := q u−univ : Spec(R) → U−+; it is a formally etale morphism which modulo

I defines the identity section of U−+. This implies that C+(0) defines indeed a versaldeformation of D+(0).

Thus we can identify naturally D(D) with Spec(R) = D(D+(0)). But as we haveD+(0) = Qp/Zp ⊕ D+, the classical Serre–Tate ordinary theory tells us that the formalscheme defined by D(D+(0)) has a canonical structure of a formal Lie group L+ overSpf(W (k)). We recall that if C is a local, artinian W (k)-algebra of residue field k, thenthe addition operation on the set D(D+(0))(C) = L+(C) is defined via the Yoneda additionlaw for short exact sequences of p-divisible groups over C that are of the following form

0 → D+C → D

+(0)C → Qp/Zp → 0.

It is well known that L+ is isomorphic to the formal Lie group of D+W (k) (see Lemma 5.3.3;

see also the property 1.1 (h.b)).

Due to the identification D(D) = D(D+(0)), the formal scheme defined by D(D) hasalso a natural structure of a formal Lie group L which can be identified with L+.

1.2.6. Sums of lifts. Let z3 be the addition of two Spf(W (k))-valued points z1 and z2 ofthe formal scheme defined by D(D) with respect to L. For i ∈ 1, 2, 3 let F 1

i be the liftof (M,ϕ) that corresponds to zi. There exists a unique element ui ∈ Ker(U−(W (k)) →U−(k)) such that we have ui(F

1) = F 1i . Let si := ui − 1M ∈ Lie(U−). Let z+i be the

Spf(W (k))-valued point of L+ defined by zi. Let u+i := q(W (k))(ui) ∈ U−+(W (k)). Thefiltered Dieudonne module associated to z+i is (EndW (k)(M)/E, u+i (HomW (k)(F

0, F 1)), ϕ),cf. the constructions of Subsection 1.2.5. To the addition of short exact sequences men-tioned in Subsection 1.2.5 (applied over quotients C of W (k)), corresponds the multiplica-tion formula u+3 = u+1 u

+2 . As q is an isomorphism, we get u3 = u1u2. In other words the

lift F 13 is the “sum” of F 1

1 and F 12 i.e., we have

(1) s3 = s1 + s2 or equivalently u3 = u1u2.

If p ≥ 3 or if p = 2 and d ∈ 0, r, then z3 is uniquely determined by u3 (cf.property 1.2.1 (a)). If p = 2 and 1 ≤ d ≤ r − 1, then it is well known that z3 is uniquelydetermined by u3 up to a 2-torsion point of L(Spf(W (k))) (this is also a direct consequenceof Proposition 9.5.1). Formula (1) guarantees that in the case when k = k, the formalLie group structure L on the formal scheme defined by D(D) is the one defined by the

11

classical Serre–Tate deformation theory (see properties 1.1 (h.a) and (h.b)). This holdseven if p = 2 and 1 ≤ d ≤ r − 1, as each automorphism of formal schemes L→L thatrespects identity sections and whose composite with the square endomorphism [2] : L → Lis an endomorphism of formal Lie groups, is in fact an automorphism of formal Lie groups.

1.2.7. Duality of language: positive versus negative. The trace form onEndW (k)(M) gives birth via restriction to a perfect bilinear form

b : Lie(U−)⊗W (k) Lie(U+) → W (k)

i.e., the natural W (k)-linear map Lie(U−)∗ → Lie(U+) defined by b is a W (k)-linearisomorphism. This W (k)-linear isomorphism allows us to identify naturally D+

W (k) with

the Cartier dual (D−W (k))

t of D−W (k), cf. properties 1.2.1 (b) and (c). As in Subsections

1.2.4 and 1.2.5, one checks that deforming D is equivalent to deforming D−(0). The mainadvantage of working with D−(0) is that it involves simpler group theoretical arguments(like we do not have to consider identifications of the form Lie(S+) = EndW (k)(M)/E)which are more adequate for suitable generalizations. The main two disadvantages ofworking with D−(0) are: (i) we have to use the bilinear form b in order to identify D+

W (k)

with (D−W (k))

t and (ii) we have to work with Ψ instead of with ϕ.

1.3. Generalized language

1.3.1. Goal. The goal of the monograph is to generalize results recalled in Section 1.1about ordinary p-divisible groups over k and their canonical lifts, to:

• the abstract context of Shimura p-divisible objects over k (see this Section andthe next one), and to

• the geometric context of good moduli spaces of principally polarized abelianschemes endowed with specializations of Hodge cycles (see Sections 1.5 and 1.6).

For generalizing the notions of ordinariness and canonical lifts to the abstract context,we do not require any deformation theory. Thus not to make this monograph too long,we postpone to future work the generalization of properties 1.1 (h) and (i) and of thedeformation theories of [dJ], [Fa2, §7], and [Zi2] to the abstract context. Moreover, inorder to avoid repetitions, the greatest part of the generalization of properties 1.1 (f) to(h) will be stated only for the geometric context. The formal deformation spaces overSpf(W (k)) we get in the geometric context are expected to have two types of naturalstructures of formal Lie groups: one commutative and one nilpotent. In this monographwe mainly restrict to the case when these two structures are expected to coincide (i.e., inthe so called commutative case): in this case we show that the formal deformation spacesdo have canonical structures of commutative formal Lie groups over Spf(W (k)) that areisomorphic to formal Lie groups of a very specific type of p-divisible groups over W (k).

The generalization of the property 1.1 (i) will rely on a comprehensive theory ofconnections on these formal deformation spaces; therefore in this monograph we mainly useconnections only to understand their Kodaira–Spencer maps. This theory of connectionswill allow us in future work to define different natural structures on the formal deformation

12

spaces (including as well the “forcing” to get formal tori) that will ease the study of theseformal deformation spaces.

See Section 1.7 for applications. See Section 1.8 for extra literature that pertains toSections 1.3 to 1.7. See Section 1.9 for our main motivation. See Section 1.10 for moredetails on how Chapters 2 to 9 are organized. For a, b ∈ Z with b ≥ a, let

S(a, b) := a, a+ 1, . . . , b.

We now introduce the language that will allow us to generalize the classical Serre–Tateordinary theory. To motivate it, let DW (k) be the p-divisible group of an abelian scheme Aover W (k) and let (M,F 1, ϕ) be its filtered Dieudonne module. Often the abelian varietyA

B(k)is endowed with a set of some type of cycles (algebraic, Hodge, crystalline, etc.)

normalized by the Galois group Gal(B(k)/B(k)) and this leads to the study of quadruplesof the form (M,F 1, ϕ, G), where G is a flat, closed subgroup scheme of GLM such thatthe Lie algebra Lie(GB(k)) is normalized by ϕ and we have a direct sum decompositionM = F 1 ⊕ F 0 with the property that the Gm subgroup scheme of GLM obtained as inSubsection 1.2.4, is contained in G. We think of G

B(k)as the identity component of the

subgroup of GLM⊗W (k)B(k)

that fixes the crystalline realizations of these cycles. In this

monograph we study the case when G is a reductive, closed subgroup scheme of GLM .Moving from the geometric context of abelian schemes to a more general and abstractcontext “related to” Shimura varieties, we have the following basic definition.

1.3.2. Basic definition. A Shimura filtered p-divisible object over k in the range [a, b] isa quadruple

(M, (F i(M))i∈S(a,b), ϕ, G),

where

– M is a free W (k)-module of finite rank,

– (F i(M))i∈S(a,b) is a decreasing and exhaustive filtration of M by direct summands,

– ϕ is a σ-linear automorphism of M [ 1p ], and

– G is a reductive, closed subgroup scheme of GLM ,

such that there exists a direct sum decompositionM = ⊕bi=aF

i(M) for which the followingthree properties hold:

(a) we have an identity F i(M) = ⊕bj=iF

j(M) for all i ∈ S(a, b) and moreover

ϕ−1(M) = ⊕bi=ap

−iF i(M) (equivalently and moreover ϕ−1(M) =∑b

i=a p−iF i(M));

(b) the cocharacter of GLM that acts on F i(M) via the −i-th power of the identitycharacter ofGm, factors throughG and this factorization µ : Gm → G is either aminusculecocharacter or factors through Z0(G) i.e., we have a direct sum decomposition

Lie(G) = ⊕j∈−1,0,1Fj(Lie(G))

such that µ(β) acts (via the adjoint representation) on F j(Lie(G)) as the multiplicationwith β−j for all j ∈ −1, 0, 1 and all β ∈ Gm(W (k));

13

(c) denoting also by ϕ the σ-linear automorphism of EndB(k)(M [ 1p ]) that takes x ∈

EndB(k)(M [ 1p ]) to ϕ x ϕ−1 ∈ EndB(k)(M [ 1p ]), we have ϕ(Lie(G)[ 1p ]) = Lie(G)[ 1p ] i.e.,

(Lie(G)[ 1p ], ϕ) is an F -subisocrystal of (EndB(k)(M [ 1p ]), ϕ).

We call the triple (M,ϕ,G) a Shimura p-divisible object over k in the range [a, b].Following [Pi], we refer to µ as a Hodge cocharacter of (M,ϕ,G). We refer to thequadruple (M, (F i(M))i∈S(a,b), ϕ, G) or to the filtration (F i(M))i∈S(a,b) (of M) as a liftof (M,ϕ,G) or as the lift of (M,ϕ,G) defined by µ. If (a, b) = (0, 1), then we do notmention F 0(M) = M and thus we also refer to (M,ϕ,G) (resp. to (M,F 1(M), ϕ, G)) asa Shimura (resp. as a Shimura filtered) F -crystal over k and to (M,F 1(M), ϕ, G) or toF 1(M) as a lift of (M,ϕ,G). Often we do not mention over k or in the range [a, b]. Fori ∈ S(a, b) let ϕi : F

i(M) → M be the restriction of p−iϕ to F i(M). Triples of the form(M, (F i(M))i∈S(a,b), ϕ) show up in [La] and [Wi]. If m ∈ N, then the reduction modulopm of (M, (F i(M))i∈S(a,b), (ϕi)i∈S(a,b)) is an object of the category MF[a,b](W (k)) usedin [FL], [Wi], and [Fa1]. This and the fact that in future work we will use analogues ofDefinition 1.3.2 for arbitrary regular, formally smooth W (k)-schemes, justifies our termi-nology “p-divisible object” (it extrapolates the terminology object used in [Fa1]). SplittingsM = ⊕b

i=aFi(M) of the lift (F i(M))i∈S(a,b) of (M,ϕ,G) show up first in [Wi].

1.3.3. Parabolic subgroup schemes. Let

F 0(Lie(G)) := F 0(Lie(G))⊕ F 1(Lie(G)).

Let P be the normalizer of (F i(M))i∈S(a,b) in G; it is a parabolic subgroup scheme of Gwhose Lie algebra is F 0(Lie(G)) (cf. Subsection 2.5.3).

By the non-negative parabolic subgroup scheme of (M,ϕ,G) we mean the parabolicsubgroup scheme

P+G (ϕ)

ofG with the properties that Lie(P+G (ϕ))[ 1p ] is normalized by ϕ and that all Newton polygon

slopes of (Lie(P+G (ϕ))[ 1p ], ϕ) (resp. of (Lie(G)[

1p ]/Lie(P

+G (ϕ))[ 1p ], ϕ)) are non-negative (resp.

are negative). By the Levi subgroup of (M,ϕ,G) we mean the unique Levi subgroup

L0G(ϕ)B(k)

of P+G (ϕ)

B(k)with the property that Lie(L0

G(ϕ)B(k)) is normalized by ϕ. In Subsections

2.6.1 and 2.6.2 we will argue the existence of the subgroup schemes P+G (ϕ) and L0

G(ϕ)B(k).In Subsection 2.6.2 we also check that all Newton polygon slopes of (Lie(L0

G(ϕ)B(k)), ϕ)

are 0 and that Lie(L0G(ϕ)B(k)) is the maximal B(k)-vector subspace of Lie(P+

G (ϕ)B(k))which has this property.

1.3.4. A new Hasse–Witt invariant. We consider the product decomposition Gadk =∏

i∈I Gi,k of Gadk into simple, adjoint groups over k, cf. [Ti1, Subsubsect. 3.1.2]. It lifts

naturally to a product decomposition

(2) Gad =∏

i∈I

Gi

14

of Gad into simple, adjoint group schemes over W (k), cf. [DG, Vol. III, Exp. XXIV,Prop. 1.21]. We have Lie(Gad)[ 1

p] = [Lie(G),Lie(G)][ 1

p]. Thus Lie(Gad) = ⊕i∈ILie(Gi) is

naturally a Lie subalgebra of Lie(G)[ 1p].

By the FSHW shift of (M,ϕ,G) we mean the σ-linear endomorphism

Ψ : Lie(Gad) → Lie(Gad)

which for i ∈ I takes x ∈ Lie(Gi) to ϕ(x) or to p(ϕ(x)) = ϕ(px) depending on the fact thatthe composite of µ : Gm → G with the natural epimorphism G ։ Gi is or is not trivial.The existence of the map Ψ is checked in Subsection 2.4.3. Let Ψ : Lie(Gad

k ) → Lie(Gadk )

be the reduction modulo p of Ψ; we call it the FSHW map of (M,ϕ,G). By the FSHWinvariant of (M,ϕ,G) we mean the following dimension

I(M,ϕ,G) := dimk(∩m∈NIm(Ψm)) = dimk(Im(Ψ

dim(Gadk )

)) ∈ N ∪ 0.

Here FSHW stands for Faltings–Shimura–Hasse–Witt. If for each element i ∈ I thecomposite of µ : Gm → G with the natural epimorphism G ։ Gi is not trivial, then Ψ isa usual Tate twist of φ (i.e., for each element x ∈ Lie(Gad) we have Ψ(x) = pφ(x)).

1.3.5. Two different notions of ordinariness and canonical lifts. We say(M, (F i(M))i∈S(a,b), ϕ, G) is a U-canonical lift (of (M,ϕ,G)) if L0

G(ϕ)B(k) is a sub-

group of PB(k). If moreover P+G (ϕ) is a (closed) subgroup scheme of P , then we say

(M, (F i(M))i∈S(a,b), ϕ, G) is an Sh-canonical lift (of (M,ϕ,G)).

We say (M,ϕ,G) is U-ordinary (resp. Sh-ordinary) if it has a lift that is a U-canonical(resp. that is an Sh-canonical) lift.

Here U stands for unique and Sh stands for Shimura. If G = GLM and 0 ≤ a ≤ b ≤ 1,then it is easy to see that the notions U-ordinary and Sh-ordinary (resp. U-canonicallift and Sh-canonical lift) coincide with the usual notion ordinary (resp. canonical lift)introduced in Section 1.1.

Next we introduce two extra concepts that pertain to Shimura p-divisible objects.

1.3.6. Definitions. (a) By an endomorphism of (M, (F i(M))i∈S(a,b), ϕ, G) (resp. of(M,ϕ,G)) we mean an element e ∈ F 0(Lie(G)) (resp. e ∈ Lie(G)) that is fixed by ϕ.

(b) Let g1, g2 ∈ G(W (k)). We say (M, g1ϕ,G) and (M, g2ϕ,G) are inner isomorphic,if there exists an element h ∈ G(W (k)) such that we have hg1ϕ = g2ϕh. We also refer tosuch an h as an inner isomorphism between (M, g1ϕ,G) and (M, g2ϕ,G).

1.3.7. Characterization of U-canonical lifts in terms of endomorphisms. Wehave L0

G(ϕ)B(k) 6 PB(k) if and only if we have an inclusion Lie(L0G(ϕ)B(k)) ⊆ Lie(PB(k)),

cf. [Bo, Ch. II, Sect. 7.1]. Thus L0G(ϕ)B(k) 6 PB(k) if and only if Lie(L0

G(ϕ)B(k)) ⊆

Lie(PB(k)). But

E := Lie(GW (k)) ∩ x ∈ Lie(L0G(ϕ)B(k))|ϕ⊗ σk(x) = x

15

is the Lie algebra over Zp of endomorphisms of (M ⊗W (k)W (k), ϕ⊗σk, GW (k)) and more-

over Lie(L0G(ϕ)B(k)) is B(k)-generated by E. Thus L0

G(ϕ)B(k) 6 PB(k) if and only if

we have E ⊆ Lie(PB(k)) and therefore if and only if the inclusion E ⊆ Lie(PW (k)) =

F 0(Lie(G))⊗W (k)W (k) holds. Thus (M, (F i(M))i∈S(a,b), ϕ, G) is a U-canonical lift if and

only if each endomorphism of (M ⊗W (k)W (k), ϕ⊗σk, GW (k)) is as well an endomorphism

of (M ⊗W (k) W (k), (F i(M)⊗W (k) W (k))i∈S(a,b), ϕ⊗ σk, GW (k)).

1.3.8. Definitions. Let M be a free W (k)-module of finite rank dM . Let (F i(M))i∈S(a,b)

be a decreasing and exhaustive filtration of M by direct summands. Let ϕ be a σ-linearautomorphism of M [ 1p ] such that we have ϕ−1(M) =

∑bi=a p

−iF i(M). If a ≥ 0, we refer

to the triple (M, (F i(M))i∈S(a,b), ϕ) as a filtered F -crystal over k.

(a) We say (M, (F i(M))i∈S(a,b), ϕ) is circular if there exists a W (k)-basis B :=

e1, . . . , edM for M such that for all j ∈ S(1, dM) we have

ϕ(ej) = pmjej+1,

where edM+1:= e1 and ej ∈ Fmj (M) \ Fmj+1(M) with F b+1(M) := 0 and with mj ∈

S(a, b). If moreover the function fB : Z/dMZ → S(a, b) defined by fB([j]) = mj is not

periodic, then we say (M, (F i(M))i∈S(a,b), ϕ) is circular indecomposable.

(b) We say (M, (F i(M))i∈S(a,b), ϕ) is cyclic if there exists a direct sum decomposition

(M, (F i(M))i∈S(a,b), ϕ) = ⊕l∈L(Ml, (Fi(M)∩ Ml)i∈S(a,b), ϕl) such that for each l ∈ L the

triple (Ml, (Fi(M) ∩Ml)i∈S(a,b), ϕl) is circular indecomposable.

(c) We assume that 0 ≤ a ≤ b ≤ 1 and that there exists a p-divisible group DW (k)

over W (k) whose filtered Dieudonne module is (M, F 1(M), ϕ). We say that DW (k) or

(M, F 1(M), ϕ) is circular indecomposable (resp. circular), if (M, (F i(M))i∈0,1, ϕ) is

circular indecomposable (resp. is circular). We say DW (k) is cyclic if it is a direct sum ofp-divisible groups over W (k) that are circular indecomposable.

1.4. Main results on Shimura p-divisible objects

In this Section we state our main results on the objects we have defined in the previousone. Let (M,ϕ,G) be a Shimura p-divisible object over k in the range [a, b]. For eachelement g ∈ G(W (k)) the triple (M, gϕ,G) is as well a Shimura p-divisible object overk. One refers to (M, gϕ,G)|g ∈ G(W (k)) as a family of Shimura p-divisible objectover k in the range [a, b]. See (2) for the product decomposition Gad =

∏i∈I Gi. We

have the following four basic results (most of them are stated in terms of the family(M, gϕ,G)|g ∈ G(W (k)), due to the sake of flexibility and of simplifying the proofs).

1.4.1. Theorem. Let g ∈ G(W (k)). The following two statements are equivalent:

(a) the triple (M, gϕ,G) is Sh-ordinary;

(b) for each element g1 ∈ G(W (k)) the Newton polygon of (M, g1ϕ) is above theNewton polygon of (M, gϕ).

16

Moreover, if (M, gϕ,G) is Sh-ordinary, then:

(c) for each element g1 ∈ G(W (k)) we have an inequality I(M, g1ϕ,G) ≤ I(M, gϕ,G).

1.4.2. Theorem. We assume that G is quasi-split i.e., it has a Borel subgroup scheme(for instance, this holds if either k is a finite field or k = k). Then the statements 1.4.1(c) and 1.4.1 (a) are equivalent if and only if for each element i ∈ I with the property thatthe simple factors of Gi,W (k) are PGLℓ+1 group schemes with ℓ ≥ 3 and for every element

h0 ∈ G(W (k)) such that (M ⊗W (k)W (k), h0(ϕ⊗ σk), GW (k)) is Sh-ordinary, the image of

the unipotent radical of P+G

W (k)(h0(ϕ⊗ σk)) in Gi,W (k) is of nilpotent class at most 2.1

1.4.3. Theorem. Let dM := rkW (k)(M). The following four statements are equivalent:

(a) the triple (M,ϕ,G) is U-ordinary;

(b) there exists a unique lift (F i0(M))i∈S(a,b) of (M,ϕ,G) such that the Levi subgroup

L0G(ϕ)B(k) of (M,ϕ,G) normalizes F i

0(M)[ 1p ] for all i ∈ S(a, b);

(c) there exists a unique Hodge cocharacter µ0 : Gm → GW (k) of (M⊗W (k)W (k), ϕ⊗

σk, GW (k)) such that there exists a W (k)-basis e1, . . . , edM for M ⊗W (k) W (k) with the

property that for all j ∈ S(1, dM) the W (k)-span of ej is normalized by Im(µ0) and wehave ϕ ⊗ σ

k(ej) = pnjeπ0(j), where nj ∈ S(a, b) and where π0 is a permutation of the set

S(1, dM);

(d) we have a direct sum decomposition of F -crystals over k

(M, p−aϕ) = ⊕γ∈Q(Mγ , p−aϕ)

with the properties that: (i) for each rational number γ, either Mγ = 0 or the only Newtonpolygon slope of (Mγ, p

−aϕ) is γ, and (ii) there exists a unique lift (F i1(M))i∈S(a,b) of

(M,ϕ,G) such that we have an identity

F i1(M) = ⊕γ∈Q(F i

1(M) ∩Mγ), ∀i ∈ S(a, b).

If (M,ϕ,G) is U-ordinary, then we have F i0(M) = F i

1(M) for all i ∈ S(a, b). If(M,ϕ,G) is U-ordinary and k = k, then (M, (F i

0(M))i∈S(a,b), ϕ) is cyclic.

1.4.4. Theorem. Let gad ∈ Gad(k) be the image of an element g ∈ G(W (k)). We assumethat (M,ϕ,G) is Sh-ordinary and that k = k. Then statements 1.4.1 (a) and (b) are alsoequivalent to any one of the following two additional statements:

(a) the Shimura p-divisible objects (M,ϕ,G) and (M, gϕ,G) are inner isomorphic;

(b) there exists an element h ∈ Gad(k) such that as σ-linear endomorphisms ofLie(Gad

k ) we have an identity hΨ = gadΨh.

1.4.5. Corollary. There exists an open, Zariski dense subscheme U of Gadk such that

for each perfect field k that contains k and for every element gk ∈ G(W (k)), the triple

(M ⊗W (k) W (k), gk(ϕ⊗ σk), GW (k)) is Sh-ordinary if and only if we have gadk

∈ U(k).

1 Based on Theorem 1.4.4 below, one needs to check this for only one such element h0.

17

If G = GLM , then b− a ∈ 0, 1 (cf. property 1.3.2 (b)) and thus up to a Tate twistwe recover the classical context of Section 1.1. Thus Subsection 1.3.7 and Theorems 1.4.1to 1.4.4 generalize the interpretations of ordinary p-divisible groups and their canonicallifts in terms of (filtered or truncated modulo p) Dieudonne modules. We now include anExample that illustrates many of the above notions and results.

1.4.6. Example. We assume that there exists a direct sum decomposition M = M0 ⊕M1 ⊕M2 with the property that there exist W (k)-bases Bi = eij |j ∈ S(1, 4) for Mi such

that we have ϕ(eij) = pni,jei+1j , where the upper index i ∈ 0, 1, 2 is taken modulo 3 and

where ni,j ∈ 0, 1 is non-zero if and only if we have j − i ≤ 1. Thus (a, b) = (0, 1). The

Newton polygon slopes of (M,ϕ) are 0, 13, 2

3, and 1, all multiplicities being 3. Let F 0(M)

be the W (k)-span of eij |0 ≤ i ≤ 2, 2 ≤ j − i ≤ 4 − i. Let F 1(M) = F 1(M) be the

W (k)-span of eij |0 ≤ i ≤ 2, 1 − i ≤ j − i ≤ 1. Let F 0(M) := M = F 0(M) ⊕ F 1(M)and G := GLM0

×W (k) GLM1×W (k) GLM2

. Let µ : Gm → G be the cocharacter that

fixes F 0(M) and that acts on F 1(M) via the inverse of the identical character of Gm. Thetriple (M,F 1(M), ϕ, G) is a Shimura filtered F -crystal.

The parabolic subgroup scheme P+G (ϕ) of G is the Borel subgroup scheme of G that

normalizes theW (k)-span of eij |0 ≤ i ≤ i, j ∈ S(1, 4) for all all i ∈ 0, 1, 2. Thus P+G (ϕ)

normalizes F 1(M). Therefore (M,ϕ,G) is Sh-ordinary and F 1(M) is its Sh-canonical lift.

As σ-linear endomorphisms of Lie(Gad), we have 1pΨ = ϕ. Thus I(M,ϕ,G) computes

the multiplicity of the Newton polygon slope −1 of (Lie(Gad), ϕ) and therefore it is 3. Fori ∈ 0, 1, 2 and j1, j2 ∈ S(1, 4), let eij1,j2 ∈ EndW (k)(Mi) be such that for j3 ∈ S(1, 4)

we have eij1,j2(eij3) = δj2,j3e

ij2. If j1 6= j2, then we have eij1,j2 ∈ Lie(Gad). We have

Ψ(ei4,1) = ei+14,1 and Ψ(ei1,4) = pei+1

1,4 . Let n− (resp. n+) be the W (k)-submodule of

Lie(Gad) generated by all elements fixed by Ψ3 (resp. by 1p3Ψ

3). We have

n− = ⊕2i=0W (k)ei4,1 and n+ = ⊕2

i=0W (k)ei1,4.

The normalizer P−G (ϕ) (resp. P+

G (ϕ)) of n− (resp. of n+) in G is the parabolic subgroupscheme of G whose Lie algebra is the W (k)-span of

eij1,j2 |0 ≤ i ≤ 2, either 1 ≤ j2 ≤ j1 ≤ 4 or (j1, j2) = (2, 3)

(resp. of eij1,j2 |0 ≤ i ≤ 2, either 1 ≤ j1 ≤ j2 ≤ 4 or (j1, j2) = (3, 2)). Thus P+G (ϕ) is a

proper subgroup scheme of P+G (ϕ). Let L0

G(ϕ) be the Levi subgroup scheme of either P−G (ϕ)

or P+G (ϕ), whose Lie algebra is the W (k)-span of eij,j , e

i2,3, e

i3,2|0 ≤ i ≤ 2, j ∈ S(1, 4).

The group scheme L0G(ϕ)

ad is a PGL32 group scheme, the cocharacter µ factors through

L0G(ϕ), and the triple (M,ϕ, L0

G(ϕ)) is also Sh-ordinary.

Let w ∈ L0G(ϕ)(W (k)) be such that it fixes eij for (i, j) /∈ (0, 2), (0, 3) and it inter-

changes e02 and e03. The Newton polygon slopes of (M,wϕ) are 0, 12 , and 1 with multi-

plicities 3, 6, and 3 (respectively). As w fixes n−, n− is W (k)-generated by all elementsof Lie(Gad) fixed by (wΨ)3. By applying this over k we get that I(M,wϕ,G) = 3. But

18

the triples (M,wϕ,G) and (M,wϕ, L0G(ϕ)) are not Sh-ordinary, as e

i3,2 ∈ Lie(P+

L0G(ϕ)

(wϕ))

and as ei3,2(F1(M)) ⊗W (k) k 6⊆ F 1(M) ⊗W (k) k. Let g ∈ G(W (k)) be such that we have

I(M, gϕ,G) = 3. Up to an inner isomorphism we can assume that g ∈ L0G(ϕ)(W (k)),

cf. property 3.3 (b). For i ∈ 0, 1, 2, the intersection L0G(ϕ) ∩ GLMi

normalizesF 1(M) ∩ Mi if and only if i 6= 1 (i.e., µ has a non-trivial image only in that PGL2

factor of L0G(ϕ)

ad which corresponds to i = 1). This implies that I(M, gϕ, L0G(ϕ)) is the

multiplicity of the Newton polygon slope −13of (Lie(L0

G(ϕ)), gϕ). In particular, we have

I(M,wϕ, L0G(ϕ)) = 0 < 3 = I(M,ϕ, L0

G(ϕ)).

In general, for g ∈ L0G(ϕ)(W (k)) the triple (M, gϕ,G) is Sh-ordinary if and only if the

triple (M, gϕ, L0G(ϕ)) is Sh-ordinary (cf. Theorem 3.7.1 (e)).

1.4.7. Remark. Example 1.4.6 points out that the limitations of Theorem 1.4.2 are ofapparent nature: always one can use a well defined sequence of FSHW invariants in orderto “detect” Sh-ordinariness. It also points out that the proofs of Theorems 1.4.1, 1.4.2, and1.4.4 will involve an induction on dim(Gk) and will appeal to opposite parabolic subgroupschemes of G, like P−

G (ϕ) and P+G (ϕ).

We now shift our attention from the abstract context of Sections 1.3 and 1.4 to ageometric context that involves abelian schemes and Shimura varieties of Hodge type.

1.5. Standard Hodge situations

We use the standard terminology of [De5] of Hodge cycles on an abelian scheme AZ

over a reduced Q–scheme Z. Thus we write each Hodge cycle v on AZ as a pair

(vdR, vet),

where vdR and vet are the de Rham and the etale (respectively) component of v. The etalecomponent vet as its turn has an l-component vlet, for each prime l ∈ N. For instance, if Zis the spectrum of a field E and if E is a fixed algebraic closure of E, then vpet is a suitableGal(E/E)-invariant tensor of the tensor algebra of H1

et(AZ ,Qp)⊕(H1et(AZ ,Qp))

∗⊕Qp(1),

where Z := Spec(E) and where Qp(1) is the usual Tate twist. If moreover E is a subfieldof C, then we also use the Betti realization of v: it corresponds to vdR (resp. to vlet)via the standard isomorphism that relates the Betti cohomology with Q–coefficients ofAZ ×Z Spec(C) with the de Rham (resp. the Ql etale) cohomology of AZ (see [De5]).

For generalities on Shimura pairs, on their adjoints, reflex fields, and canonical models,and on injective maps between them we refer to [De2], [De3], [Mi1], [Mi2], and [Va3,Subsects. 2.1 to 2.10]. For different types of Shimura varieties we refer to [De3], [Mi2],and [Va3, Subsect. 2.5]. We now introduce standard Hodge situations. They generalize thestandard PEL situations used in [Zi1], [LR], [Ko2], and [RZ]; however, the things are moretechnical than in these references due to the passage from tensors of degree 2 (polarizationsand endomorphisms) to tensors of arbitrary degree on which no special properties areimposed. Here “tensors” refer to different cohomological realizations of Hodge cycles onabelian varieties.

1.5.1. Basic notations. We start with an injective map of Shimura pairs

f : (GQ, X) → (GSp(W,ψ), S).

19

The pair (W,ψ) is a symplectic space over Q and S is the set of all homomorphismsResC/RGm → GSp(W,ψ)R that define Hodge Q–structures onW of type (−1, 0), (0,−1)and that have either 2πiψ or −2πiψ as polarizations. One calls (GSp(W,ψ), S) a Siegelmodular pair and (GQ, X) a Shimura pair of Hodge type. The group GQ is reductive overQ and is identified via f with a subgroup of GSp(W,ψ). The set X is a GQ(R)-conjugacyclass of homomorphisms ResC/RGm → GR whose composites with fR belong to S. BothS and X have canonical structures of hermitian symmetric domains, cf. [De3, Cor. 1.1.17].

Let L be a Z–lattice ofW which is self-dual with respect to ψ (i.e., such that ψ inducesa perfect form ψ : L ⊗Z L → Z). Let Z(p) be the localization of Z at its prime ideal (p).Let L(p) := L⊗Z Z(p). We assume that p and L are such that:

(*) the Zariski closure GZ(p)of GQ in GSp(L(p), ψ) is a reductive group scheme over Z(p).

Let Kp := GSp(L, ψ)(Zp). Let

Hp := GZ(p)(Zp) = GQ(Qp) ∩Kp.

Let r :=dimQ(W )

2 ∈ N. Let Af (resp. A(p)f ) be the Q–algebra of finite adeles (resp. of

finite adeles with the p-component omitted). We have Af = Qp ×A(p)f .

Let x ∈ X . Let µx : Gm → GC be the Hodge cocharacter that defines the Hodge Q–structure onW defined by x. We have a direct sum decompositionW⊗QC = F−1,0

x ⊕F 0,−1x

such that Gm acts via µx trivially on F 0,−1x and through the identity character on F−1,0

x .

The reflex field E(GQ, X) of (GQ, X) is the subfield of C that is the field of definitionof the GQ(C)-conjugacy class of cocharacters of GC defined by (any) µx; it is a numberfield. The canonical model Sh(GQ, X) of (GQ, X) over E(GQ, X) is a Shimura variety ofHodge type. Let v be a prime of E(GQ, X) that divides p and let k(v) be its residue field.Let O(v) be the localization of the ring of integers of E(GQ, X) with respect to v. As GZp

is a reductive group scheme, the prime v is unramified over p (cf. [Mi2, Cor. 4.7 (a)]).Our main data is the triple

(f, L, v)

for which (*) holds, to be called a potential standard Hodge situation.

For the notion of an integral canonical model of a quadruple of the form (GQ, X,Hp, v)we refer to [Va3, Defs. 3.2.3 6) and 3.2.6] (see also [Mi1, §2] and [Mi2]). Let

M

be the Z(p)-scheme which parameterizes isomorphism classes of principally polarizedabelian schemes over Z(p)-schemes that are of relative dimension r and that have levels symplectic similitude structure for each s ∈ N relatively prime to p. The scheme

M together with the natural action of GSp(W,ψ)(A(p)f ) on it, is an integral canonical

model of (GSp(W,ψ), S,Kp, p) (see [Va3, Example 3.2.9 and Subsect. 4.1] or [Mi1, Thm.2.10]). These structures and this action are defined naturally via (L, ψ) (see [Va3, Sub-sect. 4.1]) and the previous sentence makes sense as we can identify naturally MQ withSh(GSp(W,ψ), S)/Kp.

20

As Z(GSp(L(p), ψ))(Z(p)) is a discrete subgroup of Z(GSp(W,ψ))(A(p)f ), we have

MQ(C) = GSp(L(p), ψ)(Z(p))\(S × GSp(W,ψ)(A(p)f )) (cf. [Mi2, Prop. 4.11]). We

also have Sh(GQ, X)/Hp(C) = GZ(p)(Z(p))\[X × (GQ(A

(p)f )\Z(GZ(p)

)(Z(p)))], where

Z(GZ(p))(Z(p)) is the topological closure of Z(GZ(p)

)(Z(p)) in Z(GQ)(A(p)f ) (cf. [Mi2, Prop.

4.11]). We have a morphism Sh(GQ, X)/Hp → ME(GQ,X) of E(GQ, X)-schemes whose

pull back to C is defined by the natural embedding X×GQ(A(p)f ) → S×GSp(W,ψ)(A

(p)f )

via natural passages to quotients, cf. [De2, Cor. 5.4]. The last two sentences imply thatZ(GZ(p)

)(Z(p)) = Z(GZ(p))(Z(p)). Thus we have

Sh(GQ, X)/Hp(C) = GZ(p)(Z(p))\(X ×GQ(A

(p)f )).

We easily get that Sh(GQ, X)C/Hp is a closed subscheme of MC. Thus Sh(GQ, X)/Hp isa closed subscheme of ME(GQ,X). Let

N

be the normalization of the Zariski closure of Sh(GQ, X)/Hp in MO(v). Let

(A,ΛA)

be the pull back to N of the universal principally polarized abelian scheme over M. Let

(vα)α∈J

be a family of tensors in ∪n∈NW∗⊗n⊗QW

⊗n such that GQ is the subgroup of GLW thatfixes vα for all α ∈ J . As Z(GLW ) 6 GQ, the existence of the family (vα)α∈J is impliedby [De5, Prop. 3.1 (c)].

The choice of L and (vα)α∈J , allows a moduli interpretation of Sh(GQ, X) (see [De2],[De3], [Mi2], and [Va3, Subsect. 4.1 and Lem. 4.1.3]). For instance, the set of complexpoints Sh(GQ, X)(C) = GQ(Q)\(X × GQ(Af )) is the set of isomorphism classes of prin-cipally polarized abelian varieties over C that are of dimension r, that carry a family ofHodge cycles indexed by the set J , that have level s symplectic similitude structure forall s ∈ N, and that satisfy some extra axioms.

This interpretation endows the abelian scheme AE(GQ,X) with a family (wAα )α∈J of

Hodge cycles whose Betti realizations can be described as follows. Let

w = [xw, gw] ∈ Sh(GQ, X)/Hp(C) = GZ(p)(Z(p))\(X ×GQ(A

(p)f ))

where xw ∈ X and gw ∈ GQ(A(p)f ). Let Lgw be the Z-lattice of W such that we have

Lgw ⊗Z Z = gw(L ⊗Z Z); here we view gw as an element of GQ(Af ). Let (Aw, λAw) :=

w∗((A,ΛA)E(GQ,X)). Then Aw is the complex abelian variety whose analytic space is

Aw(C) = Lgw\W ⊗Q C/F0,−1

xw,

21

the principal polarization λAwof Aw is defined by a suitable Gm(Z(p))-multiple of ψ, and

w∗(wAα ) is the Hodge cycle on Aw whose Betti realization is the tensor vα of the tensor

algebra of W ∗ ⊕W = (L∗gw ⊗Z Q)⊕ (Lgw ⊗Z Q).

Let Lp := L ⊗Z Zp = L(p) ⊗Z(p)Zp. Let ψ∗ be the perfect form on either L∗

(p) or L∗p

that is defined naturally by ψ. We have natural identifications

H1et(Aw,Zp) = L∗

gw⊗Z Zp = L∗ ⊗Z Zp = L∗

p

under which: (i) the p-component of the etale component of w∗(wAα ) is vα, and (ii) the

perfect form on L∗⊗ZZp that corresponds naturally to λAwis a Gm(Z(p))-multiple of ψ∗.

Let ′ be the involution of EndZ(p)(L(p)) defined by the identity ψ(b(x), y) = ψ(x, b′(y)),

where b ∈ EndZ(p)(L(p)) and x, y ∈ L(p). Let B := b ∈ EndZ(p)

(L(p))|b fixed by GZ(p).

We say (f, L, v) or (f, L, v,B) is a standard PEL situation if the following axiom holds:

(PEL) B is normalized by ′, B⊗Z(p)W (F) is a product of matrix W (F)-algebras, and GQ

is the identity component of the subgroup of GSp(W,ψ) that fixes all elements of B[ 1p ].

If moreover GQ is the subgroup of GSp(W,ψ) that fixes all elements of B[ 1p], then we

also refer to (f, L, v) or to (f, L, v,B) as a standard moduli PEL situation.

1.5.2. The abstract Shimura F -crystal. The isomorphism L→L∗ induced by ψ allowsus to identify GZp

with a closed subgroup scheme of GLL∗p.

Let T0 be a maximal torus of a Borel subgroup scheme B0 of GZp. Let

µ0 : Gm → T0,W (k(v))

be an injective cocharacter such that the following condition holds (cf. [Mi2, Cor. 4.7(b)]):

(a) under a monomorphismW (k(v)) → C that extends the composite monomorphismO(v) → E(GQ, X) → C, it is GQ(C)-conjugate to the Hodge cocharacters µx : Gm → GC.

We will choose µ0 such that moreover we have the following property:

(b) if M0 := L∗p ⊗Zp

W (k(v)) = F 10 ⊕ F 0

0 is the direct sum decomposition such thatGm acts via µ0 trivially on F 0

0 and as the inverse of the identical character of Gm on F 10 ,

then B0,W (k(v)) normalizes F 10 .

We emphasize that µ0 is uniquely determined by properties (a) and (b).

Let ϕ0 := (1L∗p⊗ σk(v)) µ0(

1p ) : M0 → M0. We have ϕ−1

0 (M0) = 1pF

10 ⊕ F 0

0 and

ϕ0(Lie(GB(k(v)))) = Lie(GB(k(v))). Thus the quadruple (M0, F10 , ϕ0, GW (k(v))) is a Shimura

filtered F -crystal and the Hodge cocharacter µ0 defines the lift F 10 . We refer to the triple

C0 := (M0, ϕ0, GW (k(v)))

as the abstract Shimura F -crystal of (f, L, v). Up to a W (k(v))-linear automorphism ofM0 defined by an element of GZp

(Zp), C0 does not depend on the choices of T0 and B0 (asall pairs of the form (T0, B0) are GZp

(Zp)-conjugate, cf. Subsection 2.3.8 (d)).

22

1.5.3. Definition. The potential standard Hodge situation (f, L, v) is called a standardHodge situation if the following two conditions hold:

(a) the O(v)-scheme N is regular and formally smooth;

(b) for each perfect field k of characteristic p and for every W (k)-valued point z ∈N (W (k)), the quadruple (M,F 1, ϕ, G) is a Shimura filtered F -crystal; here (M,F 1, ϕ) isthe filtered Dieudonne module of the

p− divisible group Dz of A := z∗(A)

and G is the Zariski closure in GLM of the subgroup of GLM [ 1p ]= GLH1

dR(A/W (k))[ 1p ]

that

fixes the de Rham component tα of z∗E(GQ,X)(wAα ), for all α ∈ J .

The de Rham component tα is a tensor of the tensor algebra of M [ 1p ]⊕M∗[ 1p ].

1.5.4. Shimura F -crystals attached to points of N (k). We assume that (f, L, v) isa standard Hodge situation and we use the notations of Subsection 1.5.3. We refer to thequadruple (M,F 1, ϕ, G) as the Shimura filtered F -crystal attached to z ∈ N (W (k)). Lety ∈ N (k) be the point defined naturally by z. The triple (M,ϕ, (tα)α∈J ) depends only onthe point y ∈ N (k) and not on the lift z ∈ N (W (k) of y, cf. Proposition 6.2.7 (d). Thuswe call (M,ϕ, G) as the Shimura F -crystal attached to the point y ∈ N (k).

1.5.5. Remarks. (a) If the condition 1.5.3 (a) holds, then N is an integral canonicalmodel of (GQ, X,Hp, v) (cf. [Va3, Cor. 3.4.4]). If p ≥ 5, then the condition 1.5.3 (a)holds (cf. [Va3, Subsect. 3.2.12, Prop. 3.4.1, and Thm. 6.4.2] and its corrections in[Va6, Appendix]). See [Va3, Subsects. 6.5 and 6.6] for general ways to construct standardHodge situations. For instance, if (f, L, v) is a potential standard Hodge situation andif p ≥ max5, r, then (f, L, v) is a standard Hodge situation (cf. [Va3, Thm. 5.1, Rm.5.8.2, and Cor. 5.8.6]). In [Va8] it is proved that each potential standard Hodge situationis in fact a standard Hodge situation.

(b) A standard PEL situation is a standard Hodge situation, provided either p ≥ 3or it is a standard moduli PEL situation (see [Zi1], [LR], [Ko2], [RZ], and [Va3, Example5.6.3]; the assumption that B[ 1p ] is a Q–simple algebra used in [Ko2, §5] for p ≥ 2 and thus

also in [Va3, Example 4.3.11] with p ≥ 3, can be eliminated).

1.5.6. Shimura types. We review few things on Shimura types. Let (GadQ , X

ad) be the

adjoint Shimura pair of (GQ, X); thus Xad is the GadQ (R)-conjugacy class of the composite

of any element ResC/RGm → GR of X with the natural epimorphism GR ։ GadR . In

this Subsection we assume that the adjoint group GadQ is non-trivial. Let (GQ, X) be a

simple factor of (GadQ , X

ad); thus GQ is a simple group over Q that is a direct factor of

GadQ . Let X be the Lie type of the simple factors of GC. The existence of the cocharacters

µx : Gm → GC implies that (see [Sa], [De3], [Se1], [Pi2, Table 4.2], etc.):

(i) X is a classical Lie type;

(ii) if f is a simple factor of Lie(GadC ), then the weight of each simple f-submodule

of W ⊗Q C, is a minuscule weight.

23

We recall that can be: any fundamental weight if X = Aℓ, the fundamental weightℓ if X = Bℓ, the fundamental weight 1 if X = Cℓ, and any one of the three fundamentalweights 1, ℓ−1, ℓ if X = Dℓ with ℓ ≥ 4; see either the mentioned references or [Bou3,pp. 127–129]. If X is Aℓ, Bℓ, or Cℓ with ℓ ∈ N, then we say (GQ, X) is of X type. We

assume now that X is Dℓ with ℓ ≥ 4. Then (GQ, X) is of either DHℓ or DR

ℓ type, cf. [De3,

Table 2.3.8]. If (GQ, X) is of DHℓ (resp. of DR

ℓ ) type, then each non-compact, simple factor

of the identity component of the real Lie group GR(R) is isogenous to SO∗(2ℓ) (resp. tothe identity component of SO(2, 2ℓ− 2)) and the converse of this holds for ℓ ≥ 5 (see [He,Ch. X, §2] for the classical semisimple, real Lie groups SO∗(2ℓ) and SO(2, 2ℓ− 2)).

We mention that besides the Shimura types mentioned above, we have precisely threeextra ones. The three extra Shimura types are Dmixed

ℓ with ℓ ≥ 4, E6, and E7. AbstractTheorems 1.4.1 to 1.4.4 apply to all of them.

We have the following concrete (i.e., geometric) form of Section 1.4.

1.6. Main results on standard Hodge situations

Let (f, L, v) be a standard Hodge situation. Let (A,ΛA) be the principally polarizedabelian scheme over N introduced in Section 1.5. Let y ∈ N (k). Let z ∈ N (W (k)) be alift of y. Let the quintuple (M,F 1, ϕ, (tα)α∈J , G) be as in Definition 1.5.3; thus the triple(M,ϕ, G) is the Shimura F -crystal attached to the point y ∈ N (k).

Let C0 := (M0, ϕ0, GW (k(v))) be the abstract Shimura F -crystal of (f, L, v), cf. Sub-

section 1.5.2. Let U+0 := U+

GW (k(v))(ϕ0) be the unipotent radical of the parabolic subgroup

scheme P+GW (k(v))

(ϕ0) of GW (k(v)), cf. Subsection 1.3.3. Let

I0 ∈ N ∪ 0

be the nilpotent class of U+0 .

1.6.1. Theorem. There exists an open, Zariski dense, GQ(A(p)f )-invariant subscheme

O of Nk(v) such that for a point y ∈ N (k) we have y ∈ O(k) if and only if one of thefollowing four equivalent statements holds:

(a) the Shimura F -crystal (M,ϕ, G) attached to y is Sh-ordinary;

(b) the Newton polygon of (M,ϕ) is the Newton polygon N0 of C0 (i.e., of (M0, ϕ0));

(c) there exists a Lie isomorphism Lie(GadW (k)

)→Lie(GadW (k)

) with the properties that:

(i) it is defined naturally by a B(k)-linear isomorphism M ⊗W (k)B(k)→M0⊗W (k(v))B(k)which takes tα to vα for all α ∈ J , and (ii) its reduction modulo p defines an isomorphismbetween the FSHW maps of the triples (M ⊗W (k) W (k), ϕ ⊗ σk, GW (k)) and C0 ⊗ k :=

(M0 ⊗W (k(v)) W (k), ϕ0 ⊗ σk, GW (k));

(d) the Newton polygon of (Lie(Gad), ϕ) is the Newton polygon of (Lie(GadW (k(v))), ϕ0).

If I0 ≤ 2, then the above four statements are also equivalent to the following one:

(e) we have an equality I(M,ϕ, G) = I(C0) (equivalently, we have an inequalityI(M,ϕ, G) ≥ I(C0)).

24

Moreover, if k = k and if y, y1 ∈ N (k) are two points that factor through thesame connected component of O, then there exists an isomorphism (M,ϕ)→(M1, ϕ1) de-fined by a W (k)-linear isomorphism M→M1 that takes tα to t1,α for all α ∈ J ; here

(M1, ϕ1, (t1,α)α∈J ) is the triple that defines the Shimura F -crystal (M1, ϕ1, G1) attachedto the point y1 ∈ N (k).

1.6.2. Theorem. Let y ∈ N (k) be such that its attached Shimura F -crystal (M,ϕ, G)is U-ordinary. If p = 2 we assume that either k = k or (M,ϕ) has no integral Newtonpolygon slopes. Let ny ∈ N ∪ 0 ∪ ∞ be such that the number of W (k)-valued points

z ∈ N (W (k)) which lift y and whose attached Shimura filtered F -crystals (M,F 1, ϕ, G)are the U-canonical lift of (M,ϕ, G), is precisely ny.

Then we have ny ∈ N. If p ≥ 3 or if p = 2 and (M,ϕ) has no integral Newton polygonslopes, then ny = 1 and the resulting W (k)-valued point z ∈ N (W (k)) is the unique W (k)-valued point z1 ∈ N (W (k)) that lifts y and such that the p-divisible group Dz1 of z∗1(A)is a direct sum of p-divisible groups over W (k) whose special fibres have only one Newtonpolygon slope; if moreover k = k, then the p-divisible group Dz of z∗(A) is cyclic.

1.6.3. Theorem. We assume that k = F and that the W (k)-valued point z ∈ N (W (k))is such that (M,F 1, ϕ, G) is a U-canonical lift. We have the following four properties:

(a) Then A = z∗(A) has complex multiplication.

(b) Let eF be an endomorphism of AF whose crystalline realization cF ∈ EndW (k)(M)

is an endomorphism of (M,ϕ, G). If cF has a zero image in EndW (k)(M/4M) or if p = 2and (M,ϕ) has no integral Newton polygon slopes, then eF lifts to an endomorphism of A.

(c) Let k0 be a finite field. Let Ak0be an abelian variety over k0 such that AF is

AF = y∗(A). Let Π ∈ End(AF) be the extension to F of the Frobenius endomorphism ofAk0

. Let z0 : Spec(V ) → N be a lift of y, where V is a finite, discrete valuation ringextension of W (F). Let s ∈ N ∪ 0 and t ∈ N. We assume that the endomorphism psΠt

of AF = y∗(A) lifts to an endomorphism of z∗0(A).

If the condition W for z0 (to be defined in Subsection 6.5.5) holds, then the abelianscheme z∗0(A) has complex multiplication. Condition W for z0 holds if V =W (F) (there-fore, if V =W (F), then z∗0(A) has complex multiplication).

(d) Referring to (c), if moreover V = W (F) (resp. V = W (F), p = 2, and s = 0),then z0 is a U-canonical lift (resp. is the unique W (k)-valued point z1 ∈ N (W (k)) whichlifts y and for which the p-divisible group Dz1 of z∗1(A) is cyclic).

1.6.4. Definitions. We call every point y ∈ N (k) as in Theorem 1.6.2 as a U-ordinarypoint (of either Nk(v) or N ). We call everyW (k)-valued point z ∈ N (W (k)) as in Theorem1.6.2 as a U-canonical lift (of either y ∈ N (k) or N ). If moreover y ∈ O(k) ⊆ N (k), thenwe also refer to y as an Sh-ordinary point (of either Nk(v) or N ) and to z as an Sh-canonicallift (of either y ∈ O(k) or N ). We also refer to O as the Sh-ordinary locus of Nk(v).

1.6.5. On I0. We can have I0 ≥ 2 only if there exists a simple factor (GQ, X) of(Gad

Q , Xad) that is either of Aℓ type with ℓ ≥ 2 or of DH

ℓ type with ℓ ≥ 4 (see Subsection3.4.7 and Corollary 3.6.3 for an abstract version of this; see also the property 9.5.2 (l)). Inparticular, we always have I0 = 1 if the group Gad

Q is non-trivial and all simple factors of

25

(GadQ , X

ad) are of Bℓ (ℓ ≥ 1), Cℓ (ℓ ≥ 3), or DRℓ (ℓ ≥ 4) type. If the group Gad

Q is simple

(thus we have GQ = GadQ ) and if (Gad

Q , Xad) is of Aℓ type with ℓ ≥ 2 (resp. of DH

ℓ typewith ℓ ≥ 4), then one can check that regardless of what the prime v is, we have I0 ∈ S(1, ℓ)(resp. I0 ∈ 1, 2); in addition, we have plenty of situations in which I0 is an arbitrarynumber in S(1, ℓ) (resp. in 1, 2).

1.6.6. On W. Condition W for z0 pertains to certain Bruhat decompositions (see Subsec-tion 6.5.5) and it is easy to see that it is also a necessary condition in order that the abelianscheme z∗0(A) of the statement 1.6.3 (c) has complex multiplication. Roughly speaking,the condition W for z0 holds if there exist enough endomorphisms of (M,ϕ, G) that arecrystalline realizations of endomorphisms of z∗0(A). Condition W for z0 always holds if yis a Sh-ordinary point, cf. proof of Corollary 9.6.4.

1.7. On applications

Different examples, complements, and applications of Theorems 1.6.1 to 1.6.3 are gath-ered in Chapters 7 to 9. We will mention here only five such complements and applications;to state them, in this section we take k = k.

In Section 7.2 we define Dieudonne truncations modulo positive, integral powers of pof Shimura F -crystals attached to k-valued points of N and we include a modulo p variantof the last paragraph of Theorem 1.6.1 (see Corollary 7.2.2).

In Section 8.2 we complete for p ≥ 5 (the last step of) the proof of the existenceof integral canonical models in unramified mixed characteristic (0, p) of Shimura varietieswhose adjoints are products of simple, adjoint Shimura varieties of some DH

ℓ type withℓ ≥ 4. In other words, the Zariski density part of Theorem 1.6.1 and the statement 1.6.3(a) allow us to apply [Va3, Lem. 6.8.1 and Criteria 6.8.2] for proving (see Subsection 8.2.4)the postponed result [Va3, Thm. 6.1.2*] for these Shimura varieties.

In Sections 9.1, 9.2, and 9.4 we generalize properties 1.1 (f) to (h) to the contextof a standard Hodge situation (f, L, v). These generalizations are rooted on the notionsof Subsection 1.3.5 and on Faltings deformation theory of [Fa2, §7]. We detail on thegeneralization of the property 1.1 (h) and on complements to it.

To C0 ⊗ k = (M0 ⊗W (k(v)) W (k), ϕ0 ⊗ σk, GW (k)) we attach two formal Lie groupsover Spf(W (k)): the first one F1 is commutative and the second one F2 is nilpotent ofnilpotent class I0 (see Sections 5.3 and 5.4). The first one is the formal Lie group of thep-divisible group D+

W (k) over W (k) whose filtered Dieudonne module is

(Lie(U+0,W (k)), F

10 (Lie(GW (k))), ϕ0 ⊗ σ),

where F 10 (Lie(GW (k))) := x ∈ Lie(GW (k))|x(F

10 ⊗W (k(v)) W (k)) = 0 and x(M0 ⊗W (k(v))

W (k)) ⊆ F 10 ⊗W (k(v)) W (k); we call D+

W (k) the positive p-divisible group over W (k) of

C0 ⊗ k. The second one is a modification of F1 that pays attention to the Lie structure ofLie(U+

0,W (k)). We view F1 (resp. F2) as the etale (resp. as the crystalline) possible way

to define or to look at the formal deformation space of C0 ⊗ k. We have F1 = F2 if andonly if U+

0 is commutative. See Examples 5.2.7 to 5.2.10 and the property 9.5.2 (b) fordifferent examples that pertain to the structure of D+

W (k) (which does depend on I0).

26

Let y ∈ O(k) ⊆ N (k). Let Dy = Spf(R) be the formal scheme of the completion R ofthe local ring of the k-valued point of NW (k) defined by y. One expects that Dy has ny

natural structures of formal Lie groups isomorphic to F1 as well as ny natural structuresof formal Lie groups isomorphic to F2 (see Theorem 1.6.2 for the positive integer ny).Theorem 9.4 proves this in the so called commutative case when I0 ≤ 1 (i.e., when U+

0 iscommutative) and in Section 9.8 we deal with some specific properties of the case I0 ≥ 2.

For simplicity, in this and the next paragraph we will refer only to the commutativecase; thus we have I0 ≤ 1. If p ≥ 3, let z ∈ N (W (k)) be the Sh-canonical lift of y. Ifp = 2, then we choose a Sh-canonical lift z ∈ N (W (k)) of y (cf. Theorem 1.6.2). Then Dy

has a unique canonical formal Lie group structure defined by the following two properties:(i) the origin is defined by z, and (ii) the addition of two Spf(W (k))-valued points of Dy

is defined as in Subsection 1.2.6 via sums of lifts. If p = 2, then we have ny = 2m0(1),where m0(1) is the multiplicity of the Newton polygon slope 1 of (Lie(Gad

W (k(v)))[1p ], ϕ0)

(see Proposition 9.5.1). Also if p = 2, then the ny formal Lie group structures on Dy thatcorrespond to the ny possible choices of z, differ only by translations through 2-torsionSpf(W (k))-valued points (see property 9.5.2 (d)). For p = 2, the methods used in thismonograph can not show that we can always choose z such that the p-divisible group Dz

of z∗(A) is cyclic (see [Va8] for a proof of this); thus, if ny > 1, in this monograph we cannot single out in general any such 2-torsion Spf(W (k))-valued point and this explains whyeverywhere we mention ny canonical structures.

See Proposition 9.6.1 and Definition 9.6.2 for the Spf(V )-valued points version of sumsof lifts and see Corollary 9.6.5 and Remarks 9.6.6 for Spf(V )-valued torsion points; here Vis a finite, discrete valuation ring extension of W (k). See Subsection 5.4.1 and Proposition9.7.1 for different functorial aspects of the formal Lie groups F1 and of the mentionedcanonical formal Lie group structures on Dy. Let vad be the prime of the reflex fieldE(Gad

Q , Xad) of (Gad

Q , Xad) divided by v. If the residue field k(vad) of vad is Fp, then up

to Spf(W (k))-valued points of order two (they can exist only if p = 2), all the canonicalformal Lie group structures on Dy coincide and F1 is a formal torus (see property 9.5.2(a)); this result for the case when k(v) = k(vad) = Fp was first obtained in [No].

1.8. Extra literature

The FSHW maps were introduced in [Va1, Subsubsect. 5.5.4] following a suggestionof Faltings and this explains our terminology. They are the adjoint Lie analogues ofBarsotti–Tate groups of level 1 over k, cf. Claim of Subsection 1.2.3, the statement 1.4.4(b), etc. To our knowledge, [Va1, Subsect. 5.5 and 5.6] is the first place where Bruhatdecompositions and Weyl elements are used in the study of Ψ and thus implicitly ofBarsotti–Tate groups of level 1 over k (see property (12) of Proposition 3.5.1). The notionU-ordinariness is a new concept. It is more general than the notion Sh-ordinariness, despitethe fact that in the classical context of Section 1.1 (with G = GLM ) we do not get U-canonical lifts that are not Sh-canonical lifts. Referring to the context of Section 1.3, theU-canonical lifts typically show up when ϕ ⊗ σk gives birth to a non-trivial permutationof the simple factors of Lie(Gad

B(k)) (see Subsections 4.3.3 to 4.3.5 for several examples

and see Section 7.5 for the simplest geometric examples). The Sh-canonical lifts were firstmentioned in [Va3, Subsects. 1.6 and 1.6.2]; previously to [Va3] and [Va4], they were

27

used only either for the case of ordinary p-divisible groups or for cases pertaining to H2

crystalline realizations of hyperkahler varieties (like K3 surfaces). In the generality ofTheorem 1.6.2, for p ≥ 5 they were introduced in the first version [Va2] of this monograph(for p ≥ 2 see also [Va4]). See Remark 4.2.2 for a link between Theorem 1.4.1 and [RR].

The Shimura F -crystal C0 was introduced in [Va1, Subsect. 5.3 g)] (see also [RR,Thm. 4.2] for a general abstract version of it). For Siegel modular varieties the Zariskidensity part of Theorem 1.6.1 was checked previously in [Kob], [NO], [FC], [Va1], [We1],[Oo1], [Oo2], etc. All these references used different methods. Under slight restrictions(that were not required and are eliminated here), the Zariski density part of Theorem 1.6.1was obtained in [Va1, Thm. 5.1]. Loc. cit. is correct only when I0 ≤ 2, as it does notmake distinction between P+

G (ϕ)’s and P+G (ϕ)’s parabolic subgroup schemes as Example

1.4.6 does; but as hinted at in Remark 1.4.7, the modifications required to eliminate thisrestriction involve only a trivial induction. In [We1] a variant of the Zariski density partof Theorem 1.6.1 is obtained for standard PEL situations: it defines O via 1.6.1 (b) andfor p = 2 it deals only with standard moduli PEL situations. Paper [We2] uses a variantof 1.6.1 (c) (that pertains to Barsotti–Tate groups of level 1), in order to refine [We1] forp ≥ 3 and for the so call C case of moduli standard PEL situations. Positive p-divisiblegroups over W (k) like D+

W (k) and the nilpotent class I0 were introduced in [Va4].

We (resp. B. Moonen) reported first on Theorem 1.6.1 and [Va1] and [Va2] (resp. on[Va1, §5]) in Durham (resp. in Munster) July (resp. April) 1996.1

1.9. Our main motivation

The generalized Serre–Tate theories of Sections 1.4 and 1.6 will play for Shimura va-rieties the same role played by the classical Serre–Tate theory for Siegel modular varietiesand for p-divisible groups. For instance, the generalizations of many previous works cen-tered on ordinariness (like [De1], [FC, Ch. VII, §4], [No], etc.) are in the immediate reach.But our five main reasons to develop these two theories are the following ones.

(a) To use them in connection to the combinatorial conjecture of Langlands–Rapoport

that describes the set N (F) together with the natural actions on it of GQ(A(p)f ) and of the

Frobenius automorphism of F whose fixed field is k(v) (see [LR], [Mi1], and [Pf]). Thisconjecture is a key ingredient toward the understanding of the zeta functions of quotientsof finite type of NE(GQ,X) and of different trace functions that pertain to Ql-local systemson smooth quotients of N (see [LR], [Ko2], and [Mi1]; here l ∈ N \ p is a prime). Not tomake this monograph too long, we only list here its parts that will play roles in the proofof this conjecture. Subsections 1.6.3 (a), 4.3.1 (b), 6.2.8, 7.4.1, 8.1, 9.6.5, and 9.6.6 is allthat one requires to prove this conjecture for the subset O(F) of N (F) (i.e., generically);this will generalize [De1] and the analogous generic parts of [Zi1] and [Ko2].

(b) In future work we will use them to prove that for p ≥ 2 every potential standardHodge situation is a standard Hodge situation (see [Va8]).

1 We point out that B. Moonen’s paper “Serre-Tate theory for moduli spaces of PELtype” (see Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 2, pp. 223–269) obtains someanalogous results for standard PEL situations.

28

(c) To provide intrinsic methods that will allow us in future work to extend Theorems1.6.1 to 1.6.3 as well as (a) and (b) to all integral canonical models of Shimura varieties ofpreabelian type (a Shimura variety is said to be of preabelian type if its adjoint Shimuravariety is isomorphic to the adjoint Shimura variety of a Shimura variety of Hodge type).

(d) In future work we will use Section 1.4 to construct integral canonical models ofShimura varieties of special type (i.e., of Shimura varieties which are not of preabelian typeand thus whose adjoint Shimura varieties have simple factors of Dmixed

ℓ , E6, or E7 types).

(e) They extend to all types of polarized varieties endowed with Hodge cycles whosedifferent moduli spaces are open subvarieties of (quotients of) Shimura varieties of pre-abelian type (like polarized hyperkahler varieties; see [An] for more examples).

1.10. More on organization

In Chapter 2 we include preliminaries on reductive group schemes and Newton poly-gons. In Section 3.1 we list basic properties of Hodge cocharacters. See Section 3.2 for ageneral outline of the proofs of Theorems 1.4.1 and 1.4.2. The outline is carried out in Sec-tions 3.3 to 3.8. In Sections 4.1 and 4.2 we prove Theorems 1.4.3 and 1.4.4 (respectively).See Subsection 4.2.1 for the proof of Corollary 1.4.5. In Chapter 5 we introduce the firstand the second formal Lie groups of Sh-ordinary p-divisible objects over k. In Chapter 6we prove Theorems 1.6.1 to 1.6.3. In Chapters 7 to 9 we include examples, complements,and applications of Theorems 1.6.1 to 1.6.3 (see their beginnings and Sections 1.7 and 1.8).

Acknowledgment. We would like to thank G. Faltings for his encouragement to approachthe topics of Sections 1.3 to 1.6, for his deep insights, and for the suggestion mentioned inSection 1.8. We would like to thank R. Pink for pointing out to us the construction of non-positive solvable group schemes (like S− and S−

G(ϕ) of Subsection 1.2.4 and respectively ofSection 5.1) and the fact that a deformation theory as in [Va3, Subsect. 5.4] is all that onerequires to generalize the property 1.1 (g) to the context of a standard Hodge situation(f, L, v) (see Section 9.2 for details). We would like to thank Princeton University, FIM,ETH Zurich, University of California at Berkeley, University of Utah, University of Arizona,and Binghamton University for good conditions with which to write this monograph whichfor many years was thought as a small part of a comprehensive book (see [Va4]).

29

2 Preliminaries

We begin with a list of notations and conventions to be used throughout the mono-graph. We always use the notations and the conventions listed before Section 1.1. Let p, k,W (k), B(k), and σ := σk be as in Section 1.1. In Section 2.1 we list some basic notationsto be used in Chapters 2 to 5. In Sections 2.2 and 2.5 we include complements on Newtonpolygons. In Section 2.3 we gather elementary properties of reductive group schemes overeither fields or W (k). In Section 2.4 we introduce some Zp structures and list applicationsof them. In Subsection 2.4.3 we argue the existence of the map Ψ of Subsection 1.3.4. InSection 2.6 we introduce the main ten types of group schemes over W (k) that are intrin-sically associated to each Shimura p-divisible object over k and that will be often used inthe monograph. In particular, in Subsections 2.6.1 and 2.6.2 we argue the existence of thesubgroup schemes P+

G (ϕ) and L0G(ϕ)B(k) of Subsection 1.3.3. In Section 2.7 we introduce

an opposition principle in the form required in Chapter 3. In Section 2.8 we include someelementary crystalline complements that will be used in Chapters 4 to 9.

Let |X | be the number of elements of a finite set X . If M is a module over a com-mutative Z-algebra C, let M⊗s ⊗C M∗⊗t, with s, t ∈ N ∪ 0, be the tensor product ofs-copies of M with t-copies of M∗ taken in this order. By the essential tensor algebra ofM ⊕M∗ we mean

T (M) := ⊕s,t∈N∪0M⊗s ⊗C M∗⊗t.

Let x ∈ C be a non-divisor of 0. A family of tensors of T (M [ 1x ]) = T (M)[ 1x ] is denoted(vα)α∈J , with J as the set of indices. We emphasize that we use the same notationfor two bilinear forms or tensors (of two essential tensor algebras) that are obtained onefrom another via a scalar extension. Let M1 be another C-module and let (v1,α)α∈J bea family of tensors of T (M1[

1x]) indexed by the same set J . By an isomorphism f :

(M, (vα)α∈J )→(M1, (v1,α)α∈J ) we mean a C-linear isomorphism f :M→M1 that extendsnaturally to a C[ 1

x]-linear isomorphism T (M [ 1

x])→T (M1[

1x]) which takes vα to v1,α for all

α ∈ J . If λM and λM1are perfect alternating forms on M and M1 (respectively), then

by an isomorphism f : (M,λM , (vα)α∈J )→(M1, λM1, (v1,α)α∈J ) we mean a symplectic

C-linear isomorphism f : (M,λM)→(M1, λM1) that defines as well an isomorphism f :

(M, (vα)α∈J )→(M1, (v1,α)α∈J ).

Let F 1(M) be a direct summand of M . Let F 0(M) := M and F 2(M) := 0.Let F 1(M∗) := 0, F 0(M∗) := y ∈ M∗|y(F 1(M)) = 0, and F−1(M∗) := M∗. Let(F i(T (M)))i∈Z be the tensor product filtration of T (M) defined by the exhaustive, sepa-rated filtrations (F i(M))i∈0,1,2 and (F i(M∗))i∈−1,0,1 of M and M∗ (respectively). Bythe F 0-filtration of T (M) defined by F 1(M) we mean F 0(T (M)).

30

Let f : F → H be a homomorphism of affine group schemes over C that are of finitetype. If C is not a field, we assume F and H are smooth. Let df : Lie(F ) → Lie(H)be the Lie homomorphism over C which is the differential of f at identity sections. Ifx ∈ Lie(F ) and if f is a direct summand of Lie(F ) such that [x, f] ⊆ f, then by the actionof x on f we mean the C-linear endomorphism f → f (often called in the literature as therestriction of ad(x) to f) that takes y ∈ f to the Lie bracket [x, y]. By a Frobenius lift of acommutative Zp-algebra C we mean an endomorphism of C that lifts the usual Frobeniusendomorphism of C/pC. If C is a complete, discrete valuation ring of mixed characteristic(0, p) and of residue field k, then H(C) is called a hyperspecial subgroup of H(C[ 1p ]) (see

[Ti2]). If moreover H is a torus, then H is the unique reductive group scheme extensionof the generic fibre of H (to check this we can assume that H is split and this case isobvious); thus H(C) is the unique hyperspecial subgroup of H(C[ 1p ]).

For a commutative Fp-algebra C and for m ∈ N, let W (C) (resp. Wm(C)) be thering of Witt vectors (resp. of Witt vectors of length m) with coefficients in C. Let ΦC bethe canonical Frobenius lift of W (C), Wm(C), and C = W1(C); we have σ = σk = Φk. Ifmoreover C is an integral domain, let Cperf be the perfection of C. We refer to [BBM] and[BM] for the crystalline contravariant Dieudonne functor D. We refer to [Me, Ch. 5] and[Ka3, Ch. 1] for the Serre–Tate deformation theory. We refer to [Me, Chs. 4 and 5] forthe Grothendieck–Messing deformation theory. The divided power structure on the idealpW (k) of W (k) is the usual one from characteristic 0.

If C is a commutativeW (k)-algebra, then the crystalline sites CRIS(Spec(C)/Spec(W (k)))and CRIS(Spec(C/pC)/Spec(W (k))) will be as in [Be, Ch. III, §4].

We include an argument for the standard fact that every F -isocrystal (N,ϕ) over k is adirect sum of F -isocrystals that have only one Newton polygon slope. Let N ⊗B(k)B(k) =

⊕γ∈QW (γ)(N,ϕ) be the Newton polygon slope decomposition over k defined by ϕ ⊗ σk,

cf. [Di] or [Ma]. We recall that if n ∈ N and γ ∈ 1nZ, then there exists a B(k)-basis for

W (γ)(N,ϕ) formed by elements fixed by p−nγ(ϕ⊗ σk)n. Let e ∈ EndB(k)(N ⊗B(k) B(k))

be the semisimple element that acts on W (γ)(N,ϕ) via the multiplication with γ. TheNewton polygon slope decomposition N ⊗B(k) B(k) = ⊕γ∈QW (γ)(N,ϕ) is normalized by

Gal(k/k) = AutB(k)(B(k)) and thus e is fixed under the natural action of AutB(k)(B(k))

on EndB(k)(N ⊗B(k) B(k)) = EndB(k)(N) ⊗B(k) B(k). But B(k) is the fixed subfield of

B(k) under AutB(k)(B(k)) and thus we have e ∈ EndB(k)(N). Therefore W (γ)(N,ϕ) is

the tensorization with B(k) of a B(k)-vector subspace W (γ)(N,ϕ) of N .

Thus we have a Newton polygon slope decomposition

N = ⊕γ∈QW (γ)(N,ϕ)

defined by ϕ. For each rational number γ, let W γ(N,ϕ) (resp. Wγ(N,ϕ)) be the maximalB(k)-vector subspace of N which is normalized by ϕ and for which all Newton polygonslopes of W γ(N,ϕ) (resp. of Wγ(N,ϕ)) are greater or equal to γ (resp. are smaller orequal to γ). We have identities:

W γ(N,ϕ) = ⊕β∈[γ,∞)∩QW (β)(N,ϕ),

31

Wγ(N,ϕ) = ⊕β∈(−∞,γ]∩QW (β)(N,ϕ),

W (γ)(N,ϕ) =W γ(N,ϕ) ∩Wγ(N,ϕ).

If γ1 < γ2 < · · · < γc are all the Newton polygon slopes of (N,ϕ) listed increasingly, thenwe have a descending Newton polygon slope filtration

0 ⊆W γc(N,ϕ) ⊆ · · · ⊆ W γ1(N,ϕ) = N

and an ascending Newton polygon slope filtration

0 ⊆Wγ1(N,ϕ) ⊆ · · · ⊆Wγc

(N,ϕ) = N

For ∗ ∈ γ ,γ, (γ), the pair (W ∗ (N,ϕ), ϕ) is an F -subisocrystal of (N,ϕ). For a W (k)-

lattice M of N , we define W ∗ (M,ϕ) :=M ∩W ∗ (N,ϕ).

2.1. Standard notations for the abstract theory

We emphasize that until the end of Chapter 5 the following notations

(M, (F i(M))i∈S(a,b), ϕ, G), , Fi(M) (i ∈ S(a, b)), µ : Gm → G,

F j(Lie(G)) (j ∈ −1, 0, 1), F 0(Lie(G)), P 6 G, Gad =∏

i∈I

Gi,

Ψ : Lie(Gad) → Lie(Gad), Ψ : Lie(Gadk ) → Lie(Gad

k ), P+G (ϕ), and L0

G(ϕ)B(k)

will be as in Section 1.3; moreover, we will not use the notations of Section 1.5.

Let dM := rkW (k)(M). Let ϕ act in the usual tensor way on T (M)[ 1p]. For in-

stance, if f ∈ M∗[ 1p], then ϕ(f) := σ f ϕ−1 ∈ M∗[ 1

p]. Under the natural identification

EndB(k)(M [ 1p]) = M [ 1

p] ⊗B(k) M

∗[ 1p], the two actions of ϕ coincide (see property 1.3.2

(c)) for the first action). We denote also by ϕ the σ-linear endomorphism of each W (k)-submodule of T (M [ 1p ]) left invariant by ϕ. We consider the permutation

Γ : I→I

of the set I such that for each i ∈ I we have an identity ϕ(Lie(Gi)[1p ]) = Lie(GΓ(i))[

1p ]. If

k1 is a perfect field that contains k, then by the extension of (M,ϕ,G) to k1 we mean

(M,ϕ,G)⊗ k1 := (M ⊗W (k) W (k1), ϕ⊗ σk1, GW (k1)).

Let g := Lie(G). We have F 0(g) = F 0(g) ⊕ F 1(g). Via adjoint representations, weview Gad as a closed subgroup scheme of either GLg or GLLie(Gad). Let F

1(Lie(Gad)) and

F 1(g) be F 1(g); they are direct summands of either g or Lie(Gad). Let F 0(Lie(Gad)) :=Lie(Gad) ∩ F 0(g)[ 1

p]. Let F−1(g) := g and F−1(Lie(Gad)) := Lie(Gad).

We call the quadruple(g, F 0(g), F 1(g), ϕ)

32

(resp. the pair (g, ϕ)) as the Shimura filtered (resp. Shimura) Lie F -crystal attached to(M, (F i(M))i∈S(a,b), ϕ, G) (resp. to (M,ϕ,G)). By replacing g with Lie(Gad), we get thenotion of adjoint Shimura filtered (resp. adjoint Shimura) Lie F -crystal

(Lie(Gad), F 0(Lie(Gad)), F 1(Lie(Gad)), ϕ)

(resp. (Lie(Gad), ϕ)) attached to (M, (F i(M))i∈S(a,b), ϕ, G) (resp. to (M,ϕ,G)).

2.2. Descending slope filtration Lemma

Let (M,ϕ1) and (M,ϕ2) be two F -crystals over k with the property that for eachpositive integer m there exists an element hm ∈ GLM (W (k)) such that we have an identityϕm1 = hmϕ

m2 . Then the Newton polygons of (M,ϕ1) and (M,ϕ2) coincide and for each

number γ ∈ Q ∩ [0,∞) we have an identity W γ(M,ϕ1) = W γ(M,ϕ2).

Proof: We can assume that k = k. Let r ∈ N be such that for each i ∈ 1, 2 there exists aB(k)-basis ei1, . . . , e

idM

for M [ 1p ] which is formed by elements of M \ pM and which has

the property that we have ϕri (e

il) = pnl,ieil for all l ∈ S(1, dM), where nl,i ∈ N ∪ 0. We

choose the indices such that if l, s ∈ S(1, dM) and l < s, then nl,i ≥ ns,i. Let iM ∈ N∪0

be such that for each i ∈ 1, 2 we have piMM ⊆ ⊕dMj=1W (k)eij . For each positive integer m

the Hodge polygons of (M,ϕm2 ) and (M,ϕm

1 ) = (M,hmϕm2 ) are equal. Thus the Newton

polygon of (M,ϕi) is independent of i ∈ 1, 2, cf. [Ka2, 1.4.4]. Let S be the set of Newtonpolygon slopes of (M,ϕi). For γ ∈ S let W γ

i :=W γ(M [ 1p], ϕi); its dimension over B(k) is

independent of i ∈ 1, 2. Let W∞i := 0.

To prove the Lemma, it suffices to show that for each γ ∈ S∪∞ we have an identityW γ

1 =W γ2 . We proceed by decreasing induction on γ ∈ S∪∞. Let γ0 := inf(S∩(γ,∞)).

If γ0 = ∞, then W γ0

1 =W γ0

2 . Thus to complete the induction, we only need to show thatwe have W γ

1 =W γ2 if W γ0

1 =W γ0

2 .

Let t ∈ S(0, dM) be such that ei1, . . . , eit is a B(k)-basis forW γ0

i . Let s ∈ S(t+1, dM)be such that ei1, . . . , e

is is a B(k)-basis for W γ

i . For q ∈ S(t + 1, s) we write e1q =∑dM

l=1 cl,qe2l , where cl,q ∈ B(k). We take m = rm0, where m0 ∈ N. From the definition

of r and the identity ϕm1 = hmϕ

m2 , we get that pm0nq,1h−1

m (e1q) =∑dM

l=1 pm0nl,2σm(cl,q)e

2l .

Thus piMσm(cl,q)pm0(nl,2−nq,1) ∈ W (k) for all l ∈ S(1, dM). Taking m0 ∈ N such that

m0− iM is greater than the greatest p-adic valuation of cl,q’s, we get that for each elementl0 ∈ S(1, dM) with the property that cl0,q 6= 0, we have nl0,2 ≥ nq,1. But if u ∈ S(s+1, dM),then nu,2 < rγ = nt+1,1 = nq,1 = ns,1. Thus cl,q = 0 for each l ∈ S(s + 1, dM). Thuse1q ∈ W γ

2 for q ∈ S(t + 1, s). From this and the identity W γ0

1 = W γ0

2 , we get W γ1 ⊆ W γ

2 .Thus by reasons of dimensions we get W γ

1 =W γ2 . This completes the induction.

2.2.1. Corollary. Let (M,ϕ1, G) and (M,ϕ2, G) be two Shimura p-divisible objects overk with the property that for each positive integer m there exists an element hm ∈ G(W (k))such that we have an identity ϕm

1 = hmϕm2 . Then the Newton polygon of (Lie(G), ϕi) does

not depend on i ∈ 1, 2 and for each γ ∈ Q we have W γ(Lie(G), ϕ1) =W γ(Lie(G), ϕ2).

Proof: This follows from Lemma 2.2 applied to the F -crystals (Lie(G), p1Lie(G)ϕi)’s.

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2.3. Elementary properties of reductive group schemes

We review ten groups of elementary properties of reductive group schemes.

2.3.1. Parabolic subgroups over a field. Until Lemma 2.3.6 let P1 and P2 be twoparabolic subgroups of a reductive group H over a field L. For i ∈ 1, 2, the group Pi

is connected as well as its own normalizer in H (cf. Chevalley theorem; see [Bo, Ch. IV,Thm. 11.16]). Let Ui be the unipotent radicals of Pi. We recall four results from [Bo].

(a) If U2 6 P1, then P1 ∩ P2 is a parabolic subgroup of H (cf. [Bo, Ch. IV, Prop.14.22 (i)] applied over L).

(b) If P1 and P2 are H(L)-conjugate and if P1 ∩ P2 contains a parabolic subgroup ofH, then P1 = P2 (cf. [Bo, Ch. IV, Prop. 14.22 (iii)] applied over L).

(c) If P1,L and P2,L are H(L)-conjugate, then P1 and P2 are H(L)-conjugate (cf. [Bo,

Ch. V, Thm. 20.9 (iii)]).

(d) If either char(L) = 0 or L is perfect, then every two maximal split tori of P1 areP1(L)-conjugate (cf. [Bo, Ch. V, Thm. 15.14]).

2.3.2. Fact. Let P3 be a subgroup scheme of H that contains P1. If Lie(P1) = Lie(P3),then we have P1 = P3.

Proof: We have dim(P3) ≥ dim(P1) = dimL(Lie(P1)) = dimL(Lie(P3)) ≥ dim(P3). ThusdimL(Lie(P3)) = dim(P3) = dim(P1). As dimL(Lie(P3)) = dim(P3), the group scheme P3

is smooth. As P1 6 P3, the smooth group P3 is a parabolic subgroup of H. Thus P3 isconnected. By reasons of dimensions we conclude that P1 = P3.

2.3.3. Proposition. The following four statements are equivalent:

(a) we have P1 6 P2;

(b) we have Lie(P1) ⊆ Lie(P2);

(c) we have U2 6 U1;

(d) we have Lie(U2) ⊆ Lie(U1).

Moreover, if these four statements hold, then U2 ⊳ U1.

Proof: We can assume that L = L. The intersection P1 ∩ P2 contains a maximal torus Tof H, cf. [Bo, Ch. IV, Prop. 14.22 (ii)]. Let Φ be the root system of Lie(H) relative toT . For i ∈ 1, 2 let Φi be the subset of Φ formed by roots of Lie(Pi) relative to T . Theset Φu

i := Φi \ (Φi ∩ −Φi) is the set of roots of Lie(Ui) relative to T . We have Φ1 ⊆ Φ2

(resp. Φu2 ⊆ Φu

1) if and only if (b) (resp. (d)) holds. Also we have Φ1 ⊆ Φ2 if and only ifΦu

2 ⊆ Φu1 . But (a) holds if and only if Φ1 ⊆ Φ2, cf. the classification of parabolic subgroups

of H that contain T (see [DG, Vol. III, Exp. XXVI, Prop. 1.4] and [DG, Vol. III, Exp.XXII, Prop. 5.4.5 (ii)]). Also (c) holds if and only if Φu

2 ⊆ Φu1 , cf. [DG, Vol. III, Exp.

XXVI, Prop. 1.12 (i)]. Thus each one of the four statements (a) to (d) is equivalent to theinclusion Φ1 ⊆ Φ2. Thus the statements (a) to (d) are equivalent. If (a) holds, then U1 isa subgroup of P2 and thus it normalizes U2; from this and (c) we get that U2 ⊳ U1.

2.3.4. Lemma. If h ∈ H(L) normalizes either Lie(P1) or Lie(U1), then h ∈ P1(L).

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Proof: We have hP1h−1 = P1, cf. Proposition 2.3.3. As P1 is its own normalizer in H, we

get that h ∈ P1(L).

2.3.5. Lemma. We assume that H is adjoint and absolutely simple and that U1 iscommutative and non-trivial. If h ∈ H(L) centralizes Lie(U1), then we have h ∈ U1(L).

Proof: To prove this we can assume that L = L. We have h ∈ P1(L), cf. Lemma 2.3.4.Let T be a maximal torus of P1. We check that the inner conjugation representationT → GLLie(U1) is a closed embedding monomorphism. Let Φ and Φu

1 be as in the proofof Proposition 2.3.3. Let ∆ be a base of Φ such that there exists a non-empty subset ∆1

of ∆ with the property that Φu1 is the set of roots in Φ that have at least one element of

∆1 with a positive coefficient in their expression. As H is adjoint and simple, from thelast sentence we get that the set Φu

1 generates the (additive) group of characters of T .Therefore the representation T → GLLie(U1) is a closed embedding monomorphism.

Thus the reduced scheme Z1 of the kernel of the analogous representation P1/U1 →GLLie(U1), is a normal subgroup of P1/U1 that has trivial intersection with each torus ofP1/U1. As Z1 ⊳ P1/U1, the identity component Z0

1 of Z1 is a reductive group. As Z01 has

a trivial intersection with each maximal torus of P1/U1, its rank is 0 and thus Z01 is the

trivial group. Thus Z1 is a finite, etale, normal subgroup P1/U1 and therefore we haveZ1 6 Z(P1/U1). But Z(P1/U1) is a subgroup of every torus of P1/U1. Thus Z1 is trivialand therefore the image of h ∈ P1(L) in (P1/U1)(L) is trivial. Thus we have h ∈ U1(L).

2.3.6. Lemma. Let H be a reductive group scheme over a flat, affine, reduced Z–scheme Z. Let F1 and F2 be two smooth, closed subgroup schemes of H whose fibresin characteristic 0 are connected. If Lie(F1) ⊆ Lie(F2), then F1 6 F2. Therefore we haveLie(F1) = Lie(F2) if and only if F1 = F2.

Proof: As Z is reduced and as F1 and F2 are smooth over Z, F1 and F2 are reduced. Thusto check that F1 6 F2, it suffices to show that each fibre of F1 → Z is included in thecorresponding fibre of F2 → Z. But as Z is flat over Z and as F1 and F2 are smooth,closed subgroup schemes of H, to check this last thing it suffices to consider fibres overpoints of Z of characteristic 0. Thus we can assume that Z is the spectrum of a field ofcharacteristic 0. But in this case, the inclusion Lie(F1) ⊆ Lie(F2) implies that F1 6 F2

(cf. [Bo, Ch. II, Sect. 7.1]).

2.3.7. Descent Lemma for groups. Let Q → L → K be field extensions. Let Hbe a reductive group over L. Let F1 be a connected subgroup of HK . We assume thatthere exists a Lie subalgebra f of Lie(H) such that we have Lie(F1) = f⊗L K. Then thereexists a unique connected subgroup F of H such that we have F1 = FK . Moreover we haveLie(F ) = f.

Proof: The affine L-scheme Z := Spec(K ⊗L K) is reduced. Let q1, q2 : Z → Spec(K) bethe two natural projections. The smooth, closed subgroup schemes q∗1(F1) and q∗2(F1) ofHK⊗LK have the same Lie algebras equal to f⊗LK⊗LK. Thus we have q∗1(F1) = q∗2(F1),cf. Lemma 2.3.6. Due to this identification, we have a descent datum on the subgroup F1

of HK with respect to the faithfully flat morphism Spec(K) → Spec(L). As F1 is an affinescheme, this descent datum is effective (cf. [BLR, Ch. 6, Sect. 2.1, Thm. 6]). Thus Fexists. As Lie(F )⊗L K = Lie(F1) = f⊗L K, we get Lie(F ) = f. Obviously F is unique.

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2.3.8. On reductive group schemes over W (k). Let H be a reductive group schemeover W (k). We say H or Hk is quasi-split, if it has a Borel subgroup scheme. We list eightgroups of properties of closed subgroup schemes of either H or Hk.

(a) Let Par(H) be the projective, smooth W (k)-scheme that parameterizes parabolicsubgroup schemes of H, cf. [DG, Vol. III, Exp. XXVI, Cor. 3.5]. As Par(H) is projectiveover W (k), the Zariski closure in H of a parabolic subgroup QB(k) of HB(k), is a parabolicsubgroup scheme Q of H. As Par(H) is smooth overW (k), each parabolic subgroup of Hk

lifts to a parabolic subgroup scheme of H. Thus H is quasi-split if and only if Hk is so.

(b) Let Q be a parabolic subgroup scheme of H. Let Tk be a maximal torus of Qk,cf. [Bo, Ch. V, Thm. 18.2 (i)]. As the functor on the category of W (k)-schemes thatparameterizes maximal tori of Q is representable by a smooth W (k)-scheme (cf. [DG, Vol.II, Exp. XII, Cor. 1.10]), we get that Q has a maximal torus T that lifts Tk.

Let T be an arbitrary torus over W (k). Let l : T → H be a homomorphism. Wecheck that Im(l) is a torus of H. For this, we can assume that T is split and (cf. [DG,Vol. I, Exp. VIB, Rm. 11.11.1]) that there exists a free W (k)-module W of finite ranksuch that H = GLW . But as the T -module W is a direct sum of T -submodules whichas W (k)-modules are of free of rank 1 (cf. [Ja, Part I, 2.11]), we easily get that Ker(l) isa group scheme of multiplicative type and that Im(l) = Im(T → GLW ) = T /Ker(l) is atorus of GLW . Let F be a flat, closed subgroup scheme of H such that FB(k) is connected.

As Im(l) is a torus of H, l factors through F if and only if lB(k) factors through FB(k) and

therefore (cf. Lemma 2.3.6) if and only if Lie(Im(l)B(k)) = Im(dlB(k)) ⊆ Lie(FB(k)). Thus

l factors through F if and only if Im(dl) ⊆ Lie(FB(k)).

(c) Let T be a maximal torus of H. Let Qk be a parabolic subgroup of Hk thatcontains Tk. We claim that there exists a unique parabolic subgroup scheme Q of H thatcontains T and has Qk as its special fibre. Using Galois descent, to check this we canassume T is split. But this case follows from [DG, Vol. III, Exp. XXVI, Prop. 1.4], as theset of roots of Lie(Q) relative to T can be read out over k.

Also there exists a unique Levi subgroup scheme of Q (resp. of QB(k) or Qk) thatcontains T (resp. TB(k) or Tk), cf. [DG, Vol. III, Exp. XXVI, Prop. 1.12 (ii)].

(d) We assume that H is quasi-split. Let B1 and B2 be two Borel subgroup schemesof H that are contained in a parabolic subgroup scheme Q of H. For i ∈ 1, 2 let Ti bea maximal torus of Bi, cf. (b). We check the standard fact that there exists an elementg ∈ Q(W (k)) such that we have gB1g

−1 = B2 and gT1g−1 = T2. We can assume that

B1,k = B2,k, cf. [Bo, Ch. V, Sects. 21.11 to 21.13]. We can also assume that T1,k = T2,k,cf. [Bo, Ch. V, Thm. 19.2]. For n ∈ N let gn ∈ Ker(Q(Wn(k)) → Q(k)) be such thatwe have gnT1,Wn(k)g

−1n = T2,Wn(k), cf. [DG, Vol. II, Exp. IX, Thm. 3.6]. We can assume

that gn+1 modulo pn is gn, cf. loc. cit. Let g ∈ Q(W (k)) be the unique element that liftsall gn’s. We have gT1g

−1 = T2. Thus we also have gB1g−1 = B2, cf. (c).

(e) Let Q1 and Q2 be two parabolic subgroup schemes of H. We check the standardfact that there exists an element g ∈ G(W (k)) such that the intersection gQ1g

−1 ∩Q2 is aparabolic subgroup scheme of H. We can assume that Q1,k ∩Q2,k is a parabolic subgroupof Hk, cf. [Bo, Ch. V, Sect. 21.11 and Prop. 21.12]. Let T1 and T2 be maximal tori of

36

Q1 and Q2 such that we have T1,k = T2,k, cf. (b). By replacing Q1 with gQ1g−1, where

g ∈ Ker(G(W (k)) → G(k)), as in (d) we argue that we can also assume that T1 = T2.Let Q3 be the unique parabolic subgroup scheme of H that contains T1 = T2 and lifts theintersection Q1,k ∩ Q2,k, cf. (c). We check that Q3 6 Q1 ∩ Q2. For this we can assumek = k and this case follows from [DG, Vol. III, Exp. XXII, Prop. 5.4.5 (i)] applied withrespect to the split torus T1 = T2. Thus Q1 ∩ Q2 is a parabolic subgroup scheme of G,cf. [DG, Vol. III, Exp. XXVI, Prop. 4.4.1]. Therefore we have Q3 = Q1 ∩ Q2, cf. theuniqueness of Q3. In particular, if Q1,k 6 Q2,k (resp. if Q1,k and Q2,k areH(k)-conjugate),then there exists an element g ∈ Ker(H(W (k)) → H(k)) (resp. g ∈ H(W (k))) such thatwe have gQ1g

−1 6 Q2 (resp. gQ1g−1 = Q2).

(f) We check that every two maximal split tori T1 and T2 of a parabolic subgroupscheme Q ofH are Q(W (k))-conjugate. We can assume that T1,k = T2,k (cf. property 2.3.1(d)) and thus as in (d) we argue that there exists an element g ∈ Ker(G(W (k)) → G(k))such that we have gT1g

−1 = T2.

If U is the unipotent radical of Q, then every two Levi subgroup schemes L1 and L2

of Q are U(W (k))-conjugate (cf. [DG, Vol. III, Exp. XXVI, Cor. 1.9]).

(g) Let ∗ ∈ k, B(k),W (k). If ∗ = W (k), let ∗ := W (k). Let HW (k) := H. Twoparabolic subgroup schemes Q1 and Q2 of H∗ are called opposite if their intersection is(equivalently contains) a common Levi subgroup scheme (cf. [DG, Vol. III, Exp. XXVI,Thm. 4.3.2]). This implies that Q1 and Q2 have equal relative dimensions. If Q1 and Q2

are opposite, we also say that they are opposite with respect to every maximal torus ofQ1 ∩Q2. The opposite of Q1 with respect to a maximal torus of it is unique, cf. loc. cit.

Let s1 and s2 be two ∗-submodules of Lie(H∗) that are direct summands. We say s1and s2 are opposite with respect to a maximal torus T∗ of H∗, if they are normalized by T∗and we have s1∩Lie(T∗) = s2∩Lie(T∗) and Φs1 = −Φs2 , where Φs1 and Φs2 are the sets ofroots of s1⊗∗ ∗ and s2⊗∗ ∗ (respectively) relative to T∗; this implies that rk∗(s1) = rk∗(s2).

We consider two pairs (Q1, s1) and (Q2, s2) such that s1 is a Q1-submodule of Lie(H∗),s2 is a Q2-submodule of Lie(H∗), and s1 and s2 are direct summands of Lie(H∗). We saythat the pairs (Q1, s1) and (Q2, s2) are quasi-opposite, if the following two conditions hold:(i) there exists an element g ∈ H(∗) such that gQ1g

−1 and Q2 are opposite, and (ii) for an(any) element g ∈ H(∗) such that gQ1g

−1 and Q2 are opposite, s1 and s2 are opposite withrespect to a (any) maximal torus of gQ1g

−1 ∩ Q2. Obviously this definition is symmetricin the pairs (Q1, s1) and (Q2, s2). If Q1 and Q2 are opposite, we also say that the pairs(Q1, s1) and (Q2, s2) are opposite.

(h) We assume that H is split. For i ∈ S(1, 4) let Qi be a parabolic subgroup of Hk

and let si be a Qi-submodule of Lie(Qi). We assume that the pairs (Q1, s1) and (Q2, s2)are quasi-opposite and that the pairs (Q3, s3) and (Q4, s4) are as well quasi-opposite. Wecheck that if there exists an element h1 ∈ H(k) such that s1 = h1(s3), then there existsalso an element h2 ∈ H(k) such that s2 = h2(s4). To check this, we can assume that thereexists a split maximal torus Tk of Hk such that Q1 and Q2 are opposite with respect to Tkand Q3 and Q4 are as well opposite with respect to Tk. Thus s1 and s2 are opposite withrespect to Tk and s3 and s4 are as well opposite with respect to Tk, cf. the very definitionsof (g). We write h1 = q1wq3, where q1 ∈ Q1(k), q3 ∈ Q3(k), and w ∈ H(k) normalizes

37

Tk (cf. Bruhat decompositions; see [Bo, Ch. V, §21]). As s1 = h1(s3) = q1w(s3), we getthat s1 = q−1

1 (s1) = w(s3) and thus (due to the last two oppositions) we have s2 = w(s4).Therefore we can take h2 = w.

We have a natural variant of the previous paragraph, in which we have s1 ⊆ h1(s3)and s2 ⊆ h2(s4) (instead of s1 = h1(s3) and s2 = h2(s4)).

2.3.9. Lemma. Let V be a discrete valuation ring. Let K be the field of fractions of V .Let l : G1 → G1 be a homomorphism of reductive group schemes over V whose fibre overK is an isomorphism. Then l is an isomorphism.

Proof: Let W be a free V -module of finite rank such that G1 is a closed subgroup schemeof GLW , cf. [DG, Vol. I, Exp. V IB , Rm. 11.11.1]. For a maximal torus T1 of G1, we checkthat the restriction l|T1 of l to T1 is a closed embedding. For this, we can assume thatT1 is split. Thus the T1-module W is a direct sum of T1-submodules which as V -modulesare free of rank 1 (cf. [Ja, Part I, 2.11]) and therefore Ker(l|T1) is a group scheme ofmultiplicative type. But Ker(l|T1)K is trivial and thus the flat group scheme Ker(l|T1)is itself trivial. This implies that the composite homomorphism T1 → G1 → GLW is aclosed embedding. Thus l|T1 is a closed embedding. This implies that the kernel of thespecial fibre of l is a finite group (to be compared with the proof of Lemma 2.3.5). Thusl is a quasi-finite morphism between normal, integral schemes that have the same field offractions. From Zariski main theorem (see [Gr1, Thm. (8.12.6)]) we get that l is an openembedding. As the fibres of G1 are connected, the open embedding homomorphism l isonto and therefore it is an isomorphism.

2.3.10. Lemma. Let V be a complete discrete valuation ring which is of mixed character-istic (0,p) and whose residue field is perfect and infinite. Let K := V [ 1

p]. Let l : G2 → G1

be a homomorphism between reductive group schemes over V such that Ker(l) is a flatgroup scheme over V and G1 is the quotient G2/Ker(l) of G2 through Ker(l). Let H1 be ahyperspecial subgroup of G1(K) that contains Im(G2(V ) → G(K)). Then H1 = G1(V ).

Proof: Let G1 be the unique reductive group scheme over V that has G1,K as its genericfibre and has H1 as its group of V -valued points, cf. [Ti2, Subsubsect. 3.8.1]. Theepimorphism lK : G2,K ։ G1,K extends to a homomorphism G2 → G1, cf. [Va3, Prop.

3.1.2.1 a)]. The Zariski closure Ker(l) of Ker(lK) in G2 is included in Ker(G2 → G1). Thuswe get naturally a homomorphism l1 : G1 → G1 whose generic fibre l1,K is an isomorphism.Thus l1 is an isomorphism, cf. Lemma 2.3.9. Therefore we can identify canonically G1

with G1 and thus we have H1 = G1(V ).

2.3.11. Lemma. Let G be a split, simply connected group scheme over Z whose adjointis absolutely simple and of classical Lie type X . Let T be a maximal split torus of G. Letρ : G → GLZ be a representation associated to a minuscule weight of the root systemof the action of T on Lie(G) (thus Z is a free Z-module of finite rank, see [Hu, Ch. VII,Subsect. 27.1]). Then the special fibre ρ,Fp

of ρ is absolutely irreducible. Thus, if V is

a discrete valuation ring of mixed characteristic (0, p) and of residue field k and if ZV isa V -lattice of Z ⊗Z V [ 1p ] that is a GV -module, then the special fibre of the representationGV → GLZV

is isomorphic to ρ,k and therefore it is also absolutely irreducible.

38

Proof: Let W be the Weyl group of G with respect to T . Let W be the subgroup of Wthat fixes . We have rkZ(Z) = [W : W], cf. [Bou3, Ch. VIII, §7.3, Prop. 6] appliedto ρ,Q. The absolutely irreducible representation of GF associated to the weight hasdimension at least equal to [W : W] = dimF(Z ⊗Z F). As this last absolutely irreduciblerepresentation is isomorphic to the representation of GF on a factor of the compositionseries of the GF-module Z ⊗Z F (cf. [Ja, Part I, 10.9]), by reasons of dimensions we getthat the representation ρ,F is absolutely irreducible. Thus ρ,Fp

is also an absolutelyirreducible representation. By applying loc. cit. once more, we get the last part of theLemma.

2.3.12. Fact. Let T be a torus over W (k). Let T0,B(k) be a subtorus of TB(k). Then theZariski closure T0 of T0,B(k) in T is a subtorus of T .

Proof: To prove this we can assume that k = k. As W (k) is strictly henselian, T is a splittorus. Thus T0,B(k) is also a split torus. Let T1,B(k) be a (split) subtorus of TB(k) such thatwe can identify TB(k) with T0,B(k) ×B(k) T1,B(k). As a split torus over W (k) is uniquelydetermined by its fibre over B(k), we can also identify T with T0 ×W (k) T1, where T1 isthe Zariski closure of T1,B(k) in T . From this the Fact follows.

2.3.13. A review on centralizers. Let H be a reductive group scheme over either afield or over W (k). Then the centralizer of each torus of H in H is a reductive, closedsubgroup scheme of H, cf. [DG, Vol. III, Exp. XIX, Sect. 2.8]. If LH is a Levi subgroupscheme of a parabolic subgroup scheme of H, then LH is the centralizer of Z0(LH) in H(cf. [DG, Vol. III, Exp. XXVI, Cor. 1.13]).

2.4. Natural Zp structures and applications

Let (M,ϕ,G) and µ : Gm → G be as in Section 2.1. Let

σϕ := ϕµ(p).

It is a σ-linear automorphism ofM , cf. properties 1.3.2 (a) and (b). We have σϕ(g) ⊆ g[ 1p ],

cf. properties 1.3.2 (b) and (c). Thus, as σϕ normalizes EndW (k)(M), we have σϕ(g) = g.Let

MZp:= x ∈M |σϕ(x) = x.

If k = k, then M =MZp⊗Zp

W (k) (i.e., MZpis a Zp structure of M).

2.4.1. Lemma. We assume that k = k. Let H be a smooth (resp. reductive), closedsubgroup scheme of GLM whose Lie algebra h is normalized by σϕ and whose generic fibreHB(k) is connected. Then H is the extension toW (k) of a unique smooth (resp. reductive),closed subgroup scheme HZp

of GLMZp.

Proof: As k = k, hZp:= x ∈ h|σϕ(x) = x is a direct summand of EndZp

(MZp) and a

Zp structure of h. Thus HB(k) is the extension to B(k) of the unique connected sub-

group HQpof GLMZp [

1p ]

whose Lie algebra is hZp[ 1p], cf. Lemma 2.3.7 applied with

(L,K,H, f) = (Qp, B(k),GLM [ 1p ], hZp

[ 1p ]). As H is the Zariski closure of HB(k) in GLM ,

39

it is the extension to W (k) of the Zariski closure HZpof HQp

in GLMZp. Thus HZp

is asmooth (resp. reductive), closed subgroup scheme of GLMZp

. Obviously HZpis uniquely

determined.

2.4.2. Example. We assume that k = k. By applying Lemma 2.4.1 to G, we get that Gis the pull back to W (k) of a unique reductive, closed subgroup scheme GZp

of GLMZp.

2.4.3. On 1.3.4. We refer to Section 1.3.4. We check that Ψ(Lie(Gad)) ⊆ Lie(Gad). Wefirst check that σϕ normalizes Lie(Gad) under its action on EndB(k)(M [ 1p ]). For this we

can assume k = k. Let GZpbe as in Example 2.4.2. As σϕ fixes Lie(Gad

Zp) ⊆ Lie(GQp

) and

as Lie(Gad) = Lie(GadZp

)⊗ZpW (k), we get that σϕ normalizes Lie(Gad).

We now also refer to (2). Let i ∈ I. Let xi ∈ Lie(Gi). Let ni ∈ 0, 1 be such thatwe have ni = 0 if and only if the image of µ : Gm → G in Gi is trivial. If ni = 0, thenµ(p)(pnixi) = xi ∈ Lie(Gad). If ni = 1, then µ(p)(pnixi) ∈ Lie(Gad) (cf. property 1.3.2(b)). Thus Ψ(xi) = ϕ(pnixi) = σϕ(µ(p)(p

nixi)) ∈ Lie(Gad). As i ∈ I is arbitrary, we getthat Ψ(Lie(Gad)) ⊆ Lie(Gad). Thus the FSHW shift Ψ of (M,ϕ,G) is well defined.

2.4.4. Extensions of FSHW maps. We refer to (2). Let i ∈ I. The simple, adjointgroup Gi,k over k is of the form Reski/kHi,ki

, where ki is a finite (automatically separable)field extension of k and where Hi,ki

is an absolutely simple, adjoint group over ki (cf. [Ti1,Subsect. 3.1.2]). The images of µk in the simple factors of Gi,k =

∏j∈Homk(ki,k)

Hi,ki×ki jk

are permuted transitively by the Galois group Gal(k). This implies that if µk has a trivial(resp. a non-trivial) image in Gi,k, then for every perfect field k that contains k, µk

has a trivial (resp. a non-trivial) image in each simple factor of Gi,k. Thus, under the

identification Lie(Gadk) = Lie(Gad

k )⊗k k, the FSHW map of (M,ϕ,G)⊗ k is Ψ⊗ σk.

2.4.5. The Lie context. As σϕ normalizes g and as ϕ = σϕµ(1p ), we have ϕ−1(g) =

1pF 1(g)⊕ F 0(g)⊕ pF−1(g). Thus the quadruple (g, (F i(g))i∈−1,0,1, ϕ, G

ad) is a Shimura

filtered p-divisible object (a Hodge cocharacter of it is the composite of µ : Gm → G withthe natural epimorphism G ։ Gad). We also refer to the pair (F 0(g), F 1(g)) as a lift of(g, ϕ, Gad). A similar argument shows that the quadruple (Lie(Gad), (F i(Lie(Gad)))i∈−1,0,1,

ϕ, Gad) is as well a Shimura filtered p-divisible object. We also refer to the pair(F 0(Lie(Gad)), F 1(Lie(Gad))) as a lift of (Lie(Gad), ϕ, Gad).

2.4.6. Lemma. We have σϕG(W (k))σ−1ϕ = G(W (k)) and ϕG(B(k))ϕ−1 = G(B(k)).

Proof: We have G(W (k)) = G(W (k)) ∩ GLM (W (k)) and G(B(k)) = G(B(k)) ∩GLM (B(k)). Thus to prove the two identities we can assume that k = k. The iden-tity σϕG(W (k))σ−1

ϕ = G(W (k)) follows from the existence of the Zp structure GZpof G.

As ϕG(B(k))ϕ−1 is the σϕ-conjugate of µ( 1p)G(B(k))µ(p) = G(B(k)), it is G(B(k)).

2.4.7. Actions of σϕ and ϕ on subgroup schemes. Let S be the ordered set ofsmooth, closed subgroup schemes of GLM that have connected fibres over B(k). LetSB(k) be the ordered set of connected subgroups of GLM [ 1p ]

. As σϕ (resp. ϕ) is a σ-linear

automorphism of M (resp. of M [ 1p]), it defines naturally an isomorphism σϕ : S→S (resp.

ϕ : SB(k)→SB(k)). Explicitly: if H ∈ S (resp. if H ∈ SB(k)), then σϕ(H) (resp. ϕ(H)) is

40

the smooth, closed subgroup scheme of GLM whose fibre over B(k) is connected, (resp. isthe connected subgroup of GLM [ 1p ]

) whose group of W (k)-valued (resp. of B(k)-valued)

points is σϕH(W (k))σ−1ϕ , (resp. is ϕH(B(k))ϕ−1) and whose Lie algebra is σϕ(Lie(H))

(resp. is ϕ(Lie(H))). Thus we have σϕ(G) = G (resp. ϕ(GB(k)) = GB(k)), either cf.

Lemma 2.4.6 or cf. Lemma 2.3.6 and the fact that σϕ (resp. ϕ) normalizes g (resp. g[ 1p ]).

2.4.8. On the variation of σϕ. Let g ∈ G(W (k)). Let µ1 be another Hodge cocharacterof (M,ϕ,G). Let ϕ1 := gϕ. Let σϕ1

:= ϕ1µ1(p). Let g1 := σϕ1σ−1ϕ ; it is a W (k)-linear

automorphism of M and thus g1 ∈ GLM (W (k)). But g1 = gϕµ1(p)µ(1p )ϕ

−1 ∈ G(B(k)),

cf. Lemma 2.4.6. Thus g1 ∈ G(W (k)). As σϕ1= g1σϕ, we get that σϕ1

differs from σϕonly via a composite of it with the element g1 ∈ G(W (k)). From this and Lemma 2.4.6,by induction on m ∈ N we get that we can write σm

ϕ1= gm(σϕ)

m, where gm ∈ G(W (k)).

2.5. Lie F -isocrystals and parabolic subgroup schemes

By a Lie F -isocrystal over k we mean a pair (h, φ), where h is a Lie algebra over B(k)and where φ is a σ-linear Lie automorphism of h.

2.5.1. Lemma. Let (h, φ) be a Lie F -isocrystal over k such that h is the Lie algebra ofan adjoint group HB(k) over B(k). Then W 0(h, φ) is the Lie algebra of a unique parabolic

subgroup P+B(k) of HB(k).

Proof: As each parabolic subgroup of HB(k) is connected, the uniqueness part follows fromLemma 2.3.6. For each γ ∈ Q let cγ := W (γ)(h, φ). Let H be the set of Newton polygonslopes of (h, φ). Let n ∈ N be such that H ⊆ 1

nZ. Let ν : Gm → GLh be the Newtoncocharacter such that Gm acts on cγ as the nγ-th power of the identity character of Gm.As φ is a σ-linear Lie automorphism of h, we have

(3a) [cγ1, cγ2

] ⊆ cγ1+γ2, ∀γ1, γ2 ∈ H.

This implies that ν factors through the identity component Aut0(h) of the subgroup Aut(h)of GLh formed by Lie automorphisms of h. We view HB(k) as a subgroup of Aut0(h)via the adjoint representation. But h = Lie(HB(k)) is a semisimple Lie algebra overB(k). Thus every derivation of h is inner (cf. [Bou1, Ch. I, §6.1, Cor. 3]) i.e., we haveh = Lie(Aut0(h)). Therefore we have HB(k) = Aut0(h), cf. Lemma 2.3.6. Thus we canview ν as a cocharacter ν : Gm → HB(k). The centralizer of Im(ν) in HB(k) is a reductivegroup C0B(k) (cf. Subsection 2.3.13) whose Lie algebra is c0. Let TB(k) be a maximal torusof C0B(k); we have Im(ν) 6 TB(k). Let K be the smallest Galois extension of B(k) overwhich TB(k) splits.

Let Φ be the set of roots of h⊗B(k)K relative to TK . Let h⊗B(k)K = Lie(TK)⊕

α∈Φ hαbe the root decomposition relative to TK . Let Φ≥0 (resp. Φγ) be the subset of Φ of roots ofW 0(h, φ) (resp. of cγ) relative to TK . From (3a) we get that (Φ≥0 +Φ≥0)∩Φ ⊆ Φ≥0. Wehave an additive function J : Φ → Q that maps α ∈ Φ to the unique Newton polygon slopeγ ∈ H ⊆ Q for which we have an inclusion hα ⊆ [W γ(h, φ)⊗B(k)K \W

12n+γ(h, φ)⊗B(k)K]

(equivalently, we have an inclusion hα ⊆ cγ ⊗B(k)K). As ν factors through TB(k), we have

(3b) J(−α) = −J(α), ∀α ∈ Φ.

41

This implies that we have a disjoint union Φ = −Φ≥0 ∪ Φ≥0. From this and the inclusion(Φ≥0 + Φ≥0) ∩ Φ ⊆ Φ≥0, we get that Φ≥0 is a parabolic subset of Φ as defined andcharacterized in [Bou2, Ch. VI, §1.7]. Thus there exists a unique parabolic subgroupof HK that contains TK and whose Lie algebra is W 0(h, φ) ⊗B(k) K, cf. [DG, Vol. III,

Exp. XXVI, Prop. 1.4]. Thus P+B(k) exists, cf. Lemma 2.3.7 applied with (L,K,H, f) =

(B(k), K,HB(k),W0(h, φ)).

2.5.2. Remark. We have a similar function J : Φ → Q for every root decomposition ofh⊗B(k) K relative to a split maximal torus of P+

K (that is not necessarily TK).

2.5.3. The group scheme P . We recall that P is the normalizer of (F i(M))i∈S(a,b) in

G, cf. Subsection 1.3.3. Thus Lie(PB(k)) = F 0(g)[ 1p ] = F 0(g)[ 1p ]⊕ F 1(g)[ 1p ] and Lie(Pk) =

F 0(g) ⊗W (k) k. Moreover Z(G) 6 P . As in the proof of Lemma 2.5.1, by replacing the

role of ν with the one of the cocharacter Gm → GadB(k) defined naturally by µ : Gm →

G, we get that there exists a unique parabolic subgroup of GadB(k) whose Lie algebra is

F 0(Lie(Gad))[ 1p ] = F 0(g)[ 1p ]∩Lie(GderB(k)). This implies that there exists a unique parabolic

P ′B(k) subgroup of GB(k) whose Lie algebra is Lie(PB(k)) = F 0(g)[ 1

p]. Thus P ′

B(k) is

the identity component of PB(k) (cf. Lemma 2.3.6) and in fact we have P ′B(k) = PB(k)

(cf. Fact 2.3.2). The Zariski closure P ′ of P ′B(k) = PB(k) in G is a parabolic subgroup

scheme of G (cf. Subsection 2.3.8 (a)) as well as a subgroup scheme of P . We haveLie(P ′) = Lie(P ′

B(k)) ∩ End(M) = F 0(g). Thus Lie(P ′k) = Lie(Pk) and therefore (cf. Fact

2.3.2) we have P ′k = Pk. We conclude that P ′ = P i.e., P is the unique parabolic subgroup

scheme of G whose Lie algebra is F 0(g).

2.5.4. The unipotent radical of P . The centralizer C of Im(µ) in G is a reductive,closed subgroup scheme of G, cf. Subsection 2.3.13. It is contained in P and has F 0(g) asits Lie algebra. This implies that the nilpotent ideal F 1(g)[ 1p ] = F 1(g)[ 1p ] of Lie(PB(k)) =

F 0(g)[ 1p ]⊕ F 1(g)[ 1p ] is the nilpotent radical of Lie(PB(k)). Let U be the unipotent radical

of P . We have Lie(UB(k)) = F 1(g)[ 1p]. Thus Lie(U) = F 1(g) = g ∩ F 1(g)[ 1

p].

Let W (Lie(U)) be the vector group scheme over W (k) defined by Lie(U); thus for acommutative W (k)-algebra C we have W (Lie(U))(C) = Lie(U)⊗W (k) C. As Gm acts via

µ on [F 1(g), F 1(g)] through the second power of the inverse of the identity character ofGm, from the property 1.3.2 (b) we get that [F 1(g), F 1(g)] = 0. Thus Lie(U) = F 1(g) iscommutative. We check that the exponential W (k)-homomorphism

exp : W (Lie(U)) → U

which at the level of W (k)-valued points takes x ∈ Lie(U) to exp(x) := 1M +∑∞

i=1xi

i! ∈U(W (k)), is well defined as well as an isomorphism. This will imply that U is isomorphic

to Gdim(Uk)a (over W (k)).

To check the statement on exp, we can assume that k = k. Let ΦU be the set of rootsof Lie(U) relative to T . It is well known that expB(k) is an isomorphism. This implies thatU is commutative. Thus U =

∏α∈ΦU

Uα is a product of Ga subgroup schemes Uα of G

42

that are normalized by T , cf. [DG, Vol. III, Exp. XXVI, Prop. 1.12 (i)]. For each suchUα, we have Uα(W (k)) = exp(Vα), where Vα is a W (k)-submodule of Lie(Uα)[

1p]. But it is

easy to see that Vα is Lie(Uα) itself. From this and the mentioned product decompositionof U , we get that exp :W (Lie(U)) → U is an isomorphism.

2.5.5. The group schemes Q and N . For i ∈ S(a, b) let Fi(M) := ⊕ij=aF

j(M). LetQ be the parabolic subgroup scheme of G that is the opposite of P with respect to everymaximal torus of C. We have Lie(Q) = F 0(g)⊕ F−1(g). Moreover Q is the normalizer ofthe ascending, exhaustive filtration (Fi(M))i∈S(a,b) of M . Let N be the unipotent radical

of Q. We have Lie(N) = F−1(g) and N is isomorphic to Gdim(Uk) (over W (k)). All theseare proved as in Subsections 2.5.3 and 2.5.4 (with the role of µ being played by its inverse).

2.6. Ten group schemes

The goal of this section is to introduce ten types of flat, closed subgroup schemes of Gthat will be often used in this monograph. The ten types of group schemes are intrinsicallyassociated to the Shimura p-divisible object (M,ϕ,G) and they are:

P+G (ϕ), U+

G (ϕ), P+G (ϕ), U+

G (ϕ), L0G(ϕ), L

0G(ϕ), P

−G (ϕ), U−

G (ϕ), P−G (ϕ), and U−

G (ϕ).

2.6.1. Non-negative parabolic subgroup schemes. We recall that ϕ = σϕµ(1p).

Thus ϕ(Lie(Z0(G))) = Lie(Z0(G)). Therefore Lie(Z0(G)) ⊆W (0)(g, ϕ) ⊆W 0(g, ϕ). Thisimplies that W 0(g[ 1p ], ϕ) = Lie(Z0(G))[ 1p ] ⊕W 0(Lie(Gder)[ 1p ], ϕ). From this and Lemma

2.5.1 applied to (Lie(Gder)[ 1p], ϕ), we get the existence of a unique parabolic subgroup

P+G (ϕ)B(k) of GB(k) such that we have Lie(P+

G (ϕ)B(k)) =W 0(g[ 1p ], ϕ). The Zariski closure

P+G (ϕ) of P+

G (ϕ)B(k) in G is a parabolic subgroup scheme of G (cf. Subsection 2.3.8 (a))whose Lie algebra is W 0(g, ϕ).

Let U+G (ϕ) be the unipotent radical of P+

G (ϕ). Based on (3a) and the inclusionLie(Z0(G)) ⊆ W (0)(g, ϕ), we get that for every rational number γ we have an inclusion[W 0(g[ 1p ], ϕ),W

γ(g[ 1p ], ϕ)] ⊆ W γ(g[ 1p ], ϕ). Thus the identity component of the normalizer

ofW γ(g[ 1p], ϕ) in GB(k) has a Lie algebra that contains Lie(P+

G (ϕ)B(k)) = W 0(g[ 1p], ϕ) and

therefore (cf. Lemma 2.3.6) it contains P+G (ϕ)B(k). This implies that P+

G (ϕ) normalizesW γ(g, ϕ) for all rational numbers γ.

2.6.2. Levi subgroups. Let L0Gad(ϕ)B(k) be the unique reductive subgroup of P+

Gad(ϕ)B(k)

that has the same rank as P+Gad(ϕ)B(k) and we have Lie(L0

Gad(ϕ)B(k)) =W (0)([g, g][ 1p ], ϕ) =

W (0)(Lie(Gad)[ 1p], ϕ) (see Lemma 2.3.6 for the uniqueness part; see the reductive group

C0B(k) of the proof of Lemma 2.5.1 for the existence part). The inverse image of

L0Gad(ϕ)B(k) in P+

G (ϕ)B(k) is a subgroup L0G(ϕ)B(k) of P+

G (ϕ)B(k) whose Lie algebra is

W (0)(g[ 1p ], ϕ) and which contains a maximal torus of GB(k). Thus Z(GB(k)) 6 L0G(ϕ)B(k)

and L0G(ϕ)B(k) is generated by its identity component and by such a maximal torus of

GB(k). This implies that L0G(ϕ)B(k) is connected.

Let γ+ ∈ Q ∩ (0,∞) be such that W 0(g[ 1p ], ϕ) = Lie(L0G(ϕ)B(k)) ⊕ W γ+(g[ 1p ], ϕ).

As W γ+(g[ 1p ], ϕ) is a nilpotent ideal of W 0(g[ 1p ], ϕ), Lie(L0G(ϕ)B(k)) is a Levi subalgebra

43

of W 0(g[ 1p ], ϕ) = Lie(P+G (ϕ)B(k)). This implies that L0

G(ϕ)B(k) is a Levi subgroup of

P+G (ϕ)B(k) and thatW γ+(g[ 1

p], ϕ) is the nilpotent radical of Lie(P+

G (ϕ)B(k)) =W 0(g[ 1p], ϕ).

This last thing implies that

Lie(U+G (ϕ)) =W γ+(g, ϕ) = ⊕γ∈Q∩(0,∞)W (γ)(g, ϕ).

Let LB(k) be a Levi subgroup of P+G (ϕ)B(k) such that Lie(LB(k)) is normalized by ϕ. The

natural Lie isomorphism Lie(LB(k))→W 0(g[ 1p ], ϕ)/Wγ+(g[ 1p ], ϕ) is compatible with the σ-

linear Lie automorphisms induced by ϕ. Thus all Newton polygon slopes of (Lie(LB(k)), ϕ)are 0 and therefore we have an inclusion Lie(LB(k)) ⊆ Lie(L0

G(ϕ)B(k)). By reasons ofdimensions we get that Lie(LB(k)) = Lie(L0

G(ϕ)B(k)) and thus we have LB(k) = L0G(ϕ)B(k),

cf. Lemma 2.3.6.

Thus P+G (ϕ) and L0

G(ϕ)B(k) have the properties listed in Section 1.3.3.

Let L0G(ϕ) be the Zariski closure of L

0G(ϕ)B(k) in P

+G (ϕ). The following Lemma points

out that in general L0G(ϕ) is not a Levi subgroup scheme of P+

G (ϕ).

2.6.3. Lemma. The flat, closed subgroup scheme L0G(ϕ) of P+

G (ϕ) is a Levi subgroupscheme if and only if we have a direct sum decomposition M = ⊕γ∈QW (γ)(M,ϕ).

Proof: Let n ∈ N be such that all Newton polygon slopes of (M [ 1p ], ϕ) belong to 1nZ. Let

νB(k) : Gm → GLM [ 1p ]be the Newton cocharacter such that for every rational number γ

it acts on W (γ)(M [ 1p ], ϕ) via the nγ-th power of the identity character of Gm. We claim

that νB(k) factors through GB(k). To check this, we can assume that k = k. Due to the

existence of the Qp structure (MZp[ 1p], GQp

) of (M [ 1p], GB(k)) (see Section 2.4), the claim

follows from [Ko1, §3].

But L0G(ϕ)B(k) is the centralizer of Im(νB(k)) in GB(k), cf. proof of Lemma 2.5.1 and

the construction of L0G(ϕ)B(k). Thus, if L

0G(ϕ) is a Levi subgroup scheme of P+

G (ϕ), thenthe natural factorization νB(k) : Gm → Z0(L0

G(ϕ)B(k)) of νB(k) extends to a cochar-acter ν : Gm → Z0(L0

G(ϕ)); this implies that we have a direct sum decompositionM = ⊕γ∈QW (γ)(M,ϕ).

If we have a direct sum decomposition M = ⊕γ∈QW (γ)(M,ϕ), then the naturalfactorization νB(k) : Gm → GB(k) of νB(k) extends to a cocharacter ν : Gm → G andthus the centralizer of Im(ν) in G is a reductive, closed subgroup scheme C0 of G (cf.Subsection 2.3.13) whose generic fibre is L0

G(ϕ)B(k). This implies that C0 = L0G(ϕ). The

natural homomorphism C0 = L0G(ϕ) → P+

G (ϕ)/U+G (ϕ) is an isomorphism, cf. Lemma

2.3.9. Thus L0G(ϕ) is a Levi subgroup scheme of P+

G (ϕ).

2.6.4. Example. If L0G(ϕ) is a Levi subgroup scheme of P+

G (ϕ), then from Lemma 2.6.3we get that L0

GLM(ϕ) =

∏γ∈Q GLW (γ)(M,ϕ). Moreover

Lie(P+GLM

(ϕ)) = ⊕γ1,γ2∈Q,γ2≥γ1HomW (k)(W (γ1)(M,ϕ),W (γ2)(M,ϕ))

and Lie(P−GLM

(ϕ)) = ⊕γ1,γ2∈Q,γ2≥γ1HomW (k)(W (γ2)(M,ϕ),W (γ1)(M,ϕ)).

44

2.6.5. Tilde non-negative parabolic subgroup schemes. For i ∈ I let δi ∈ [0,∞)∩Qbe the greatest number such that W (δi)(Lie(Gi)[

1p], ϕ) 6= 0. Let P+

G (ϕ)B(k) be the nor-

malizer of the direct sum ⊕i∈IW (δi)(Lie(Gi)[1p], ϕ) in GB(k). We have δi = δΓ(i) and

ϕ takes W (δi)(Lie(Gi)[1p], ϕ) isomorphically onto W (δΓ(i))(Lie(GΓ(i))[

1p], ϕ); thus ϕ nor-

malizes ⊕i∈IW (δi)(Lie(Gi)[1p ], ϕ). The Lie algebra Lie(P+

G (ϕ)B(k)) is the normalizer of

⊕i∈IW (δi)(Lie(Gi)[1p ], ϕ) in g[ 1p ] and thus it is normalized by ϕ. As Lie(P+

G (ϕ)B(k)) nor-

malizes ⊕i∈IW (δi)(Lie(Gi)[1p ], ϕ) = ⊕i∈IW

δi(g[ 1p ], ϕ) ∩ Lie(Gi)[1p ] (cf. end of Subsection

2.6.1), we have Lie(P+G (ϕ)B(k)) ⊆ Lie(P+

G (ϕ)B(k)). Thus P+G (ϕ)B(k) is a subgroup of

(the identity component of) P+G (ϕ)B(k) (cf. Lemma 2.3.6) and therefore P+

G (ϕ)B(k) is a

parabolic subgroup of GB(k). Thus the Zariski closure P+G (ϕ) of P+

G (ϕ)B(k) in G is a

parabolic subgroup scheme of G (cf. Subsection 2.3.8 (a)) that contains P+G (ϕ). We refer

to P+G (ϕ) as the tilde non-negative parabolic subgroup scheme of (M,ϕ,G). Let U+

G (ϕ)

be the unipotent radical of P+G (ϕ).

Let L0G(ϕ)B(k) be the unique Levi subgroup of P+

G (ϕ)B(k) that contains (an arbitrarymaximal torus of) L0

G(ϕ)B(k) (cf. Subsection 2.3.8 (c)); we call it the tilde Levi subgroup

of (M,ϕ,G). Let L0G(ϕ) be the Zariski closure of L0

G(ϕ)B(k) in P+G (ϕ). We have L0

G(ϕ) 6

L0G(ϕ). We emphasize that (cf. Lemma 2.6.3), in general L0

G(ϕ) is not a Levi subgroupscheme of P+

G (ϕ).

2.6.6. Non-positive parabolic subgroup schemes. Similar arguments show thatthere is a unique parabolic subgroup scheme P−

G (ϕ) (resp. P−G (ϕ)) of G such that

we have Lie(P−G (ϕ)) = W0(g, ϕ) (resp. such that P−

G (ϕ)B(k) is the normalizer of

⊕i∈IW (−δi)(Lie(Gi)[1p ], ϕ) in GB(k)). We refer to P−

G (ϕ) (resp. to P−G (ϕ)) as the non-

positive (resp. as the tilde non-positive) parabolic subgroup scheme of (M,ϕ,G). The groupL0G(ϕ)B(k) is also a Levi subgroup of P−

G (ϕ)B(k). Thus P−G (ϕ)B(k) and P

+G (ϕ)B(k) are op-

posite parabolic subgroups of GB(k). Let U−G (ϕ) and U−

G (ϕ) be the unipotent radicals of

P−G (ϕ) and P−

G (ϕ) (respectively). As in the end of Subsection 2.6.1 we argue that for everyrational number γ, P−

G (ϕ) normalizes Wγ(g, ϕ).

2.6.7. Extensions. Let k1 be a perfect field that contains k. In what follows we willuse without any extra comment the fact that the constructions of our ten types of groupschemes are compatible with extensions to k1. In other words, we have the followingformulas P+

G (ϕ)W (k1) = P+GW (k1)

(ϕ⊗ σk1), L0

G(ϕ)W (k1) = L0GW (k1)

(ϕ⊗ σk1), etc.

2.7. The opposition principle

In this Section we assume that G is quasi-split. Let g ∈ G(W (k)) be such that theintersection g(P+

G (ϕ))g−1∩P−G (ϕ) contains a maximal torus T of a Borel subgroup scheme

of G contained in P−G (ϕ), cf. Subsection 2.3.8 (e). Let h ∈ G(W (k)) be such that it

normalizes T and hgP+G (ϕ)g−1h−1 is the opposite of P−

G (ϕ) with respect to T (cf. [Bo,Ch. V, Prop. 21.12] for the existence of h modulo p).

2.7.1. Basic quasi-oppositions. We check that the last opposition at the level of Liealgebras respects the Newton polygon slope filtrations i.e., that for all rational numbers

45

γ the direct summand hg(W γ(g, ϕ)) of g is the opposite of W−γ(g, ϕ) with respect to T .

Let h ∈ P−G (ϕ)(B(k)) be such that hTB(k)h

−1 is a maximal torus TB(k) of L0G(ϕ)B(k).

As hhgP+G (ϕ)B(k)g

−1h−1h−1 and P+G (ϕ)B(k) are both parabolic subgroups of GB(k) that

are opposite to P−G (ϕ)B(k) with respect to TB(k), we have hhgP+

G (ϕ)B(k)g−1h−1h−1 =

P+G (ϕ)B(k). Thus hhg ∈ P+

G (ϕ)(B(k)) and therefore hhg normalizes W γ(g, ϕ)[ 1p ], cf. endof Subsection 2.6.1.

To check that hg(W γ(g, ϕ))g−1h−1 is the opposite of W−γ(g, ϕ) with respect to T ,we can assume that k = k and that (cf. Subsection 2.4.5) G is adjoint and it suf-fices to show that W γ(g, ϕ)[ 1p ] and W−γ(g, ϕ)[

1p ] are opposite with respect to the split

torus TB(k). But this last thing is implied by (3b) applied with (HB(k), K, φ, TB(k)) =

(GB(k), B(k), ϕ, TB(k)).

Thus for each rational number γ the pairs (P+G (ϕ),W γ(g, ϕ)) and (P−

G (ϕ),W−γ(g, ϕ))are quasi-opposite in the sense of Subsection 2.3.8 (g). Therefore we have an identityrkW (k)(W

γ(g, ϕ)) = rkW (k)(W−γ(g, ϕ)). This implies that for every rational number γ wealso have rkW (k)(W (γ)(g, ϕ)) = rkW (k)(W (−γ)(g, ϕ)).

2.7.2. Corollary. We assume that G is adjoint and quasi-split and that the Newtonpolygon slope δi introduced in Subsection 2.6.5 does not depend on i ∈ I; let δ := δi. Thenthe pairs (P+

G (ϕ),W (δ)(g, ϕ)) and (P−G (ϕ),W (−δ)(g, ϕ)) are quasi-opposite. Therefore

for each element i ∈ I, the pairs (P+G (ϕ)k ∩ Gi,k, (W (δ)(g, ϕ) ∩ Lie(Gi)) ⊗W (k) k) and

(P−G (ϕ)k ∩Gi,k, (W (−δ)(g, ϕ) ∩ Lie(Gi))⊗W (k) k) are also quasi-opposite.

Proof: We have W (δ)(g, ϕ) =W δ(g, ϕ) and W (−δ)(g, ϕ) =W−δ(g, ϕ). Thus the fact thatthe pairs (P+

G (ϕ),W (δ)(g, ϕ)) and (P−G (ϕ),W (−δ)(g, ϕ)) are quasi-opposite follows from

the last paragraph of Subsection 2.7.1 applied with γ = δ.

2.7.3. The opposition of tilde parabolic subgroup schemes. From (3b) we get that⊕i∈IW (δi)(Lie(Gi)[

1p], ϕ) and ⊕i∈IW (−δi)(Lie(Gi)[

1p], ϕ) are opposite with respect to the

maximal torus TB(k) of L0G(ϕ)B(k). This implies that the parabolic subgroups P+

G (ϕ)B(k)

and P−G (ϕ)B(k) of GB(k) are opposite with respect to TB(k); thus their intersection is

L0G(ϕ)B(k). Therefore L0

G(ϕ) is also the Zariski closure of L0G(ϕ)B(k) in P−

G (ϕ). The

opposite of hg(P+G (ϕ))g−1h−1 with respect to T and P−

G (ϕ) are two parabolic subgroupschemes of G that contain P−

G (ϕ). But their fibres over B(k) are G(B(k))-conjugate and

thus (cf. property 2.3.1 (b)) they coincide. Thus hg(P+G (ϕ))g−1h−1 and P−

G (ϕ) are alsoopposite with respect to T .

2.8. Elementary crystalline complements

In this section we include six crystalline complements of different natures.

2.8.1. Fact. (a) Let (M1, F11 , ϕ1) be a filtered Dieudonne module over k that is cyclic in

the sense of Definition 1.3.8 (c). Then there exists a unique p-divisible group over W (k)that is cyclic and whose filtered Dieudonne module is (M1, F

11 , ϕ1).

(b) Let E be a p-divisible group over W (k). If p = 2, then we assume that either Eor Et is connected. Then E is cyclic (resp. circular or circular indecomposable) if andonly if its filtered Dieudonne module is cyclic (resp. circular or circular indecomposable).

46

Proof: To prove (a) we can assume (M1, ϕ1) has a unique Newton polygon slope. Butin this case, (a) follows from properties 1.2.1 (b) and (c). Part (b) is also implied byproperties 1.2.1 (b) and (c).

2.8.2. Lemma. Let the triple (M, (F i(M))i∈S(a,b), ϕ) be as in Definition 1.3.8. If this

triple is cyclic, then there exists a maximal torus T big of GLM such that the quadruple

(M, (F i(M))i∈S(a,b), ϕ, Tbig) is a Shimura filtered p-divisible object.

Proof: We can assume (M, (F i(M))i∈S(a,b), ϕ) is circular. We will use the notations of

Definition 1.3.8 (a). Let T big be the maximal torus of GLM that normalizes W (k)ejfor all j ∈ S(1, dM). For each number j ∈ S(1, dM), let ej,j ∈ EndW (k)(M) be suchthat for all numbers j′ ∈ S(1, dM) we have ej,j(ej′) = δj,j′ej′ . As ϕ(ej) = pmjej+1, wehave ϕ(ej,j) = ej+1,j+1 for each number j ∈ S(1, dM), where edM+1,dM+1 := e1,1. Thus

Lie(T big) = ⊕dMj=1W (k)ej,j is normalized by ϕ. Let µbig : Gm → T big be the cocharacter

such that Gm acts via µbig on W (k)ej through the −mj ’s power of the identity character

of Gm. The triple (M, ϕ, T big) is a Shimura filtered p-divisible object that has µbig as itsunique Hodge cocharacter. Moreover the filtration of M defined by µbig is (F i(M))i∈S(a,b)

itself.

2.8.3. Lemma. Let the triple (M, (F i(M))i∈S(a,b), ϕ) be as in Definition 1.3.8. To ease

notations we assume a ≥ 0. If (M, (F i(M))i∈S(a,b), ϕ) is circular indecomposable, thenthere exists no non-trivial decomposition of it into circular filtered F -crystals over k.

Proof: Let B = e1, . . . , edM be a W (k)-basis for M as in Definition 1.3.8 (a); thus the

function fB : Z/dMZ → S(a, b) that takes [j] to mj is non-periodic. We show that theassumption that there exists a non-trivial decomposition

(M, (F i(M))i∈S(a,b), ϕ) = ⊕d∈D(Md, (Fi(M) ∩ Md)i∈S(a,b), ϕ)

into circular filtered F -crystals, leads to a contradiction. Here by non-trivial we meanthat the set D has at least two elements and that each Md (with d ∈ D) is a non-zeroW (k)-module.

Let M0 := ⊕dMj=1W (F

pdM)ej ; it is W (F

pdM)-submodule of M formed by all elements

fixed by p−∑

dM

j=1mj ϕdM . We have a direct sum decomposition M0 = ⊕d∈DM0,d, where

for d ∈ D we define M0,d := Md ∩ M0. Let l ∈ (dM !)N. For each x ∈ M0 \ pM0 wedefine a function fx : Z/lZ → Z via the inductive rule: for every number s ∈ S(1, l), the

element p−∑s

u=1fx([u])ϕs(x) belongs to M0 \ pM0. For two elements x, y ∈ M0 \ pM0 we

say that fx < fy if there exists a number s ∈ S(1, l) such that we have fx([u]) = fx([u])for all numbers u ∈ S(1, s− 1) and moreover fx([s]) < fy([s]). The non-periodicity of the

function fB implies that: for each x ∈ B, the period of the function fx is [dM ] ∈ Z/lZ.

Let x0 ∈ (B⋃

∪d∈DM0,d) \ pM0 be such that the function fx0is maximal i.e., such

that there exists no element y ∈ (B⋃∪d∈DM0,d) \ pM0 with the property that fx0

< fy.

If x0 ∈ M0,d for some d ∈ D, then let j0 ∈ S(1, dM) be such that in the writing of

47

x0 =∑dM

j=1 cj(x0)ej as a linear combination of elements of B, we have cj0 ∈W (k)\pW (k).It is easy to see that this implies that fx0

≤ fej0 and therefore (due to the maximalityproperty of fx0

) that fx0= fej0 .

Thus we can assume that x0 ∈ B. Let d0 ∈ D be such that the projection x1 of x0 ∈ Bon Md belongs to M0,d0

\ pM0,d0. It is easy to see that this implies that fx0

≤ fx1and

therefore (due to the maximality property of fx0) that fx0

= fx1. But the period [dx1

] offx1

is defined by a divisor dx1of rkW (k)(Md0

). Thus [dx1] is not the period [dM ] of fx0

.Contradiction.

2.8.4. Connections. Let C be a commutative W (k)-algebra that is p-adically complete.Let MC be a free C-module of finite rank. Let Ω∧

C/W (k) be the p-adic completion of the

C-module ΩC/W (k) of relative differentials of C with respect to W (k). By a connection onMC we mean a map ∇ : MC → MC ⊗C Ω∧

C/W (k) that satisfies the usual Leibniz rule: if

x ∈ MC and y ∈ C, then ∇(yx) = y∇(x) + x ⊗ dy. Let ∇Ctriv be the flat connection on

M ⊗W (k) C that annihilates x⊗ 1 for all x ∈M .

Let m ∈ N. Let R = W (k)[[x1, . . . , xm]] be the commutative W (k)-algebra of formalpower series with coefficients in W (k) in the independent variables x1, . . . , xm. We haveΩ∧

R/W (k)= ⊕m

i=1Rdxi. Let I := (x1, . . . , xm) ⊆ R. Let ΦR be a Frobenius lift of R that is

compatible with σ. Let dΦR be the differential of ΦR. Let M be a free R-module of finite

rank. Let ΦM : M → M be a ΦR-linear endomorphism. We say ΦM is horizontal with

respect to a connection ∇ on M , if we have an identity of maps

(4) ∇ ΦM = (ΦM ⊗ dΦR) ∇ : M → M ⊗R (⊕mi=1Rdxi).

2.8.5. Fact. We assume that we have ΦR(xi) = xpi for each i ∈ S(1, m) and that M [ 1p ]

is R[ 1p]-generated by Im(ΦM ). Then there exists at most one connection on M such that

ΦM is horizontal with respect to it.

Proof: Let ∇1 and ∇2 be two connections on M such that ΦM is horizontal with respect to

each one of them. For each i ∈ S(1, m) we have dΦR(xi) = pxp−1i ∈ Ip−1. From this and

from (4), we get by induction on l ∈ N that ∇1 −∇2 ∈ EndR(M)⊗R (⊕mi=1I

l(p−1)[ 1p ]dxi).

As R is complete with respect to the I-topology, we conclude that ∇1−∇2 is the 0 elementof EndR(M)⊗R (⊕m

i=1Rdxi). Thus ∇1 = ∇2.

2.8.6. PD hulls. Let R and I be as in Subsection 2.8.4; we have R/I =W (k). Let RPD

be the p-adic completion of the divided power hull of the maximal ideal (p, I) of R. Theformal scheme Spf(RPD) is defined by the ideal pRPD. If ΦR is a fixed Frobenius lift of

R, let ΦRPD be the Frobenius lift of RPD defined naturally by ΦR. We emphasize that

ΦRPD is not the pull back of ΦR via the W (k)-monomorphism R → RPD and thus thenotation ΦRPD does not respect the conventions of the paragraphs before Section 1.1. For

l ∈ N we denote by I [l] the l-th ideal of the divided power filtration of RPD defined byI. We recall that I [l] is the p-adic closure in RPD of the ideal of RPD that is generated

by elements of the form∏

j∈Ej

anjj

nj !, where nj’s are non-negative integers whose sum is at

48

least l, where Ej is a finite set, and where aj’s are elements in I. As (p, I)pm

is an ideal of

R included in pRPD, we have a natural morphism Spf(RPD) → Spf(R) of formal schemesover Spf(W (k)).

2.8.7. Ramified complements. Let V be a totally ramified, finite, discrete valuationring extension of W (k). Let eV := [V : W (k)]. Let x be an independent variable. LetReV be theW (k)-subalgebra of B(k)[[x]] formed by formal power series

∑∞n=0 anx

n, wherean ∈ B(k) and the sequence (bn)n∈N defined by bn := an[

neV

]! is formed by elements ofW (k) and converges to 0. The W (k)-algebra ReV is p-adically complete. Let Spf(ReV ) bethe formal scheme defined by the ideal pReV of ReV . Let IeV :=

∑∞n=0 anx

n ∈ ReV |a0 =0; it is an ideal of ReV such that we have ReV /IeV = W (k). Let ΦReV be the Frobeniuslift of ReV that is compatible with σ and takes x to xp. Let cV : Spec(V ) → Spec(ReV )and dV : Spf(V ) → Spf(ReV ) be the closed embedding and the formal closed embedding(respectively) defined naturally by a W (k)-epimorphism hV : ReV ։ V that takes x to afixed uniformizer πV of V . The kernel Ker(hV ) has a natural divided power structure.

Each element of IeV is nilpotent modulo pReV . Thus each W (k)-homomorphismhV : R → V lifts to a W (k)-homomorphism h : R → ReV . Let e ∈ N be such thate ≥ peV . Let [e] : ReV → ReV be the W (k)-endomorphism that takes x to xe. We have[e](IeV ) ⊆ pReV . Thus ([e] h)(xi) ∈ pReV for all i ∈ S(1, m). This implies the existenceof a W (k)-homomorphism h(e) : RPD → ReV such that we have a commutative diagram

R −→ RPD

yh

yh(e)

ReV[e]−→ ReV

of W (k)-homomorphisms, where R → RPD is the natural W (k)-monomorphism.

49

3 Abstract theory, I: proofs of Theorems 1.4.1 and 1.4.2

Let (M, (F i(M))i∈S(a,b), ϕ, G) be a Shimura filtered p-divisible object over k in therange [a, b]. We recall that we use the notations of Section 1.3 (see Section 2.1). Thusµ : Gm → G is a Hodge cocharacter of (M,ϕ,G) that defines (F i(M))i∈S(a,b), P is theparabolic subgroup scheme of G that is the normalizer of (F i(M))i∈S(a,b) in G, we denote

g = Lie(G), etc. For g ∈ G(W (k)), we will often use one of the ten group schemes P+G (gϕ),

U+G (gϕ), P+

G (gϕ), U+G (gϕ), L0

G(gϕ), L0G(gϕ), P

−G (gϕ), U−

G (gϕ), P−G (gϕ), and U−

G (gϕ) thatare obtained as in Section 2.6 but working with gϕ instead of ϕ.

In Section 3.1 we include some simple properties of Hodge cocharacters that are oftenused in the rest of the monograph. In particular, in Subsection 3.1.8 we prove the impli-cation 1.4.3 (a) ⇒ 1.4.3 (b). In Sections 3.2 to 3.8 we prove Theorems 1.4.1 and 1.4.2. SeeSection 3.9 for complements on Hasse–Witt invariants.

3.1. On Hodge cocharacters

For i ∈ S(a, b) we have (pi(ϕ−1(M))) ∩ M =∑i−a

j=0 pjF i−j(M), cf. property

1.3.2 (a). Using this, by decreasing induction on i ∈ S(a, b) we get that the filtra-tion (F i(M)/pF i(M))i∈S(a,b) of M/pM is uniquely determined by (M,ϕ,G). Thus theparabolic subgroup Pk of Gk does not depend on the choice of the Hodge cocharacterµ : Gm → G of (M,ϕ,G). Therefore the group

L := g ∈ G(W (k))|g modulo p belongs to Pk(k)

is intrinsically attached to the family (M, gϕ,G)|g ∈ G(W (k)) of Shimura p-divisibleobjects.

Let N and Q be as in Subsection 2.5.5. Thus N is the unique closed subgroup schemeof G that is isomorphic to a product of Ga group schemes and whose Lie algebra isF−1(g). Also N is the unipotent radical of the parabolic subgroup scheme Q of G that isthe opposite of P with respect to every maximal torus of P that contains Im(µ).

3.1.1. Basic Lemma. If two Hodge cocharacters µ1 and µ2 of (M,ϕ,G) commute, thenthey coincide. Similarly, if the special fibres of two cocharacters of (M,ϕ,G) commute,then they coincide.

Proof: For s ∈ 1, 2 let M = ⊕bi=aF

is(M) be the direct sum decomposition defined by µs.

As µ1 and µ2 commute, we have a direct sum decomposition

M = ⊕bi=a ⊕b

i=a[F i

1(M) ∩ F i2(M)].

50

We have to show that for all numbers i ∈ S(a, b) the identity F i1(M) = F i

2(M) holds i.e.,

that we have F i1(M) ∩ F i

2(M) = 0 if i 6= i. But we have F i1(M) ∩ F i

2(M) = 0 if and

only if (F i1(M)/pF i

1(M)) ∩ (F i2(M)/pF i

2(M)) = 0. Thus to prove the Lemma we are leftto show that if µ1,k and µ2,k commute, then they coincide. For each number i ∈ S(a, b)

the direct sum ⊕bj=iF

js (M)/pF j

s (M) does not depend on s ∈ 1, 2 and we have a direct

sum decomposition M/pM = ⊕bi=a ⊕b

i=a[(F i

1(M)/pF i1(M)) ∩ (F i

2(M)/pF i2(M))]. Thus

by decreasing induction on i ∈ S(a, b), we get that F i1(M)/pF i

1(M) = F i2(M)/pF i

2(M).Therefore we have µ1,k = µ2,k.

3.1.2. Lemma. Each element h ∈ L normalizes ϕ−1(M) and therefore hµh−1 is a Hodgecocharacter of (M,ϕ,G).

Proof: The intersection P ∩Q is a Levi subgroup scheme of Q and therefore it has trivialintersection with N . Thus the intersection N ∩P is the identity section of G. Therefore, asthe natural product morphism N ×W (k) P → G is an open embedding around the identitysections (cf. [DG, Vol. III, Exp. XXII, Prop. 4.1.2]), it is in fact an open embedding.

Thus we can write h = u1h1, where u1 ∈ N(W (k)) is congruent to 1M modulo pand where h1 ∈ P (W (k)). As µ acts via the identical character of Gm on F−1(g) and asN is isomorphic to the vector group scheme over W (k) defined by Lie(N) = F−1(g) (seeSubsections 2.5.4 and 2.5.5), we have µ( 1p )u1µ(p) ∈ N(W (k)). Thus, as σϕ = ϕµ(p) is a

σ-linear automorphism of M that normalizes G(W (k)) (cf. Section 2.4 and Lemma 2.4.6),we get that

ϕu1ϕ−1 = σϕµ(

1

p)u1µ(p)σ

−1ϕ ∈ σϕN(W (k))σ−1

ϕ 6 G(W (k)).

Therefore ϕu1ϕ−1 normalizes M . Thus u1 normalizes ϕ−1(M). As for all numbers i ∈

S(a, b) we have h1(Fi(M)) = F i(M), h1 normalizes

∑bi=a p

−iF i(M) = ϕ−1(M). There-

fore h = u1h1 normalizes ϕ−1(M). Thus h(ϕ−1(M)) = ⊕bi=ap

−ih(F i(M)) = ϕ−1(M).But hµh−1 acts on the direct summand h(F i(M)) of M via the −i-th power of the iden-tity character of Gm. From this and the identity ϕ−1(M) = ⊕b

i=ap−ih(F i(M)) we get

that hµh−1 is a Hodge cocharacter of (M,ϕ,G).

3.1.3. Lemma. Let µ1 be another Hodge cocharacter of (M,ϕ,G). Then there exists anelement g1 ∈ L such that we have g1µg

−11 = µ1.

Proof: Let (F i1(M))i∈S(a,b) be the lift of (M,ϕ,G) defined by µ1. Let P1 be the parabolic

subgroup scheme ofG that is the normalizer of (F i1(M))i∈S(a,b) in G. As we have Pk = P1,k,

by conjugating µ with an element of P (W (k)) (cf. Lemma 3.1.2), we can assume that µk

and µ1,k commute and therefore (cf. Lemma 3.1.1) that they coincide. Thus there exists anelement g1 ∈ Ker(G(W (k)) → G(k)) ⊳ L such that we have g1µg

−11 = µ1 (to be compared

with Subsection 2.3.8 (d) and [DG, Vol. II, Exp. IX, Thm. 3.6]).

3.1.4. Corollary. We fix a lift (F i(M))i∈S(a,b) of (M,ϕ,G). For each lift (F i1(M))i∈S(a,b)

of (M,ϕ,G), there exists a unique element n1 ∈ Ker(N(W (k)) → N(k)) such that we haveF i1(M) = n1(F

i(M)) for all numbers i ∈ S(a, b). Thus the set of lifts L of (M,ϕ,G) has

51

a natural structure of a free W (k)-module Ker(N(W (k)) → N(k)) of rank dim(Nk), the 0element being (F i(M))i∈S(a,b).

Proof: The set L is in one to one correspondence with the elements of the quotient groupL/P (W (k)), cf. Lemmas 3.1.2 and 3.1.3. As the product morphism N ×W (k) P → G isan open embedding, we have L/P (W (k)) = Ker(N(W (k)) → N(k)). This implies the

existence and the uniqueness of n1 ∈ Ker(N(W (k)) → N(k)). As N→Gdim(Nk)a , we can

identify naturally L with the free W (k)-module Ker(N(W (k)) → N(k)) of rank dim(Nk)in such a way that (F i(M))i∈S(a,b) is the 0 element.

3.1.5. Corollary. Under the process of associating adjoint Shimura filtered Lie F -crystals, we get a bijection between the set L of lifts of (M,ϕ,G) and the set Lad oflifts of (Lie(Gad), ϕ, Gad). Thus (M,ϕ,G) is U-ordinary (resp. Sh-ordinary) if and only if(Lie(Gad), ϕ, Gad) is U-ordinary (resp. Sh-ordinary).

Proof: As N is naturally a closed subgroup scheme of Gad, we can identify naturallyLad with Ker(N(W (k)) → N(k)) (see proof of Corollary 3.1.4). Thus the natural mapL → Lad is a bijection. The group L0

G(ϕ)B(k) (resp. the group scheme P+G (ϕ)) is the

extension of L0Gad(ϕ)B(k) (resp. of P+

Gad(ϕ)) by Z(G)B(k) (resp. by Z(G)). Therefore we

have L0G(ϕ)B(k) 6 P+

G (ϕ)B(k) if and only if we have L0Gad(ϕ)B(k) 6 P+

Gad(ϕ)B(k). Fromthis and the definitions of Subsection 1.3.5 we get the last part of the Corollary.

3.1.6. Fact. Let h ∈ L. Then h2 := ϕhϕ−1 ∈ G(W (k)). Thus (M,hϕh−1, G) is of theform (M, gϕ,G), where g := hh−1

2 ∈ G(W (k)).

Proof: Let u1 and h1 be as in the proof of Lemma 3.1.2. We have h2 ∈ G(B(k)) (cf.Lemma 2.4.6) and thus h2 ∈ G(W (k)) if and only if h2(M) =M . Using the property 1.3.2(a), we compute

h2(M) = ϕhϕ−1(⊕bi=aϕ(p

−iF i(M))) = ϕh(⊕bi=ap

−iF i(M))

= ϕu1h1(⊕bi=ap

−iF i(M)) = ϕu1(⊕bi=ap

−iF i(M)) = ϕu1ϕ−1(M) =M

(the last equality holds as u1 normalizes ϕ−1(M)). Thus h2(M) =M .

3.1.7. Canonical split cocharacters. Let µW : Gm → GLM be the inverse of thecanonical split cocharacter of (M, (F i(M))i∈S(a,b), ϕ) defined in [Wi, p. 512]. We check

that µW factors through G. For this we can assume that k = k. Let σϕ and MZpbe as

in Section 2.4. See Example 2.4.2 for GZp. Let (tα)α∈J be a family of tensors of T (MZp

)such that GQp

is the subgroup of GLMZp [1p ]

that fixes tα for all α ∈ J , cf. [De5, Prop. 3.1

(c)]. As the tensor tα ∈ T (M) = T (MZp) ⊗Zp

W (k) is fixed by σϕ and µ, it is also fixedby ϕ. As tα is fixed by µ and ϕ, it is also fixed by µW (cf. the functorial aspects of [Wi,p. 513]). Thus µW,B(k) factors through GB(k) and therefore µW factors through G.

The factorization of µW through G will be also denoted as µW : Gm → G and will bealso called as the inverse of the canonical split cocharacter of (M, (F i(M))i∈S(a,b), ϕ); inparticular, µW : Gm → G is a Hodge cocharacter of (M,ϕ,G) that defines (F i(M))i∈S(a,b)

(this is the most basic property of inverses of canonical split cocharacters, cf. [Wi, p. 512]).

52

3.1.8. Proof of 1.4.3 (a) ⇒ 1.4.3 (b). We assume that (M,ϕ,G) is U-ordinary. For s ∈1, 2 let (F i

s(M))i∈S(a,b) be a lift of (M,ϕ,G) such that L0G(ϕ)B(k) normalizes F i

s(M)[ 1p]

for all i ∈ S(a, b). Let µs : Gm → G be the inverse of the canonical split cocharacter of(M, (F i

s(M))i∈S(a,b), ϕ); it defines the lift (F is(M))i∈S(a,b) of (M,ϕ,G).

To show that 1.4.3 (a) ⇒ 1.4.3 (b), we have to check that (F is(M))i∈S(a,b) does

not depend on s ∈ 1, 2. For this it suffices to show that µ1 = µ2. To check thatµ1 = µ2, we can assume that k = k. As k = k, Lie(L0

G(ϕ)B(k)) is B(k)-generatedby endomorphisms of (M,ϕ,G) (i.e., by elements of g that are fixed by ϕ). Thus, asL0G(ϕ)B(k) normalizes F i

s(M)[ 1p ], each endomorphism of (M,ϕ,G) is also an endomorphism

of (M, (F is(M))i∈S(a,b), ϕ, G). From this and the functorial aspects of [Wi, p. 513] we get

that µs fixes all endomorphisms of (M,ϕ,G). Thus µs,B(k) factors through the subgroupZ(L0

G(ϕ)B(k)) of GB(k) that fixes these endomorphisms. Therefore µ1 and µ2 commute.Thus we have µ1 = µ2, cf. Lemma 3.1.1. Therefore indeed 1.4.3 (a) ⇒ 1.4.3 (b).

3.1.9. Corollary. We assume that (M,ϕ,G) is U-ordinary, that (M, (F i(M))i∈S(a,b), ϕ, G)is a U-canonical lift, and that the cocharacter µ : Gm → G is the inverse of the canonicalsplit cocharacter of (M, (F i(M))i∈S(a,b), ϕ). Then there exists a smallest torus T (0) of G

through which µ factors and such that ϕ normalizes Lie(T (0)) (thus the triple (M,ϕ, T (0))

is a Shimura p-divisible object). Moreover, we have T(0)B(k) 6 Z0(L0

G(ϕ)B(k)).

Proof: We know that σϕ := ϕµ(p) normalizes both M and Lie(Z0(L0G(ϕ)B(k))) (see Sec-

tion 2.4). Also µ(p) normalizes Lie(Z0(L0G(ϕ)B(k))). Thus for m ∈ N ∪ 0 we have

ϕm(Lie(Im(µ))) = σmϕ (Lie(Im(µ))) and therefore ϕm(Lie(Im(µ))) is the Lie algebra of the

Gm subgroup scheme σmϕ (Im(µ)) of G = σm

ϕ (G) (see Subsection 2.4.7) whose fibre overB(k) is a subgroup of Z0(L0

G(ϕ)B(k)) (cf. Lemma 2.3.6). The Gm subgroup schemesσmϕ (Im(µ)) with m ∈ N∪0 commute (as their fibres over B(k) commute) and thus their

centralizer C(0) in GLM is a finite product of GL group scheme. Thus the flat, closed sub-group scheme T (0) of G generated by the images of these commuting Gm subgroup schemesof G, is a flat, closed subgroup scheme of Z(C(0)) whose fibre over B(k) is a subtorus of

both C(0)B(k) and Z0(L0

G(ϕ)B(k)). From Fact 2.3.12 we get that T (0) is a subtorus of C(0)

and thus also a torus of G. Therefore T (0) has all the desired properties.

3.1.10. Definition. By a positive (resp. a negative) Hodge cocharacter of (M,ϕ,G)we mean a Hodge cocharacter Gm → G of (M,ϕ,G) that factors through P+

G (ϕ) (resp.through P−

G (ϕ)).

3.1.11. Lemma. There exist positive (resp. negative) Hodge cocharacters of (M,ϕ,G).

Proof: The intersection Pk ∩P+G (ϕ)k contains a maximal split torus Tk of Gk, cf. [Bo, Ch.

V, Props. 21.12 and 21.13 (i)]. Based on the property 2.3.1 (d), there exists an element

h ∈ P (k) such that hµkh−1

factors through Tk. Let h ∈ L be such that its reductionmodulo p is h. Let µ1 := hµh−1 : Gm → G. It is a Hodge cocharacter of (M,ϕ,G)(cf. Lemma 3.1.2) whose special fibre factors through Tk. Let T be a maximal torus ofP+G (ϕ) that lifts Tk, cf. Subsection 2.3.8 (b). Let S be the Gm subgroup scheme of T

whose special fibre is Im(µ1,k). By replacing h with a Ker(G(W (k)) → G(k))-multiple

53

of it (cf. Lemma 3.1.2), we can assume that Im(µ1) is S itself (cf. [DG, Vol. II, Exp.IX, Thm. 3.6]). In particular, µ1 factors through every Levi subgroup scheme of P+

G (ϕ)

that contains T and thus it is a positive Hodge cocharacter of (M,ϕ,G). By workingwith P−

G (ϕ) instead of P+G (ϕ), a similar argument shows that there exist negative Hodge

cocharacters of (M,ϕ,G).

3.1.12. Practical σ-linear automorphisms ofM . Let g ∈ G(W (k)). Let µ1 : Gm → Gbe a positive (resp. a negative) Hodge cocharacter of (M, gϕ,G). Let ϕ1 := gϕ. As inSubsection 2.4.8 we define σϕ1

:= ϕ1µ1(p) :M→M . As µ1 normalizes Lie(P+G (gϕ)) (resp.

Lie(P−G (gϕ))) and as ϕ1 normalizes Lie(P+

G (gϕ))[ 1p ] (resp. Lie(P−G (gϕ))[ 1p ]), the σ-linear

automorphism σϕ1ofM normalizes Lie(P+

G (gϕ)) (resp. Lie(P−G (gϕ))). A similar argument

shows that σϕ1normalizes Lie(P+

G (gϕ)) (resp. Lie(P−G (gϕ))).

3.1.13. Descent Lemma for lifts. Let k1 be a perfect field that contains k and suchthat we have k = x ∈ k1|τ(x) = x ∀τ ∈ Autk(k1). Let (F i

1(M ⊗W (k)W (k1)))i∈S(a,b) be alift of (M,ϕ,G)⊗ k1 that is fixed under the natural action of AutW (k)(W (k1)) = Autk(k1)on M ⊗W (k) W (k1). Then (F i

1(M ⊗W (k) W (k1)))i∈S(a,b) is the tensorization with W (k1)(over W (k)) of a unique lift of (M,ϕ,G).

Proof: Let µW,1 : Gm → GW (k1) be the inverse of the canonical split cocharacter of(M⊗W (k)W (k1), (F

i1(M⊗W (k)W (k1)))i∈S(a,b), ϕ⊗σk1

), cf. Subsection 3.1.7. We considerthe semisimple element l1 ∈ EndW (k)(M)⊗W (k)W (k1) that is the image under dµW,1 of thestandard generator of Lie(Gm). As the lift (F i

1(M ⊗W (k)W (k1)))i∈S(a,b) of (M,ϕ,G)⊗ k1is fixed under the natural action of AutW (k)(W (k1)) = Autk(k1) on M ⊗W (k) W (k1), theelement l1 is fixed by the natural action of AutW (k)(W (k1)) on EndW (k)(M)⊗W (k)W (k1).As k = x ∈ k1|τ(x) = x ∀τ ∈ Autk(k1), we have W (k) = x ∈ W (k1)|τ(x) = x ∀τ ∈AutW (k)(W (k1)). Thus l1 ∈ EndW (k)(M). This implies that µW,1 : Gm → GW (k1) is thepull back of a cocharacter of G that is a Hodge cocharacter of (M,ϕ,G). From this weget that the lift (F i

1(M ⊗W (k) W (k1)))i∈S(a,b) of (M,ϕ,G)⊗ k1 is the tensorization withW (k1) (over W (k)) of a unique lift of (M,ϕ,G).

3.2. A general outline

We start the proofs of Theorems 1.4.1 and 1.4.2. Fact 3.2.1 and Subsection 3.2.2present simple properties. In Subsections 3.2.3 to 3.8.4 we will assume that G is quasi-split. In Subsection 3.2.3 we introduce a standard example for Sh-ordinariness. The mainingredients for the proofs of Theorems 1.4.1 and 1.4.2 are stated in Section 3.3. Section3.4 introduces the required language of roots and Newton polygon slopes; it prepares thebackground for translating properties 3.3 (a) and (b) into properties of parabolic subgroupsschemes of either G or Gk. The proofs of properties 3.3 (a) and (b) are carried out inSections 3.5 to 3.7. Section 3.8 completes the proofs of Theorems 1.4.1 and 1.4.2.

3.2.1. Facts. (a) Let k1 be a perfect field that contains k. We assume that (M,ϕ,G)⊗k1is U-ordinary. Then the U-canonical lift (F i

1(M ⊗W (k) W (k1)))i∈S(a,b) of (M,ϕ,G) ⊗ k1is the tensorization with W (k1) (over W (k)) of a unique lift of (M,ϕ,G). Thus (M,ϕ,G)is also U-ordinary (and moreover it is Sh-ordinary if (M,ϕ,G)⊗ k1 is Sh-ordinary).

54

(b) There exists a unique open, Zariski dense subscheme Z of Gadk such that for each

field k1 as in (a) and for every two elements g1 and g2 ∈ G(W (k1)) with g1 mapping to ak1-valued point of Z, the strict inequality

I(M ⊗W (k) W (k1), g1(ϕ⊗ σk1), GW (k1)) > I(M ⊗W (k) W (k1), g2(ϕ⊗ σk1

), GW (k1))

holds if and only if g2 does not map to a k1-valued point of Z.

Proof: To check (a) we can assume that k1 = k1. Thus k is the subfield of k1 fixed byAutk(k1). The U-canonical lift (F i

1(M ⊗W (k) W (k1)))i∈S(a,b) of (M,ϕ,G) ⊗ k1 is unique(as 1.4.3 (a) ⇒ 1.4.3 (b); see Subsection 3.1.8) and thus it is fixed by the natural action ofAutW (k)(W (k1)) = Autk(k1) on lifts of (M,ϕ,G) ⊗ k1. From this and Lemma 3.1.13 weget that (a) holds.

We check (b). Let iC : Spec(C) → Gadk be a morphism of affine k-schemes. We denote

also by iC the Lie automorphism (over C) of Lie(Gadk )⊗k C defined by inner conjugation

through iC . For each geometric point yk : Spec(k) → Spec(C), let ik := iC yk ∈ Gad(k)and

I(ik) := ∩m∈NIm((ik(Ψ⊗ σk))m) = Im((ik(Ψ⊗ σk))

dim(Gadk )).

Let IC be the C-submodule of Lie(Gadk )⊗k C generated by Im((iC(Ψ⊗ΦC))

dim(Gadk )) (we

recall that ΦC is the Frobenius endomorphism of C). Let QC := (Lie(Gadk )⊗k C)/IC .

We now take iC : Spec(C) → Gadk to be an isomorphism of k-schemes. Let Z be the

maximal open, Zariski dense subscheme of Gadk over which the OGad

k-module R defined

by QC is free (cf. [Gr1, Prop. (8.5.5)]). A point p of Gadk belongs to Z if and only if

the dimension of the fibre of R over p is equal to (equivalently it is not greater than)the dimension of the fibre of R over the generic point of Gad

k . Thus, as we have I(ik) =

dim(Gadk )− dimk(QC ⊗C k), we easily get that (b) holds for Z.

3.2.2. The trivial situation. It is easy to see that the following four statements areequivalent:

(a) we have F 1(g) = 0;

(b) we have F 0(g) = g;

(c) the Hodge cocharacter µ : Gm → G of (M,ϕ,G) factors through Z0(G);

(d) the G(W (k))-conjugacy class [µ] of µ has a unique representative.

Let g ∈ G(W (k)). Let σϕ = ϕµ(p) be as in Section 2.4. If the statement (c) holds, thenϕG(W (k))ϕ−1 = σϕµ(

1p )G(W (k))µ(p)σ−1

ϕ = σϕG(W (k))σ−1ϕ = G(W (k)) (cf. Lemma

2.4.6 for the last identity) and therefore by induction on m ∈ N we get that we have anidentity (gϕ)m = gmϕ

m for some element gm ∈ G(W (k)); thus the Newton polygon of(M, gϕ) does not depend on g (cf. Lemma 2.2 applied to the F -crystals (M, p−agϕ) and(M, p−aϕ)). If the statement (c) holds, then Ψ is a σ-linear automorphism of Lie(Gad);thus I(M, gϕ,G) = dim(Gad

k ) does not depend on g. If the statement (b) holds, thenP = G (cf. Lemma 2.3.6) and thus (M, (F i(M))i∈S(a,b), gϕ,G) is an Sh-canonical lift.This proves Theorems 1.4.1 and 1.4.2, if statements (a) to (d) hold.

55

3.2.3. Standard example. We emphasize that until Subsection 3.8.5 we will assumethat G is quasi-split. Let B be a Borel subgroup scheme of G contained in P , cf. Subsection2.3.8 (e). Let T be a maximal torus of B. Every two maximal tori of Bk are B(k)-conjugate(cf. [Bo, Ch. V, Thm. 19.2]) and thus from the structure of Borel subgroups of Gk (see[Bo, Ch. V, Sect. 21.11 and Prop. 21.12]), we get that Tk contains a maximal split torusof Gk. Thus T contains a maximal split torus of G. As µ factors through P , up to aP (W (k))-conjugation we can assume that µ factors through T (cf. Subsection 2.3.8 (f)and Lemma 3.1.2). Therefore we also view µ as a cocharacter of T .

Let t := Lie(T ) and b := Lie(B). We have inclusions t ⊆ b ⊆ F 0(g) and F 1(g) ⊆ b.Thus ϕ(t) ⊆ ϕ(b) ⊆ g. Let σϕ := ϕµ(p) : M→M be as in Section 2.4 and Subsection2.4.7. We have σϕ(G) = G, cf. Subsection 2.4.7. The maximal torus σϕ(T ) of G = σϕ(G)has ϕ(t) = σϕ(t) as its Lie algebra. The Borel subgroup scheme σϕ(B) of G = σϕ(G)contains σϕ(T ) and its Lie algebra is σϕ(b). Let g0 ∈ G(W (k)) be such B = g0σϕ(B)g−1

0

and T = g0σϕ(T )g−10 , cf. Subsection 2.3.8 (d). Thus g0ϕ normalizes t and takes b to

g0σϕµ(1p )(b) ⊆ g0σϕ(b) = b. We emphasize that in order not to introduce extra notations

by always replacing g ∈ G(W (k)) with gg0 ∈ G(W (k)), until Section 3.8 we will assumethat g0 = 1M . Thus T = σϕ(T ) and B = σϕ(B) and therefore we have

(5) ϕ(t) = t and ϕ(b) ⊆ b.

As ϕ(t) = t, we have t ⊆W (0)(g, ϕ). Thus T is a maximal torus of L0G(ϕ) (cf. Lemma

2.3.6) and therefore also of L0G(ϕ) (cf. Subsection 2.6.5). Thus L0

G(ϕ) (resp. L0G(ϕ)) is the

unique (cf. Subsection 2.3.8 (c)) Levi subgroup scheme of P+G (ϕ) or P−

G (ϕ) (resp. P+G (ϕ)

or P−G (ϕ)) that contains T .

3.3. The essence of the proofs of Theorems 1.4.1 and 1.4.2

We recall that G is quasi-split and that (M,ϕ,G) is such that (5) holds. The mainpart of the proofs of Theorems 1.4.1 and 1.4.2 is to check the following two properties:

(a) For all elements g ∈ G(W (k)) we have an inequality I(M, gϕ,G) ≤ I(M,ϕ,G).

(b) Let g ∈ G(W (k)). We have an equality I(M, gϕ,G) = I(M,ϕ,G) if and only ifwe can write gϕ = g1rϕg

−11 , where r ∈ L0

G(ϕ)(W (k)) and g1 ∈ L.

The proof of the property (a) ends with Section 3.5 while the proof of the property(b) ends with Subsection 3.7.2. Until the end of Section 3.4 we will include preliminarymaterial that will be used in Sections 3.5 to 3.7.

3.3.1. Lemma. Let g2 ∈ G(W (k)) be such that P+G (g2ϕ) is a subgroup scheme of P .

If g1 ∈ L and h1 ∈ P+G (g2ϕ)(W (k)), then (M, g2ϕ) and (M, g1h1g2ϕg

−11 ) have the same

Newton polygons and moreover we have an identity P+G (g1h1g2ϕg

−11 ) = g1P

+G (g2ϕ)g

−11 .

Proof: We can assume that g1 = 1M . To check that P+G (h1g2ϕ) = P+

G (g2ϕ) it suf-fices to show that Lie(P+

G (h1g2ϕ)) = Lie(P+G (g2ϕ)) (cf. Lemma 2.3.6) and thus that

W 0(g, h1g2ϕ) = W 0(g, g2ϕ). As g2ϕ normalizes Lie(P+G (g2ϕ))[

1p], we have an identity

g2ϕP+G (g2ϕ)B(k)ϕ

−1g−12 = P+

G (g2ϕ)B(k) (cf. Lemma 2.3.6 and cf. the existence of the

action ϕ of Subsection 2.4.7). Thus g2ϕh1 = h2g2ϕ, where h2 ∈ P+G (g2ϕ)(B(k)). But we

56

have h2 ∈ G(W (k)), cf. Fact 3.1.6 applied to h1 ∈ P+G (g2ϕ)(W (k)) 6 P (W (k)) 6 L. Thus

h2 ∈ P+G (g2ϕ)(W (k)). By induction on m ∈ N\1 we get that (h1g2ϕ)

m = h1hm(g2ϕ)m,

where hm ∈ P+G (g2ϕ)(W (k)). Therefore from Lemma 2.2 and Corollary 2.2.1 applied to

(M, p−ah1g2ϕ,G) and (M, p−ag2ϕ,G), we get that (M, g2ϕ) and (M, g1h1g2ϕg−11 ) have

the same Newton polygons and that W 0(g, h1g2ϕ) =W 0(g, g2ϕ).

3.3.2. The non-trivial situation. If statements 3.2.2 (a) to (d) hold, then we haveP+G (ϕ) = P+

G (ϕ) = L0G(ϕ) = G and (cf. Subsection 3.2.2) I(M,ϕ,G) = I(M, gϕ,G);

obviously the properties 3.3 (a) and (b) hold. Thus until the end of this Chapter 3 we willassume that the statements 3.2.2 (a) to (d) do not hold. Therefore G is not a torus.

3.4. The split, adjoint, cyclic context

Sections 3.4 to 3.6 are the very heart of the proof of Theorem 1.4.1. Their maingoals are to prove the property 3.3 (a) and to prepare the background for the proof of theproperty 3.3 (b). Let T , B, t, and b be as in Subsection 3.2.3. For the proof of the property3.3 (a) we can assume that G is split, adjoint (see Subsection 2.4.5). Therefore we havea product decomposition G =

∏i∈I Gi into absolutely simple, adjoint group schemes (cf.

(2) of Subsection 1.3.4) and the torus T is split. The set I is non-empty, cf. Subsection3.3.2. For each element i ∈ I let

gi := Lie(Gi).

To check the property 3.3 (a) we can also assume that I = S(1, n) and that thepermutation Γ : I→I is the n-th cycle (12 · · ·n); thus we have ϕ(gi[

1p ]) = gi+1[

1p ] if

i ∈ S(1, n − 1) and ϕ(gn[1p ]) = g1[

1p ]. Only in Subsection 3.4.4 it will become clear why

we take the set I to be S(1, n) and not Z/nZ. The Lie type of Gi does not depend on theelement i ∈ I and thus we denote it as

X .

Letℓ ∈ N

be the rank of X . Let l0 ∈ g = ⊕i∈Igi be the image through dµ of the standard generatorof Lie(Gm); we have l0(x) = −ix for all pairs (x, i) ∈ F i(M)× S(a, b). For each elementi ∈ I let

li ∈ gi

be the component of l0 in gi; we have l0 =∑

i∈I li. If li = 0, then gi ⊆ F 0(g). Ifli 6= 0, then li is a semisimple element of gi and the eigenvalues of the action of li ongi are precisely −1, 0, and 1 (cf. property 1.3.2 (b)). More precisely, we have identitiesF 1(g) = F 1(g) = x ∈ g|[l0, x] = −x, F 0(g) = x ∈ g|[l0, x] = 0, and F−1(g) = x ∈g|[l0, x] = x. Thus F 1(gi) := F 1(g) ∩ gi = F 1(g) ∩ gi = x ∈ gi|[li, x] = −x. Similarly,F 0(g) = x ∈ g|[l0, x] ∈ 0,−x and thus F 0(gi) := F 0(g)∩gi = x ∈ gi|[li, x] ∈ 0,−x.Based on Subsection 3.3.2, we can choose the identification I = S(1, n) such that we havel1 6= 0; therefore F 1(g1) 6= 0.

3.4.1. Newton Polygon slopes. Let g ∈ G(W (k)). Let Hg (resp. H) be the setof Newton polygon slopes of the Lie F -isocrystal (g[ 1p ], gϕ) (resp. (g[ 1p ], ϕ)). As (gϕ)n

57

normalizes each one of gi[1p ]’s, we have

W (γ)(g, gϕ) = ⊕i∈I(W (γ)(g, gϕ)∩ gi), ∀γ ∈ Hg.

As gϕ maps W (γ)(g[ 1p ], gϕ) ∩ gi[1p ] bijectively onto W (γ)(g[ 1p ], gϕ) ∩ gi+1[

1p ], the rank of

W (γ)(g, gϕ)∩ gi does not depend on i ∈ I.

3.4.2. Parabolic decompositions. Each parabolic subgroup scheme Q of G is a productQ =

∏i∈I Q ∩ Gi of parabolic subgroup schemes of the Gi’s. Let ∗ ∈ +,−. If ∗ = +

(resp. if ∗ = −) let µ1 be a positive (resp. a negative) Hodge cocharacter of (M, gϕ,G). Letσϕ1

:= gϕµ1(p). The σ-linear automorphism σϕ1: M→M normalizes both Lie(P ∗

G(gϕ))

and Lie(P ∗G(gϕ)), cf. Subsection 3.1.12. We get:

(a) For each i ∈ I we have identities σi−1ϕ1

(Lie(P ∗G(gϕ) ∩ G1)) = Lie(P ∗

G(gϕ) ∩ Gi)

and σi−1ϕ1

(Lie(P ∗G(gϕ) ∩ G1)) = Lie(P ∗

G(gϕ) ∩ Gi). Thus the following two dimensions

dim(P ∗G(gϕ)k ∩Gi,k) and dim(P ∗

G(gϕ)k ∩Gi,k) do not depend on i ∈ I.

By taking ∗ = + in the property (a) we get that Lie(P+G (gϕ)) = Lie(P+

G (gϕ)) if and

only if Lie(P+G (gϕ) ∩G1) = Lie(P+

G (gϕ) ∩G1). From this and Lemma 2.3.6 we get:

(b) We have P+G (gϕ) = P+

G (gϕ) if and only if Lie(P+G (gϕ)∩G1) = Lie(P+

G (gϕ)∩G1).

We now check the following simple property:

(c) If P+G (gϕ)k ∩G1,k and P+

G (ϕ)k ∩G1,k are G1(k)-conjugate, then for each element

i ∈ I the parabolic subgroups P+G (gϕ)k∩Gi,k and P+

G (ϕ)k∩Gi,k ofGi,k areGi(k)-conjugate.

To check the property (c) we first recall that σϕ = ϕµ(p). As T 6 L0G(ϕ) 6 P+

G (ϕ),µ : Gm → G is a positive Hodge cocharacter of (M,ϕ,G). Arguments similar to the onesused to get the property (a), show that σi−1

ϕ (Lie(P+G (ϕ) ∩ G1)) = Lie(P+

G (ϕ) ∩ Gi). But

we have σi−1ϕ1

= hi−1σi−1ϕ for some element hi−1 ∈ G(W (k)), cf. Subsection 2.4.8. Let

g1 ∈ G1(W (k)) be such that P+G (gϕ)∩G1 = g1(P

+G (ϕ)∩G1)g

−11 , cf. the hypothesis of (c)

and Subsection 2.3.8 (e). Thus we compute that

Lie(P+G (gϕ) ∩Gi) = hi−1σ

i−1ϕ (Lie(P+

G (gϕ) ∩G1)) = hi−1σi−1ϕ g1(Lie(P

+G (ϕ) ∩G1))

= hi−1σi−1ϕ g1σ

1−iϕ σi−1

ϕ (Lie(P+G (ϕ) ∩G1)) = hi−1σ

i−1ϕ g1σ

1−iϕ (Lie(P+

G (ϕ) ∩Gi)).

As ji−1 := hi−1σi−1ϕ g1σ

1−iϕ ∈ G(W (k)) (cf. Lemma 2.4.6) we get that Lie(P+

G (gϕ)∩Gi) =

Lie(ji−1(P+G (gϕ) ∩ Gi)j

−1i−1). Thus P+

G (gϕ) ∩ Gi = ji−1(P+G (gϕ) ∩ Gi)j

−1i−1, cf. Lemma

2.3.6. In this last equality we can replace ji by its component in Gi(W (k)) and thus theproperty (c) holds.

3.4.3. Root decompositions. Let ti := t∩ gi and bi := b∩ gi. Let Ti := T ∩Gi. Let Φi

be the set of roots of gi relative to the torus Ti. Let ∆i be the base of Φi that correspondsto bi. In what follows Φ+

i (resp. Φ−i ) is the set of positive (resp. negative) roots in Φi

with respect to ∆i. We have ∆i ⊆ Φ+i . Let

g = ⊕i∈I ti⊕

i∈I, α∈Φi

gi(α)

58

be the root decomposition of g relative to T . As ϕ(t) = t, there exists a permutation ΓΦ

of the disjoint union ∪i∈IΦi such that we have

(6a) ϕ(gi(α)) = psi(α)gΓ(i)(ΓΦ(α)), ∀i ∈ I and ∀α ∈ Φi.

Here si(α) ∈ −1, 0, 1 is such that µ(β) acts on gi(α) as the multiplication with β−si(α),for all β ∈ Gm(W (k)) (cf. property 1.3.2 (b)). We have (cf. (5))

(6b) ΓΦ(∪i∈IΦ+i ) = ∪i∈IΦ

+i and thus also ΓΦ(∪i∈IΦ

−i ) = ∪i∈IΦ

−i .

As si(α) ≤ 0 (resp. as si(α) ≥ 0) if α ∈ Φ−i (resp. if α ∈ Φ+

i ), we get inclusions

t ⊆ b ⊆ Lie(P+G (ϕ)) ⊆ Lie(P ) = F 0(g) = F 0(g)⊕F 1(g) and Lie(P−

G (ϕ)) ⊆ F 0(g)⊕F−1(g).

In particular, we get that T 6 B 6 P+G (ϕ) 6 P (cf. Lemma 2.3.6). Thus (M,ϕ,G) is

Sh-ordinary.

3.4.4. The order e. Let ΓΦ1be the permutation of Φ1 defined by Γn

Φ. It normalizes ∆1,Φ+

1 , and Φ−1 . Let e be the order of ΓΦ1

. Let

I := S(1, en).

We think of ge := g⊕· · ·⊕g (e summands) as a direct sum ⊕i∈Igi. Here for s ∈ S(1, e−1)and i ∈ I, we define gi+sn := gi, ti+sn := ti, ∆i+sn := ∆i, and bi+sn := bi. Below theright lower index en + 1 is identified with 1 (i.e., we do identify naturally S(1, en) withZ/enZ). It is convenient to work also with

(ge, ϕ(e)),

where the σ-linear endomorphism ϕ(e) of ge[ 1p ] acts for each element i ∈ I on gi[1p ] as ϕ

but the image ϕ(e)(gi[1p ]) belongs to gi+1[

1p ]. In a similar way we define gϕ(e). The set

of Newton polygon slopes of (ge, gϕ(e)) is the set of Newton polygon slopes of (g, gϕ), allmultiplicities being e times more. The permutation of ∆1 defined by (ϕ(e))en is 1∆1

. Fors ∈ S(1, e − 1) and i ∈ I let li+sn := (ϕ(e))sn(li) ∈ gi+sn. Thus the semisimple elementli ∈ gi is well defined for all i ∈ I1.

We have e ∈ 1, 2, 3. If e = 2, then X ∈ Aℓ|ℓ ≥ 2 ∪ Dℓ|ℓ ≥ 4 ∪ E6, and ife = 3, then X = D4. This follows from the fact that the group of automorphisms of theDynkin diagram of Φ1 is either trivial or Z/2Z or S3. The order e is independent of thechoice of an element i ∈ I to define it (above we chose 1 ∈ I i.e., we worked with ∆1).

3.4.5. Simple roots. We recall that Gm acts on g through µ via the trivial, identical,and the inverse of the identical characters of Gm (cf. property 1.3.2 (b)). Thus, as thehighest roots of the E8, F4, and G2 Lie types have all coefficients greater than 1 (see [Bou2,plates VII to IX]), the Lie type X can not be E8, F4, or G2 (cf. also [Sa], [De3], [Se1],or the classification of minuscule characters in [Bou3, Ch. VIII, §7.4, Rm.]). For each

59

element i ∈ I let α1(i), . . . , αℓ(i) be a numbering of the roots in ∆i such that the followingtwo things hold:

(a) we have ϕ(e)(gi(αs(i))[1p ]) = gi+1(αs(i+ 1))[ 1p ], and

(b) the numbering for i = 1 matches to the one of [Bou2, plates I to VI] (thus thisnumbering is unique up to an automorphism of Φ1 that normalizes ∆1).

We will drop the lateral right index (1) of αi(1)’s. Therefore ∆1 = αi|i ∈ S(1, ℓ).

3.4.6. Highest root. The highest root αh of Φ+1 depends on X . It is (cf. [Bou2, plates

I to VI]):

(a) α1 + α2 + · · ·+ αℓ if X = Aℓ, with ℓ ≥ 2;

(b) α1 + 2α2 + · · ·+ 2αℓ if X = Bℓ, with ℓ ≥ 1;

(c) 2α1 + 2α2 + · · ·+ 2αℓ−1 + αℓ if X = Cℓ, with ℓ ≥ 3;

(d) α1 + 2α2 + 2α3 + · · ·+ 2αℓ−2 + αℓ−1 + αℓ if X = Dℓ, with ℓ ≥ 4;

(e6) α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 if X = E6;

(e7) 2α1 + 2α2 + 3α3 + 4α4 + 3α5 + 2α6 + α7 if X = E7.

In what follows, depending on the possibilities of X listed in (a) to (e7), we will referto the case (a), to the case (b), etc. We note down that we consider Bℓ with ℓ ≥ 1 and Aℓ

with ℓ ≥ 2.

3.4.7. The η’s. Let i ∈ I. If li = 0 let ηi := 0. If li 6= 0 let ηi ∈ S(1, ℓ) be the uniquenumber such that li acts (under left commutation) as −1 (resp. as 0) on gi(αs(i)) fors = ηi (resp. for s ∈ S(1, ℓ) \ ηi). Let

I1 := i ∈ I|ηi 6= 0 and I1 := i ∈ I|ηi 6= 0.

As I ⊆ I, we have I1 ⊆ I1. We have 1 ∈ I1, cf. sentence before Subsection 3.4.1. Incases (b), (c), and (e7) the number η1 is 1, ℓ, and 7 (respectively). In the case (a) we haveη1 ∈ S(1, ℓ). In the case (d) (resp. (e6)) we have η1 ∈ 1, ℓ− 1, ℓ (resp. η1 ∈ 1, 6). Allthese can be read out from the set of those roots in ∆1 that show up with coefficient 1 inthe highest root αh.

For α ∈ Φ1 let gα := g1(α). For i ∈ I1 let pηibe the parabolic Lie subalgebra of g1

that contains b1 and whose nilpotent radical uηiis the Lie subalgebra of g1 generated by

all gα’s with α ∈ Φ+1 having αηi

in its expression. We have pη1= F 0(g1) and uη1

= F 1(g1).Let qηi

be the parabolic Lie subalgebra of g1 that is opposite to pηiwith respect to T1.

Let nηibe the nilpotent radical of qηi

. We have direct sum decompositions of t1-modules

g1 = pηi⊕ nηi

= qηi⊕ uηi

, where i ∈ I1.

3.4.8. The sets E and E. Let

E := ηi|i ∈ I1 ⊆ S(1, ℓ).

60

We have ∩i∈I1nηi

= ∩x∈Enx. If |E| ≥ 2, then we are in cases (a), (d), or (e6). Inthe case (d) we have |E| ≤ 3. In the case (e6) we have |E| ≤ 2. In the case (a) we haveE ⊆ S(1, ℓ) and thus

⋂x∈E nx = nx1

∩ nx|E|, where x1 := min(E) and x|E| := max(E); this

is so as every positive root that contains both αx1and αx|E|

in its expression as sums ofelements of ∆1, also contains αx for all x ∈ S(x1, x|E|) (see [Bou2, plate I]). Next we definea subset

E ⊆ E .

If X 6= Aℓ let E := E and if X = Aℓ let E := x1, x|E|. We have

⋂

i∈I1

nηi=

⋂

x∈E

nx =⋂

x∈E

nx.

3.4.9. Types. We refer to the en+ 2-tuple

(X , e, η1, η2, . . . , ηen)

as the type of (M,ϕ,G) and to the en-tuple (η1, η2, . . . , ηen) as the η-type of (M,ϕ,G).

If e = 1, then (η1, . . . , ηn) is uniquely determined up to a cyclic permutation and arenumbering in the statement 3.4.5 (b). If e = 2 and X 6= D4 (resp. if e ∈ 2, 3 andX = D4), then (η1, . . . , ηen) is uniquely determined up to a cyclic permutation (resp. upto a cyclic permutation and a renumbering in the statement 3.4.5 (b)).

3.4.10. The smallest Newton polygon slope. From (6a) and the numberings inSubsection 3.4.5 we get that

(7) ϕen(gα) = pnαgα and ϕen(g−α) = p−nαg−α, ∀α ∈ Φ+1 ,

where nα ∈ N ∪ 0 is the number of elements i ∈ I1 such that we have gα ⊆ uηi(or

equivalently such that we have g−α ⊆ nηi). Let

δ :=|I1|

n> 0.

3.4.11. Fact. The sets H and Hg are subsets of the interval [−δ, δ]. They are symmetricwith respect to the mid-point 0 of [−δ, δ] and we have δ ∈ H. Thus −1, 1 ⊆ H if and onlyif |I1| = n (i.e., if and only if the image of µ : Gm → G in every simple factor Gi of G isnon-trivial).

Proof: We have (gϕ)(e)(gi) = gi+1 if i ∈ I \ I1 and pgi+1 ⊆ (gϕ)(e)(gi) ⊆1pgi+1 if i ∈ I1.

This implies that

(8) pnδgi ⊆ (gϕ)n(gi) ⊆ p−nδgi, ∀i ∈ I.

From (8) we get that Hg ⊆ [−δ, δ], cf. also Mazur theorem of [Ka2, 1.4.1] applied to(g1, p

nδ(gϕ)n). A similar argument shows that H ⊆ [−δ, δ]. From (3b) we get that H and

61

Hg are symmetric with respect to the mid-point 0 of [−δ, δ]. We have nαh=

∣∣∣I1∣∣∣ = e |I1|.

Thus we have gαh⊆W (δ)(g, ϕ) ∩ g1, cf. (7). Therefore δ ∈ H.

3.4.12. Multiplicities of −δ as ranks. From (7) we get that

W (−δ)(g, ϕ) ∩ g1 = ∩i∈I1nηi

= ∩x∈Enx.

Thus m−δ := rkW (k)(W (−δ)(g, ϕ)) is n times the rank of the intersection ∩x∈Enx, cf.Subsection 3.4.1. A similar argument shows that the multiplicity of the Newton polygonslope −δ for (g[ 1p ], gϕ) is (cf. Subsection 3.4.1) n times the rank of

W−i (g) :=W (−δ)(g, gϕ) ∩ gi, where i ∈ I.

3.4.13. FSHW invariants as multiplicities of −δ. Let g ∈ G(k) be the reduction ofg ∈ G(W (k)) modulo p. The pair (g, gΨ) is an F -crystal. It is well known that I(M, gϕ,G)is the multiplicity of the Newton polygon slope 0 of (g, gΨ). By very definitions we have(gΨ)n = p|I1|(gϕ)n = pnδ(gϕ)n. Thus

W (γ)(g, gΨ) =W (γ − δ)(g, gϕ), ∀γ ∈ Q ∩ [0, 2δ].

Therefore I(M, gϕ,G) is the multiplicity of the Newton polygon slope −δ of (g, gϕ) andmoreover we have identities

W−i (g)⊗W (k) k = (W (0)(g, gΨ)∩ gi)⊗W (k) k = ∩m∈N(gΨ)mn(gi ⊗W (k) k).

3.4.14. Normalizer of W (δ)(g, gϕ). As W (δ)(g, gϕ) is a direct summand of g we canspeak about its normalizer P+

G (gϕ)′ in G. If W (δ)(g, gϕ) 6= 0, then for each element i ∈ Ithe greatest number δi ∈ [0,∞) ∩ Q such that we have W (δi)(g, gϕ) ∩ gi 6= 0 is δ itself(cf. Fact 3.4.11 and Subsection 3.4.1) and thus we have P+

G (gϕ)′B(k) = P+G (gϕ)B(k) (cf.

Subsection 2.6.5); thus P+G (gϕ) is a closed subgroup scheme of P+

G (gϕ)′ and this explains

our notation P+G (gϕ)′.

The group scheme P+1 (g) := P+

G (gϕ)′ ∩G1 is the normalizer of

W+1 (g) :=W (δ)(g, gϕ) ∩ g1

in G1. We have (cf. the rank parts of Subsections 3.4.1 and 2.7.1)

rkW (k)(W+1 (g)) = rkW (k)(W

−1 (g)).

Let S+

1 (g) be the reduced group of the special fibre P+1 (g)k of P+

1 (g).

3.4.15. Lemma. We have W+1 (g)⊗W (k) k ⊆ uη1

:= uη1⊗W (k) k = F 1(g1)⊗W (k) k.

Proof: We can assume that k = k. Let s := 2enδ ∈ N. The rank of the Zp-module

E1(g) := x ∈W+1 (g)|(gΨ)en(x) = psx is rkW (k)(W

+1 (g)). Let W+

1 (g) be the W (k)-span

of E1(g). As (gΨ)n = pnδ(gϕ)n, from (8) we get that psg1 ⊆ (gΨ)en(g1). If x ∈ g1 \ pg1,

62

then (gΨ)en(x) /∈ ps+1g1; moreover, if (gΨ)en(x) ∈ psg1, then (gΨ)(x) = (gϕ)(x)must be divisible by p and thus x modulo p must belong to uη1

. Therefore all Hodgenumbers of the σen-linear endomorphism (gΨ)en of W+

1 (g) are at most s. But thelength of W+

1 (g)/(gΨ)en(W+1 (g)) and W+

1 (g)/(gΨ)en(W+1 (g)) as artinian W (k)-modules

are equal. Thus all these Hodge numbers must be s. Therefore for each x ∈ W+1 (g)

we have (gΨ)en(x) ∈ psg1 and thus x modulo p belongs to uη1. Therefore we have

W+1 (g)⊗W (k) k ⊆ uη1

.

As p−s(gΨ)en is a σen-linear automorphism of W+1 (g) we get:

3.4.16. Corollary. If k = k, then W+1 (g) = W+

1 (g).

3.4.17. Lemma. The group S+

1 (g)∩ (Pk ∩G1,k) is a parabolic subgroup of G1,k and thusit contains a Borel subgroup of G1,k.

Proof: As P+G (gϕ) ∩ G1 6 P+

1 (g), S+

1 (g) contains P+G (gϕ)k and thus it is a parabolic

subgroup of G1,k. As the unipotent radical of Pk ∩ G1,k is commutative and has uη1=

F 1(g1) ⊗W (k) k as its Lie algebra (cf. Subsection 2.5.4), it centralizes uη1and therefore

(cf. Lemma 3.4.15) it also centralizes W+1 (g) ⊗W (k) k. Thus this unipotent radical is a

subgroup of S+

1 (g). Therefore the Lemma follows from the property 2.3.1 (a).

3.4.18. Lemma. We assume that W+1 (g) 6= 0. Then the following two pairs (P+

G (gϕ)k ∩G1,k,W

+1 (g)⊗W (k) k) and (P−

G (gϕ)k ∩G1,k,W−1 (g)⊗W (k) k) are quasi-opposite.

Proof: As G is split, adjoint and as δ is the greatest number δi ∈ [0,∞)∩Q such that wehave W (δi)(g, gϕ) ∩ gi 6= 0 (cf. Subsection 3.4.14), the hypotheses of Corollary 2.7.2 holdin the context of (M, gϕ,G). Thus the Lemma follows from Corollary 2.7.2 applied withi = 1 to (M, gϕ,G).

3.4.19. Notations and Weyl elements. For i ∈ I1 let

Pηiand Qηi

be the parabolic subgroup schemes of G1 whose Lie algebras are pηiand qηi

(respectively).They are unique (see Proposition 2.3.3) and therefore they are normalized by T1. Thus wehave T1 6 Pηi

∩Qηi. The parabolic subgroup schemes Pηi

and Qηiof G1 are opposite with

respect to T1. We have Pη1= P∩G1. From the very definitions of li’s and ηi’s and from the

numbering in the statement 3.4.5 (a), we get that F 1(gi) = σi−1ϕ (uηi

), F 0(gi) = σi−1ϕ (pηi

),

and F−1(gi) := gi ∩ F−1(g) = σi−1ϕ (nηi

). From this and Lemma 2.3.6 we easily get thatPηi

and Qηiare isomorphic to P ∩Gi and Q∩Gi (respectively). From this and Subsections

2.5.4 and 2.5.5, we get that the unipotent radicals of Pηiand Qηi

are commutative. Let

P ηiand Qηi

be the special fibres of Pηiand Qηi

(respectively).

For i ∈ I1 let pηi:= pηi

⊗W (k) k, qηi:= qηi

⊗W (k) k, nηi:= nηi

⊗W (k) k, and uηi:=

uηi⊗W (k) k. For i ∈ I and α ∈ Φ1 let gi := gi ⊗W (k) k and gα := gα ⊗W (k) k. We have

63

direct sum decompositions of t1 ⊗W (k) k-modules

g1 = pηi⊕ nηi

= qηi⊕ uηi

= t1 ⊗W (k) k⊕

α∈Φ1

gα.

We will call each element of G1(k) that normalizes T1,k, a Weyl element.

3.5. The proof of the property 3.3 (a)

We recall that G is split, adjoint, that I = S(1, n), and that Γ = (1 · · ·n). With thenotations of Subsections 3.4.7, 3.4.10, 3.4.13, and 3.4.19 and based on Subsection 3.4.13,to prove the property 3.3 (a) it suffices to show that

(9) rkW (k)(W−1 (g)) ≤ rkW (k)(∩x∈Enx) =

m−δ

n.

The proof of inequality (9) starts now and ends before Section 3.6. For i ∈ I let

Πηi∈ Endk(g1)

be the projector of g1 defined as follows. If i ∈ I1, then Πηiis the projection on nηi

along pηi. For i ∈ I \ I1 let Πηi

:= 1g1 . Let σϕ be the reduction modulo p of the σ-linear automorphism σϕ = ϕµ(p) of either M or g. As for each element i ∈ I1 we haveF 0(gi) = σi−1

ϕ (pηi), we get that

(10a) Ker(gΨ) = Ker(Ψ) = ⊕i∈I1(F0(gi)⊗W (k) k) = ⊕i∈I1σ

i−1ϕ (pηi

).

Moreover for each element i ∈ I1 we have

(10b) gΨ(x) = gσϕ(x), ∀x ∈ σi−1ϕ (nηi

).

Let s ∈ N. We use formulas (10a) and (10b) to follow the image of g1 under the s-thiterate of gΨ: by identifying (σϕ)

u−1(g1) with g1 for each u ∈ S(1, s) (this is allowed aswe deal here only with images), we come across k-vector spaces of the following form

(11) hsΠηis

(hs−1Πηis−1

(· · ·

(h2Πηi2

(h1nη1) · · ·

))),

where hu ∈ G1(k) for each u ∈ S(1, s). We emphasize that, here and in what followsuntil Section 3.7, the left upper index hu means inner conjugation by hu; it is used forconjugating subgroups as well as Lie subalgebras or elements of g1. Except for elementsor for intersections we do not use parentheses i.e., we write h1nη1

instead of h1(nη1), etc.

There exist formulas that express each hu in terms of g (like h1 is the component of g inG1(k), etc.) but they are not required in what follows.

Taking s ∈ e |I1|N to be big enough, the k-vector space of (11) becomesW−1 (g)⊗W (k)k

(cf. end of Subsection 3.4.13). We allow such an s ∈ e |I1|N to vary.

64

3.5.1. Proposition. We consider a k-linear endomorphism of g1 of the form ΠxhΠy,

where x, y ∈ E and h ∈ G1(k). Then there are elements h, l ∈ G1(k) and there exists aWeyl element w ∈ G1(k) such that we have

(12) ΠxhΠy = hΠx

wΠyl.

We also have the following elimination property

(13) Im(ΠxhΠx) ⊆ Im(Πx).

In addition, for every Weyl element w ∈ G1(k) we have the following inversion property

(14) ΠxwΠy = wΠy

w−1

Πxw.

Proof: Property (14) can be checked easily using the root decomposition of g1 with respectto T1,k. Property (13) is trivial. To prove the property (12) we need few preliminaries.

Let w0 ∈ G1(k) be a Weyl element such that w0B1,k is the opposite Bopp1,k of B1,k

with respect to T1,k. Let Πw−10 (x) be the projector of g1 on w−1

0 nx along w−10 px. We have

Πx = w0Πw−10 (x)

w−10 . The Bruhat decomposition forG1,k allows us to write w−1

0 h = bxwqy,

where bx ∈ Bopp1,k (k), w ∈ G1(k) is a Weyl element, and qy ∈ Qy(k) (see [Bo, Ch. V, §21]).

For α ∈ Φ1, let Ga,α,k be the Ga subgroup of G1,k that is normalized by T1,k andwhose Lie algebra is gα (cf. [DG, Vol. III, Exp. XXII, Thm. 1.1]). We write bx =

cxb(0)x b

(2)x , where cx ∈ T1(k), where b

(0)x =

∏α∈Φ−

1 ,gα⊆w

−10 qx

dx,α with dx,α ∈ Ga,α,k(k),

and where b(2)x =

∏α∈Φ−

1 ,gα⊆w

−10 ux

dx,α with dx,α ∈ Ga,α,k(k) (cf. [DG, Vol. III, Exp.

XXII, Prop. 5.6.1]). The element b(2)x ∈ Bopp

1,k (k) is also a k-valued point of the unipotent

radical of w−10 P x. Let b

(1)x := cxd

(1)x ∈ Bopp

1,k (k) ∩w−10 Qx(k). We have bx = b

(1)x b

(2)x . As

Bopp1,k 6 w−1

0 Px and Ker(Πw−10 (x)) = w−1

0 px = Lie(w−10 P x), the elements bx, b

(1)x , and b

(2)x

normalize Ker(Πw−10 (x)). Thus Πw−1

0 (x)b(1)x = b(1)x Πw−1

0 (x). For ux ∈ w−10 nx we have

(15) b(2)x (ux) = ux + vx, where vx ∈ w−10 px.

To check (15) we can assume that ux ∈ gβ1and b

(2)x ∈ Ga,β2,k(k), where β1 ∈ Φ and

β2 ∈ Φ−1 are such that gβ1

⊆ w−10 nx and gβ2

⊆ w−10 ux. To check (15) we can also assume

that k is a finite field. It is well known that we have (for instance, see [Va5, proof of Prop.3.6])

(16) b(2)x (ux)− ux ∈ ⊕s∈N, β1+sβ2∈Φ1gβ1+sβ2

.

As we have ⊕s∈N, β1+sβ2∈Φ1gβ1+sβ2

⊆ w−10 px, property (15) follows from the property

(16).

65

As b(2)x normalizes Ker(Πw−1

0 (x)) and due to the property (15), we have an identity

Πw−10 (x)

b(2)x = Πw−10 (x). Thus

Πw−10 (x)

bx = Πw−10 (x)

b(1)x b(2)x = b(1)x Πw−10 (x)

b(2)x = b(1)x Πw−10 (x).

We write qy = lyry, where ry is a k-valued point of the unipotent radical of Qy and

where ly is a k-valued point of the Levi subgroup P y ∩ Qy of Qy. As this last unipotentradical is commutative (cf. Subsection 3.4.19), ry fixes ny = Im(Πy) and thus we haveqyΠy = lyΠy = Πy

ly . Using the above formulas for Πx, Πw−10 (x)

bx , and qyΠy, we get that

ΠxhΠy = w0Πw−1

0 (x)w−1

0 hΠy = w0Πw−10 (x)

bxwqyΠy = w0Πw−10 (x)

bxwΠyly

= w0b(1)x Πw−1

0 (x)wΠy

ly = w0b(1)x w−1

0 w0Πw−10 (x)

w−10 w0wΠy

ly = w0b(1)x w−1

0 Πxw0wΠy

ly .

Thus we can take w := w0w, h := w0b(1)x w−1

0 , and l = ly. This proves the property (12).

3.5.2. Theorem. We assume that X 6= E6 and that E = x, y has two elements. Then

dimk(Πx(hny)) ≤ dimk(nx ∩ ny).

Thus, if h is a Weyl element, then the dimension dimk(Πx(hny)) is maximal if and only

if the dimension dimk(hpy ∩ px) is maximal (i.e., if and only if the intersection hpy ∩ px

contains the Lie algebra of a Borel subgroup of G1,k).

Proof: We can assume that h is a Weyl element, cf. property (12). We have

dimk(Πx(hny)) = dimk(ny)− dimk(

hny ∩ px) = dimk(ny)− dimk(px) + dimk(hpy ∩ px).

We denote also by h the image of h in the Weyl group WX of G1,k of automorphisms ofeither Φ1 or T1,k defined by Weyl elements of G1(k).

In this paragraph we check that if hpy∩px contains the Lie algebra of a Borel subgroup

of G1,k, then dimk(hpy∩px) = dimk(py∩px). To check this we can assume that Lie(B1,k) ⊆

hpy∩px and thus that B1,k 6 hP y∩P x (cf. Proposition 2.3.3). From the property 2.3.1 (b)

we get that hP y = P y and thus we have hpy = py as well as dimk(hpy∩px) = dimk(py∩px).

Based on the last two paragraphs, to end the proof of the Theorem it suffices tocheck: (i) that dimk(

hny ∩ nx) ≤ dimk(ny ∩ nx), and (ii) that if we have an equalitydimk(

hny ∩ nx) = dimk(ny ∩ nx), then we can write h = h1h2 with h1 as h2 as Weylelements of G1(k) that normalize px and py (respectively), and therefore hpy ∩ px containsthe Lie algebra of a Borel subgroup of G1,k.

Below, if either h1 or h2 is the identity element, then we will not mention it. Onechecks that a suitable Weyl element h1 (resp. h2) normalize px (resp. py) by an easy anddirect verification that it normalizes nx (resp. ny). We consider three Cases.

66

Case 1: X = Aℓ. We can assume that 1 ≤ y < x ≤ ℓ, cf. property (14). We use thenotations of [Bou2, plate I] for Φ1; thus ε1, . . . , εℓ+1 is the standard R-basis for Rℓ+1

and Φ1 = εi − εj |1 ≤ i, j ≤ ℓ+ 1, i 6= j. For η ∈ E ⊆ S(1, ℓ), the subset of Φ1 of roots ofnη relative to T1,k is εj − εi|1 ≤ i ≤ η < j ≤ ℓ+1. We denote also by h the permutationof S(1, ℓ+ 1) such that for all i, j ∈ S(1, ℓ+ 1) with i 6= j we have hgei−ej = gh(ei)−h(ej).

Let ny,x(h) :=∣∣(i, j) ∈ S(1, ℓ+ 1)2|i ≤ y < j and h(i) ≤ x < h(j)

∣∣. We have an equalitydimk(

hny ∩ nx) = ny,x(h). But ny,x(h) = yx, where y := |i ∈ S(1, y)|h(i) ≤ x| ∈ S(0, y)and where x := |j ∈ S(y + 1, ℓ+ 1)|x < h(j)| ∈ S(0, ℓ+ 1− x).

Thus ny,x(h) ≤ y(ℓ+1− x) = dimk(ny ∩ nx). The equalities ny,x(h) = y(ℓ+1− x) =dimk(ny ∩ nx) hold if and only if h(S(1, y)) ⊆ S(1, x). If h(S(1, y)) ⊆ S(1, x), then leth−11 be a permutation of S(1, ℓ+1) that normalizes S(1, x) and that takes h(S(1, y)) ontoS(1, y); if h2 := h−1

1 h, then h2 as its own turn normalizes S(1, y) and thus every Weylelement that defines h2 (resp. h1), normalizes py (resp. px).

Case 2: X = Dℓ and E = ℓ − 1, ℓ. We use the notations of [Bou2, plate IV] for Φ1;thus ε1, . . . , εℓ is the standard R-basis for Rℓ. We have:

– the subset of Φ1 of roots of nℓ−1 relative to T1,k is −εi+εℓ|1 ≤ i < ℓ∪−εi−εj |1 ≤i < j < ℓ and the subset of Φ1 of roots of nℓ relative to T1,k is −εi − εj |1 ≤ i < j ≤ ℓ.

We can assume that (x, y) = (ℓ−1, ℓ), cf. property (14). If h(εℓ) = εℓ, thenhny∩nx ⊆

ny∩nx. Moreover, if h(εℓ) = εℓ, then we have hny∩nx = ny∩nx if and only if h normalizesε1, . . . , εℓ−1; this implies that h1 := h normalizes both px and py. Thus we can assumethat h(εℓ) 6= εℓ.

For the last part of Case 2 we will write dimk(hny ∩ nx) = d1 + d2, where d1 (resp.

d2) is the number of vectors −εi − εj , 1 ≤ i < j ≤ ℓ, mapped by h to vectors of the form−εi + εℓ, 1 ≤ i < ℓ (resp. of the form −εi − εj , 1 ≤ i < j < ℓ).

If we have h(εℓ) = −εℓ, then dimk(hny ∩ nx) = s+ (s−1)s

2, where

s := |i ∈ S(1, ℓ− 1)|h(εi) ∈ ε1, . . . , εℓ−1| .

If s = ℓ − 1, then −εℓ = h(ε1), . . . , h(εℓ) ∩ −ε1, . . . ,−εℓ. Thus h /∈ WDℓ(cf. the

structure of WDℓas a semidirect product; see [Bou2, plate IV]) and therefore the equality

s = ℓ− 1 leads to a contradiction. Thus we have s ≤ ℓ− 2. If s = ℓ− 2 let h2 ∈ G1(k) bea Weyl element whose image in WDℓ

is the transposition (εℓεi0), where i0 ∈ S(1, ℓ− 1) isthe unique number such that we have h(εi0) ∈ −ε1, . . . ,−εℓ−1; we have h(εi0) = −εi0 .Thus h2 (resp. h1 := hh−1

2 ) normalizes py (resp. px).

If we have h(εℓ) ∈ −ε1, . . . ,−εℓ−1, then dimk(hny ∩ nx) = s + (s−1)s

2 , where agains := |i ∈ S(1, ℓ− 1)|h(εi) ∈ ε1, . . . , εℓ−1|. As h(−εℓ) ∈ ε1, . . . , εℓ−1, we have s ≤ℓ− 2. If s = ℓ− 2, then h1 := h normalizes px.

If we have h(εℓ) = εj , with j ∈ S(1, ℓ− 1), then dimk(hny ∩ nx) = s + (s−1)s

2 , wheres := |i ∈ S(1, ℓ− 1)|h(εi) ∈ ε1, . . . , εj−1, εj+1, . . . , εℓ−1|. Thus again we have s ≤ ℓ−2.If s = ℓ− 2, then h2 := h normalizes ε1, . . . , εℓ and therefore also py.

Thus this Case 2 follows from the relations dimk(ny ∩ nx) = (ℓ−2)(ℓ−1)2 = (ℓ − 2) +

(ℓ−3)(ℓ−2)2 ≥ s+ (s−1)s

2 .

67

Case 3: X = Dℓ and either E = 1, ℓ− 1 or E = 1, ℓ. We have:

– the subset of Φ1 of roots of n1 relative to T1,k is −ε1 ± εj |2 ≤ j ≤ ℓ.

We can assume that y = 1, cf. property (14). It suffices to consider the case whenE = 1, ℓ−1; thus x = ℓ−1. If h(ε1) ∈ −ε1,−ε2, . . . ,−εℓ−1, εℓ, then hn1∩nℓ−1 = 0 andwe are done. We now assume that h(ε1) ∈ ε1, ε2, . . . , εℓ−1,−εℓ. We have dimk(

hn1 ∩nℓ−1) = ℓ− 1 = dimk(n1 ∩ nℓ−1). Let h1 ∈ G1(k) be a Weyl element whose image in WDℓ

is the element (ε1h(ε1)); it is either the identity or a transposition. If h2 := h−11 h, then

h2(ε1) = ε1. Thus h2 (resp. h1) normalizes p1 (resp. pℓ−1).

We conclude that in all three Cases, sentences (i) and (ii) of the beginning paragraphof the proof of the Theorem are true.

3.5.3. Proof of (9) for X = E6 and |E| = 2. We consider the case when X = E6

and the set E = x, y has two elements. We have x, y = 1, 6. The structure of theWeyl group WE6

is not given in [Bou2, plate V], though [Bou2, Ch. VI, §4.12 or Ch. VI,Excs. for §4, Exc. 2)] gives some information on it. One can check using computers thatTheorem 3.5.2 also holds for X = E6. We will prove inequality (9) using a new approach.

We can assume that W+1 (g) 6= 0 and therefore that W (δ)(g, ϕ) 6= 0. Thus we have

P+G (gϕ)k ∩ G1,k 6 S

+

1 (g), cf. Subsection 3.4.14. From Lemma 3.4.17 we get that by

replacing gϕ with g1gϕg−11 , where g1 ∈ P (W (k)), we can assume that B1,k 6 S

+

1 (g).Let T 1(g) be a maximal torus of B1,k ∩ P+

G (gϕ)k. Let B1(g) be a Borel subgroup of

P+G (gϕ)k ∩G1,k that contains T 1(g). As P+

G (gϕ)k ∩G1,k 6 S+

1 (g), there exists an element

h+(g) ∈ S

+

1 (g)(k) such that under inner conjugation it takes B1(g) to B1,k and T 1(g)to T1,k. We consider an additive function J1(g) : Φ1 → Q that has the following threeproperties (cf. Remark 2.5.2):

(a) its image is the set of Newton polygon slopes of (g[ 1p ], gϕ);

(b) if α ∈ Φ1, then we have J1(g)(α) ≥ 0 if and only if gα ⊆ Lie(h+(g)(P+

G (gϕ)k∩G1,k))

(thus, if α ∈ Φ+1 , then from the relations B1,k 6 h

+(g)(P+

G (gϕ)k ∩ G1,k) 6 S+

1 (g) we getthat J1(g)(α) ≥ 0);

(c) if Φ1(δ) is the subset of Φ1 formed by those roots α for which we have an inclusiongα ⊆ W+

1 (g) ⊗W (k) k, then we have J1(g)(α) = δ for all α ∈ Φ1(δ) (the subset Φ1(δ) is

well defined as W+1 (g)⊗W (k) k is a T1,k-module)

Strictly speaking in order to get J1(g) we would have to: (i) lift T 1(g) to a maximaltorus T1(g) of P

+G (gϕ)∩G1, (ii) choose a maximal torus T (g) of P+

G (gϕ) whose intersectionwith G1 is T1(g), and (iii) get an additive function J(g) as in the proof of Lemma 2.5.1 butin the context of (g[ 1

p], gϕ) and of T (g)B(k) 6 P+

G (gϕ)B(k). Then J1(g) is obtained from

J(g) via inner conjugation through an element h+I (g) ∈ G(W (k)) that lifts h+(g) and that

satisfies the identity h+I (g)T (g)h+I (g)

−1 = T and via a natural restriction to the direct

factor Φ1 of ∪i∈IΦi. Property (c) is implied by the fact that h+(g) ∈ S

+

1 (g)(k) normalizesW+

1 (g)⊗W (k) k.

68

If J1(g)(αη1) > 0, then from (b) we get that J1(g)(α) < 0 for every root α ∈ Φ−

1 suchthat gα ⊆ nη1

(i.e., such that gα 6⊆ pη1); from this and the property (b) we get an inclusion

Lie(h+(g)(P+

G (gϕ)k∩G1,k)) ⊆ pη1= Lie(P η1

) and therefore (cf. Proposition 2.3.3) we haveh+(g)(P+

G (gϕ)k ∩G1,k) 6 P η1.

We first assume that h+(g)(P+

G (gϕ)k ∩ G1,k) is not a subgroup of P η1= Pk ∩ G1,k.

Therefore we have J1(g)(αη1) = 0, cf. previous paragraph. Thus we have

(17) (Φ1(δ)− αη1) ∩ Φ1 ⊆ Φ1(δ).

The first basic property of Φ1(δ) is that each root in Φ1(δ) is a linear combination ofα1, . . . , α6 with non-negative integers such that the coefficient of αη1

is 1, cf. property(c) and Lemma 3.4.15. From this and the inclusion (17) we get the second basic propertyof Φ1(δ): we have (Φ1(δ) − αη1

) ∩ Φ1 = ∅. These two basic properties imply that Φ1(δ)has at most 6 elements. More precisely, by assuming to fix the ideas that η1 = 1 andby using [Bou, plate V], from these two basic properties of Φ1(δ) we get the inclusionΦ1(δ) ⊆ α1, α1 + α2 + 2α3 + 2α4 + α5 ∪ α0 + α1 + · · · + α6|α0 ∈ U0, where the setU0 := α3 + α4, α3 + α4 + α5, α3 + 2α4 + α5, α2 + α3 + 2α4 + α5 has four roots.

As dimk(n1∩n6) = 8, to prove the inequality (9) we can assume that rkW (k)(W+1 (g)) =

rkW (k)(W−1 (g)) ≥ 8. Therefore h

+(g)(P+

G (gϕ)k ∩G1,k) must be a subgroup of P η1= Pk ∩

G1,k, cf. previous paragraph. Thus we have T1,k 6 B1,k 6 h+(g)(P+

G (gϕ)k ∩ G1,k) 6 P η1.

In addition, we have h+(g)(W+

1 (g) ⊗W (k) k) = W+1 (g) ⊗W (k) k ⊆ uη1

(cf. Lemma 3.4.15

and the relation h+(g) ∈ S1(g)(k)).

Let h−(g) ∈ G1(k) be such that h

−(g)(P−

G (gϕ)k∩G1,k) is the opposite ofh+(g)(P+

G (gϕ)k∩

G1,k) with respect to T1,k, cf. Subsection 2.7.1. As h+(g)(P+

G (gϕ)k ∩G1,k) 6 P η1, we have

h−(g)(P−

G (gϕ)k ∩ G1,k) 6 Qη1. For the rest of this paragraph we check that we have

an inclusion h−(g)(W−

1 (g) ⊗W (k) k) ⊆ nη1. We consider four pairs: (P η1

, uη1), (Qη1

, nη1),

(h+(g)(P+

G (gϕ)k∩G1,k),h+(g)(W+

1 (g)⊗W (k)k)), and (h−(g)(P−

G (gϕ)k∩G1,k),h−(g)(W−

1 (g)⊗W (k)

k)). The first two pairs are opposite with respect to T1,k and (cf. Lemma 3.4.18) the last

two pairs are quasi-opposite. Thus, as h+(g)(P+

G (gϕ)k∩G1,k) andh−(g)(P−

G (gϕ)k∩G1,k) are

opposite with respect to T1,k, we get thath+(g)(W+

1 (g)⊗W (k)k) andh−(g)(W−

1 (g)⊗W (k)k)are opposite with respect to T1,k (cf. the very definition of quasi-opposition). Ash+(g)(W+

1 (g) ⊗W (k) k) ⊆ uη1, via the opposition with respect to T1,k we get that

h−(g)(W−

1 (g)⊗W (k) k) ⊆ nη1.

For simplicity, let h1 := h−(g)−1 ∈ G1(k). We have P−

G (gϕ)k ∩ G1,k 6 h1Qη1and

W−1 (g) ⊗W (k) k ⊆ h1nη1

. Working with an arbitrary element i ∈ I1 such that we have(η1, ηi) = (1, 6) and due to the circular aspect of either property 3.4.2 (a) or sequence (11), asimilar argument shows that there exists h2 ∈ G1(k) such that P−

G (gϕ)k∩G1,k 6h2 Q6 andW−

1 (g)⊗W (k)k ⊆ h2nηi= h2n6. Thus P

−G (gϕ)k∩G1,k 6 h1Q1∩

h2Q6 andW−1 (g)⊗W (k)k ⊆

h1n1 ∩ h2n6. Let h3 ∈ G1(k) be such that h3(h1Q1 ∩h2Q6) contains a Borel subgroup of

69

Q1 ∩ Q6. From the property 2.3.1 (b) we get that h3h1Q1 = Q1 and that h3h2Q6 = Q6.Thus h3h1 ∈ Q1(k) and h3h2 ∈ Q6(k). But we can replace h1 (resp. h2) by an arbitraryelement of h1Q1(k) (resp. of h2Q6(k)). Thus we can assume that h1 = h2 = h−1

3 andtherefore we have h1n1∩h2n6 = h1(n1∩n6). Thus rkW (k)(W

−1 (g)) ≤ dimk(

h1(n1∩n6)) = 8.Therefore the inequality (9) holds if X = E6 and |E| = 2.

For a later use we mention that if rkW (k)(W−1 (g)) = 8, then we have an identity

W−1 (g)⊗W (k) k = h1(n1 ∩ n6).

3.5.4. Proof of (9). We now prove inequality (9) in three Steps.

Step 1. If E = x, then inequality (9) holds as rkW (k)(W−1 (g)) = rkW (k)(W

+1 (g)) ≤

rkW (k)(ux) = rkW (k)(nx) (cf. Lemma 3.4.15 for the inequality part).

Step 2. We assume that E = x, y has two elements. If X = E6, then the inequality(9) holds (cf. end of Subsection 3.5.3). Let now X 6= E6. Each sequence (11) that“computes” W−

1 (g) ⊗W (k) k has a subsequence ΠxhΠy with h ∈ G1(k). Therefore we

have rkW (k)(W−1 (g)) ≤ dimk(Πx(

hny)) ≤ dimk(nx ∩ ny) (cf. Theorem 3.5.2 for the lastinequality). Thus inequality (9) holds.

Step 3. We assume that |E| ≥ 3. Using repeatedly properties (12), (13), and (14), weget that there exists a sequence (11) that “computes” W−

1 (g) ⊗W (k) k and that has asubsequence:

– of the form Πx|E|

hΠx1if we are in the case (a) and x1 and x|E| are as in Subsection

3.4.8, or

– of the form ΠℓhΠℓ−1

hΠ1 if we are in the case (d) and E = 1, ℓ− 1, ℓ.

In the case (a), as in Step 2 we argue that rkW (k)(W−1 (g)) ≤ dimk(Πx|E|

hΠx1) ≤

dimk(nx|E|∩ nx1

) = dimk(∩x∈Enx); thus the inequality (9) holds.

We now assume we are in the case (d) with E = 1, ℓ− 1, ℓ. We check that

(18) dimk(Im(ΠℓhΠℓ−1

hΠ1)) ≤ ℓ− 2 = dimk(r1 ∩ rℓ−1 ∩ rℓ)

and that if equality holds in (18), then pℓ ∩hpℓ−1 ∩ hp1 contains the Lie algebra of a

Borel subgroup of G1,k. We can assume that h ∈ G1(k) is the identity element, cf. Case

3 of Theorem 3.5.2. We can also assume that h is a Weyl element (the proof of this isthe same as the proof of property (12), the role of the projector Πy being replaced by

the one of the new projector Πℓ−1Π1). Thus Im(ΠℓhΠℓ−1

hΠ1) = nℓ ∩ h(nℓ−1 ∩ n1). Weuse the notations of Cases 2 and 3 of Theorem 3.5.2. If h(ε1) ∈ −ε1, . . . ,−εℓ, then

nℓ ∩ h(nℓ−1 ∩ n1) = 0. If h(ε1) ∈ ε1, . . . , εℓ, then dimk(nℓ ∩ h(nℓ−1 ∩ n1)) is the numbers of elements of the set −ε2, . . . ,−εℓ−1, εℓ sent by the element of WDℓ

defined by hto the set −ε1, . . . ,−εℓ−1,−εℓ. As in the Case 2 of the proof of Theorem 3.5.2, thestructure of WDℓ

implies that s ≤ ℓ − 2. Therefore (18) holds. Thus rkW (k)(W−1 (g)) ≤

dimk(Im(ΠℓhΠℓ−1

hΠ1)) ≤ dimk(r1 ∩ rℓ−1 ∩ rℓ). Therefore the inequality (9) holds.

To check the part on the Lie algebra of a Borel subgroup of G1,k, we can assume that

the equality holds in (18) and thus that h(ε1) ∈ ε1, . . . , εℓ and s = ℓ− 2. If h(ε1) = εℓ,

70

then h normalizes pℓ. If h(ε1) = εs with s ∈ S(1, ℓ − 1), then as s = ℓ − 2 there exists anumber s0 ∈ S(1, ℓ) such that h(εℓ) = −εs0 . Let h2 ∈ G1(k) be a Weyl element such thatwe have h2(εℓ) = −εs0 and h2(εi) = εi for all i ∈ S(1, ℓ−1)\s0. Then h1 := hh−1

2 (resp.h2) normalizes pℓ (resp. nℓ−1+n1 and therefore also pℓ−1∩p1). Thus, if the equality holds

in (18), then pℓ ∩h(pℓ−1 ∩ p1) contains a Borel Lie subalgebra of Lie(G1,k).

This ends the proofs of inequalities (18) and (9). It also ends the proof of the property3.3 (a).

3.6. Formulas for group normalizers

See Lemma 3.4.15 and Corollary 3.4.16 for the meanings ofW+1 (g), P+

1 (g), and S+

1 (g).In Subsection 3.6.1 we recall some known facts that are needed in Theorem 3.6.2 to prove

a key result on S+

1 (g). See Corollaries 3.6.3 and 3.6.4 for two applications of Theorem3.6.2 that describe the intersections P+

G (ϕ)∩G1 and P+G (ϕ)∩G1 and that present as well

a variant of Theorem 3.6.2. Let the sets E and E be as in Subsection 3.4.8.

3.6.1. Chevalley rule. Let α, β ∈ Φ1 with α 6= −β. If α+ β /∈ Φ1 let gα+β := 0 andgα+β := 0. We have (cf. Chevalley rule of [DG, Vol. III, Exp. XXIII, Cor. 6.5]):

(a) We always have inclusions [gα, gβ] ⊆ gα+β and [gα, gβ ] ⊆ gα+β . These inclusionsare identities if α+β ∈ Φ1 and if α and β generate an A2 root subsystem of Φ1. Moreoverwe always have an identity [gα[

1p], gβ[

1p]] = gα+β[

1p].

In what follows we will apply the property (a) in the following concrete context:

(b) If |E| ≥ 2, then X ∈ Aℓ|ℓ ≥ 2∪ Dℓ|ℓ ≥ 4∪ E6 and thus all roots of Φ1 haveequal length. Thus, if α, β, α + β ∈ Φ1, then α and β generate an A2 root subsystem ofΦ1 and therefore (cf. property (a)) we have identities [gα, gβ ] = gα+β and [gα, gβ ] = gα+β.

(c) Let s1 and s2 be two B(k)-vector subspaces of g[ 1p] that are opposite with respect

to TB(k). It is easy to see that the last part of the property (a) implies that [s1, s1] = 0 ifand only if [s2, s2] = 0.

3.6.2. Theorem. We assume that (5) holds, that G is split, adjoint, that I = S(1, n), andthat Γ = (1 · · ·n). We also assume that I(M, gϕ,G) = I(M,ϕ,G). Then there exists anelement h ∈ P η1

(k) such that we have W+1 (g)⊗W (k) k =h (∩x∈Eux). Moreover the reduced

scheme S+

1 (g) of the normalizer P+1 (g)k ofW+

1 (g)⊗W (k)k in G1,k, is the parabolic subgrouph(∩x∈EP x) of G1,k.

1

Proof: We check that if S+

1 (g) = h(∩x∈EP x), then h ∈ P η1(k). The group h(∩x∈EP x)

contains a Borel subgroup of P η1, cf. Lemma 3.4.17. Let h ∈ P η1

(k) be such that

B1,k 6 hh(∩x∈EP x). We have ∩x∈EP x = hh(∩x∈EP x), cf. property 2.3.1 (b). Thus

hh ∈ (∩x∈EP x)(k) 6 P η1(k). Therefore h ∈ P η1

(k). Thus, once we show the existence

of h, to check the last statement of the Theorem (on S+

1 (g)) we can assume that h is the

1 It is not difficult to check that P+1 (g)k is h(∩x∈EP x), except when p = 2 and X = B1.

71

identity element. As I(M, gϕ,G) = I(M,ϕ,G), we have rkW (k)(W−1 (g)) = rkW (k)(∩x∈Enx)

(cf. Subsections 3.4.12 and 3.4.13). Thus we also have rkW (k)(W+1 (g)) = rkW (k)(∩x∈Eux),

cf. Subsection 3.4.14. We consider four Cases.

Case 1: E = x. Thus η1 = x. We have rkW (k)(W+1 (g)) = rkW (k)(ux). Therefore

W+1 (g) ⊗W (k) k = ux (cf. Lemma 3.4.15) and thus the element h exists and we have

P x 6 S+

1 (g) 6 P+1 (g)k. We have S

+

1 (g)(k) = P x(k), cf. Lemma 2.3.4 applied over k.

Thus S+

1 (g) = P x.

Case 2: E = x, y. We have rkW (k)(W+1 (g)) = rkW (k)(uy ∩ ux). There exists an element

h ∈ G1(k) such that W−1 (g) ⊗W (k) k is included in a G1(k)-conjugate of hny ∩ nx (see

beginning of Subsection 3.5). If X 6= E6, then by reasons of ranks we can assume that

W−1 (g)⊗W (k)k is hny∩nx and that h is a Weyl element such that hpy∩px contains the Lie

algebra of a Borel subgroup of G1,k (cf. Theorem 3.5.2). Up to P x(k)-conjugation we can

assume that this Borel subgroup is B1,k and thus we have hpy = py (cf. property 2.3.1 (b)

and Proposition 2.3.3); as h is a Weyl element, we also get that hny∩nx = ny∩nx. Thus we

have W−1 (g)⊗W (k) k = h−

(nx ∩ ny) for some element h− ∈ G1(k). This last property also

holds for X = E6, cf. end of Subsection 3.5.3. The pairs (P+G (gϕ)k ∩ G1,k,W

+1 (g) ⊗W (k)

k) and (P−G (gϕ)k ∩ G1,k,W

−1 (g) ⊗W (k) k) are quasi-opposite, cf. Lemma 3.4.18. The

pairs (P y ∩ Px, uy ∩ ux) and (Qy ∩ Qx, ny ∩ nx) are opposite with respect to T1,k. As

W−1 (g)⊗W (k) k = h−

(nx ∩ ny), there exists an element h := h+ ∈ G1(k) such that we have

W+1 (g)⊗W (k) k =h (ux ∩ uy) (cf. Subsection 2.3.8 (h)).

We are left to show that the reduced normalizer P x,y of ux∩uy in G1,k is P x∩P y. Tofix the ideas, we can assume that x < y. As P x and P y normalize ux and uy (respectively),we have P x ∩ P y 6 P x,y. Thus P x,y is a parabolic subgroup of G1,k. From the structureof parabolic subgroups of G1,k (see [Bo, Ch. IV, Prop. 14.22 (ii)]), we get that the only

parabolic subgroups of G1,k that contain P x ∩ P y are P x ∩ P y, P x, P y, and G1,k.

We show that the assumption P x,y 6= P x ∩ P y leads to a contradiction. This as-sumption implies that Lie(P x,y) contains either Lie(Px) or Lie(P y) and thus it containsg−α0

, where α0 ∈ αx, αy. If X = Dℓ and E = ℓ − 1, ℓ, let αx,y = αℓ−2 + αℓ−1 + αℓ;

otherwise, let αx,y :=∑y

s=x αs. We have gαx,y⊆ ux ∩ uy. Let β := αx,y − α0 ∈ Φ+

1 .We have gβ 6⊆ ux ∩ uy. From the property 3.6.1 (b) we get that [gαx,y

, g−α0] = gβ. The

last two sentences imply that Lie(Px,y) does not normalize ux ∩ uy. Contradiction. ThusP x,y = P x ∩ P y.

Case 3: |E| ≥ 3 and X = Aℓ. This Case follows from Case 2, cf. Step 3 of Subsection3.5.4.

Case 4: |E| = 3 and X = Dℓ. Based on Step 3 of Subsection 3.5.4, as in Case 2 we arguethat W+

1 (g) ⊗W (k) k is h(u1 ∩ uℓ−1 ∩ uℓ), where h ∈ G1(k). We are left to show that the

reduced normalizer P 1,ℓ−1,ℓ of u1 ∩ uℓ−1 ∩ uℓ in G1,k is P 1 ∩ P ℓ−1 ∩ P ℓ. As in Case 2 weargue that P 1 ∩ P ℓ−1 ∩ P ℓ 6 P 1,ℓ−1,ℓ and thus that P 1,ℓ−1,ℓ is one of the following eightparabolic subgroups of G1,k: P 1∩P ℓ−1∩P ℓ, P 1∩P ℓ−1, P 1∩P ℓ, P ℓ−1∩P ℓ, P 1, P ℓ−1, P ℓ,

72

and G1,k. We show that the assumption P 1,ℓ−1,ℓ 6= P 1∩P ℓ−1∩P ℓ leads to a contradiction.This assumption implies that Lie(P 1,ℓ−1,ℓ) contains either Lie(P 1 ∩P ℓ−1) or Lie(P 1 ∩P ℓ)or Lie(P ℓ−1 ∩ P ℓ). Thus Lie(P 1,ℓ−1,ℓ) contains g−α0

, where α0 ∈ α1, αℓ−1, αℓ. Let

α1,ℓ−1,ℓ :=∑ℓ

s=1 αs. As β := α1,ℓ−1,ℓ − α0 ∈ Φ+1 and as gβ 6⊆ u1 ∩ uℓ−1 ∩ uℓ, as in Case 2

we reach a contradiction. Thus P 1,ℓ−1,ℓ = P 1 ∩ P ℓ−1 ∩ P ℓ.

We now list two Corollaries of Theorem 3.6.2.

3.6.3. Corollary. We assume that (5) holds, that G is split, adjoint, that I = S(1, n),and that Γ = (1 · · ·n). We have

P+G (ϕ) ∩G1 = ∩x∈EPx and P+

G (ϕ) ∩G1 = ∩x∈EPx.

Thus the nilpotent classes of U+G (ϕ) and U+

G (ϕ) are |E| and |E | (respectively). Moreover

P−G (ϕ) ∩G1 = ∩x∈EQx and P−

G (ϕ) ∩G1 = ∩x∈EQx

and the nilpotent classes of U−G (ϕ) and U−

G (ϕ) are |E| and |E | (respectively).

Proof: We will check here only the + part, as the − part will implicitly follow from the+ part via oppositions with respect to T1 and T . From (7) we get the following identityLie(P+

G (ϕ)∩G1) = ∩x∈Epx. Thus Lie(P+G (ϕ)∩G1) = Lie(∩x∈EPx). Therefore the identity

P+G (ϕ) ∩G1 = ∩x∈EPx follows from Lemma 2.3.6.

From Theorem 3.6.2 applied with g = 1M and h = 1M/pM , we get that the special

fibres of P+G (ϕ) ∩ G1 and ∩x∈EPx coincide. As these two parabolic subgroup schemes of

G1 contain T1, they coincide (cf. Subsection 2.3.8 (c)).

For the part on nilpotent classes, it is enough (cf. property 3.4.2 (a)) to compute thenilpotent classes of the Lie algebras of the unipotent radicals UE and UE of the intersections∩x∈EPx and ∩x∈EPx (respectively). But Lie(UE) (resp. Lie(UE)) is the direct sum of those

gα that correspond to roots α ∈ Φ+1 that contain at least one of the roots αx with x ∈ E

(resp. with x ∈ E) in their expressions. Thus the nilpotent class of Lie(UE) (resp. ofLie(UE)) is |E| (resp. |E |), as one can easily check based on the property 3.6.1 (a).

3.6.4. Corollary. We assume that (5) holds, that G is split, adjoint, that I =S(1, n), and that Γ = (1 · · ·n). Then for each element i ∈ I the reduced normalizer of(W (−δ)(g, ϕ)⊗W (k) k) ∩ Lie(Gi,k) in Gi,k is P−

G (ϕ)k ∩Gi,k.

Proof: We have (W (−δ)(g, ϕ) ⊗W (k) k) ∩ Lie(G1,k) = ∩x∈Enx. Thus by interchanging inthe proof of Theorem 3.6.2 the roles of positive and negative Newton polygon slopes, weget that the reduced normalizer of (W (−δ)(g, ϕ)⊗W (k) k) ∩ Lie(G1,k) in G1,k is ∩x∈EQx.

From this and the expression of P−G (ϕ) ∩G1 in Corollary 3.6.3, we get that the Corollary

holds for i = 1. Let σϕ be as in the second paragraph of Section 3.5. As σϕ normalizes

both W (−δ)(g, ϕ)⊗W (k) k and (cf. end of Subsection 3.4.2) Lie(P−G (ϕ)k), the normalizer

of (W (−δ)(g, ϕ)⊗W (k) k) ∩ Lie(Gi,k) in Gi,k is σi−1ϕ (P−

G (ϕ)k ∩ G1,k)σ1−iϕ . We know that

σi−1ϕ (P−

G (ϕ)k ∩G1,k)σ1−iϕ and P−

G (ϕ)k∩Gi,k have equal Lie algebras (cf. last paragraph of

73

Subsection 3.4.2) and thus from Proposition 2.3.3 we get that σi−1ϕ (P−

G (ϕ)k ∩G1,k)σ1−iϕ =

P−G (ϕ)k ∩Gi,k. The Corollary follows from the last two sentences.

3.7. The proof of the property 3.3 (b)

We come back to Section 3.3. Thus the reductive group scheme G is quasi-split butit is not necessarily split or adjoint. Moreover, the permutation Γ does not necessarily acttransitively on I. The proof of the property 3.3 (b) requires first a Theorem.

3.7.1. Theorem. We recall that (5) holds. Let g ∈ P+G (ϕ)(W (k)). We write g = ur,

where u ∈ U+G (ϕ)(W (k)) and r ∈ L0

G(ϕ)(W (k)). Then the following six properties hold:

(a) The triples (M,ϕ, L0G(ϕ)) and (M, rϕ, L0

G(ϕ)) are Shimura p-divisible objects.

(b) The group scheme U+G (ϕ) is a closed subgroup scheme of U+

G (gϕ).

(c) There exists an element u ∈ U+G (ϕ)(W (k)) such that we have ugϕu−1 = rϕ.

(d) We have P+G (gϕ) = P+

G (ϕ).

(e) We have I(M, gϕ,G) = I(M,ϕ,G).

(f) The triple (M, gϕ,G) is Sh-ordinary if and only if (M, rϕ, L0G(ϕ)) is Sh-ordinary.

Proof: As ϕ normalizes Lie(P+G (ϕ)B(k)) and t, it also normalizes Lie(L0

G(ϕ))[1p]. Therefore

(a) is implied by the fact that the Hodge cocharacter µ : Gm → G of (M,ϕ,G) factorsthrough T and thus also through L0

G(ϕ).

To prove (b), (d), and (e) we can assume that G is split, adjoint, that I = S(1, n),and that Γ = (1 · · ·n). As U+

G (ϕ) is a unipotent, smooth group scheme, there exist anon-negative integer v and a unique ideal filtration

0 = i0 ⊆ i1 ⊆ . . . ⊆ iv = Lie(U+G (ϕ))

of Lie(U+G (ϕ)) such that by induction on j ∈ S(0, v− 1) we have that the ideal ij+1 of iv is

maximal with the properties that it is normalized by U+G (ϕ), that it contains ij , and that

moreover U+G (ϕ) acts trivially on ij+1/ij. As iv is normalized by P+

G (ϕ) and left invariant

by ϕ, this ideal filtration is also normalized by P+G (ϕ) and left invariant by ϕ. Thus this

filtration is also left invariant by gϕ. To check (b) it suffices to show that iv ⊆ Lie(U+G (gϕ)),

cf. Lemma 2.3.6. But we have iv ⊆ Lie(U+G (gϕ)) if and only if all Newton polygon

slopes of (iv, gϕ) are positive, cf. formula for Lie(U+G (gϕ)) in Subsection 2.6.2. As for

j ∈ S(0, v − 1), the natural actions of gϕ and rϕ on ij+1/ij coincide, to check (b) we can

assume that g = r ∈ L0G(ϕ)(W (k)) (and accordingly we can forget about the mentioned

ideal filtration). As g = r ∈ P−G (ϕ)(W (k)), it normalizes W (−δ)(g, ϕ) = W−δ(g, ϕ)

(cf. Subsection 2.6.6). Thus gΨ is a σ-linear automorphism of W (−δ)(g, ϕ). ThereforeW (−δ)(g, gϕ) = W (0)(g, gΨ) ⊇ W (0)(g,Ψ) = W (−δ)(g, ϕ) and thus from the property3.3 (a) we get by reasons of ranks that in fact

W (−δ)(g, gϕ) =W (0)(g, gΨ) =W (0)(g,Ψ) =W (−δ)(g, ϕ).

Thus I(M, gϕ,G) = I(M,ϕ,G), cf. Subsection 3.4.12. We consider a split maximal torusT0B(k) of L0

L0G(ϕ)

(gϕ)B(k) (as g = r, this group makes sense due to (a)). As we have

74

W (−δ)(g, gϕ) = W (−δ)(g, ϕ), we get (via formula (3b) applied with (h, HB(k), TB(k)) =

(g[ 1p ], GB(k), T0B(k)) and with φ ∈ gϕ, ϕ) that we also haveW (δ)(g, gϕ)[ 1p ] =W (δ)(g, ϕ)[ 1p ].

Thus W (δ)(g, gϕ) = W (δ)(g, ϕ) and therefore P+G (gϕ) = P+

G (ϕ). Thus U+G (gϕ) = U+

G (ϕ)is a closed subgroup scheme of the unipotent radical U+

G (gϕ) (resp. U+G (ϕ)) of the parabolic

subgroup scheme P+G (gϕ) (resp. P+

G (ϕ)) of G contained in P+G (gϕ) = P+

G (ϕ), cf. Proposi-tion 2.3.3 applied over B(k). Thus (b) holds; moreover, (d) and (e) also hold if g = r.

We check that (c) holds without any additional assumption on either G or g. Letu1 := u. For s ∈ N we inductively define (cf. Lemma 2.4.6)

us+1 := (rϕ)us(rϕ)−1 ∈ G(B(k)).

We have U+G (ϕ) 6 U+

G (ϕ) 6 P+G (ϕ) 6 P (see Subsection 3.4.3 for the last inclusion).

We check by induction on s ∈ N that we have us ∈ U+G (ϕ)(W (k)) 6 P (W (k)) 6 L.

The case s = 1 is obvious and the passage from s − 1 ≥ 1 to s ≥ 2 goes as follows.As U+

G (ϕ) is a closed subgroup scheme of P and as Lie(U+G (ϕ))[ 1p ] is normalized by rϕ

, from Subsection 2.4.7 applied with rϕ instead of ϕ we get that rϕU+G (ϕ)B(k)(rϕ)

−1 =

U+G (ϕ)B(k). Thus, as us−1 ∈ U+

G (ϕ)(B(k)) we have us ∈ U+G (ϕ)(B(k)) 6 P (B(k)). But as

us−1 ∈ L, we have us ∈ G(W (k)) (cf. Fact 3.1.6). Thus us ∈ U+G (ϕ)(W (k)) 6 P (W (k)) 6

L. This ends the induction.

All Newton polygon slopes of (Lie(U+G (ϕ)), rϕ) are positive, cf. proof of (b). Thus

the sequence (us)s∈N converges to 1M in the p-adic topology of U+G (ϕ)(W (k)). Therefore

the element u := (∏

s∈N us)−1 ∈ U+

G (ϕ)(W (k)) is well defined. We compute

(19) ugϕu−1(rϕ)−1 = · · ·u−13 u−1

2 u−11 u(rϕ)u1u2u3 · · · (rϕ)

−1

= · · ·u−13 u−1

2 u−11 u(rϕ)u1(rϕ)

−1(rϕ)u2(rϕ)−1(rϕ)u3 · · · = · · ·u−1

3 u−12 u−1

1 u1u2u3 · · · = 1M .

Thus (c) holds. To check (d) and (e), we can assume that g = r (cf. (c)) and this case wasalready checked above. Therefore (d) and (e) also hold.

To prove (f) we can assume that g = r (cf. (c)) and that k = k (cf. Fact 3.2.1(a)). In this paragraph we check the “only if” part; thus (M, rϕ,G) is Sh-ordinary. AsL0G(ϕ) is a Levi subgroup scheme of P+

G (ϕ) (see end of Subsection 3.2.3), it is the cen-

tralizer of Z0(L0G(ϕ)) in G (see Subsection 2.3.13). As T 6 L0

G(ϕ) (see end of Subsection3.2.3), we also have Z0(L0

G(ϕ)) 6 T . Thus ϕ normalizes Lie(Z0(L0G(ϕ))) and therefore

all Newton polygon slopes of (Lie(Z0(L0G(ϕ))), rϕ) are 0. Thus every Hodge cocharacter

of (M, rϕ,G) that fixes all endomorphisms of (M, rϕ,G), commutes with Z0(L0G(ϕ)) and

therefore it is also a Hodge cocharacter of (M, rϕ, L0G(ϕ)). Let (F

ir(M))i∈S(a,b) be the Sh-

canonical lift of (M, rϕ,G). Each endomorphism of (M, rϕ,G) is also an endomorphismof (M, (F i

r(M))i∈S(a,b), rϕ,G) (cf. either Subsection 1.3.7 or Subsection 3.1.8) and thusit is fixed by the inverse of the canonical split cocharacter of (M, (F i

r(M))i∈S(a,b), rϕ) (cf.the functorial aspects of [Wi, p. 513]). Thus this last Hodge cocharacter is also a Hodgecocharacter of (M, rϕ, L0

G(ϕ)). Thus (M, (F ir(M))i∈S(a,b), rϕ, L

0G(ϕ)) is a Shimura filtered

p-divisible object. As P+

L0G(ϕ)

(rϕ) 6 P+G (rϕ) and as P+

G (rϕ) normalizes (F ir(M))i∈S(a,b), we

75

get that (M, (F ir(M))i∈S(a,b), rϕ, L

0G(ϕ)) is an Sh-canonical lift. Therefore (M, rϕ, L0

G(ϕ))is Sh-ordinary.

In this paragraph we check the “if” part; thus (M, rϕ, L0G(ϕ)) is Sh-ordinary. Let

(F ir(M))i∈S(a,b) be the Sh-canonical lift of (M, rϕ, L0

G(ϕ)). Let l0 ∈ L ∩ L0G(ϕ)(W (k))

be such that we have (F ir(M))i∈S(a,b) = (l0(F

i(M)))i∈S(a,b), cf. Corollary 3.1.4 applied

to (M, rϕ, L0G(ϕ)). As U+

G (ϕ) is a subgroup scheme of P normalized by l0, it normalizes

(F ir(M))i∈S(a,b). The group scheme P+

G (rϕ) is contained in P+G (ϕ) (cf. (d)) and contains

U+G (ϕ) (cf. (b)). Thus P+

G (rϕ) is generated by U+G (ϕ) and by P+

L0G(ϕ)

(rϕ) = L0G(ϕ) ∩

P+G (rϕ); therefore P+

G (rϕ) normalizes (F ir(M))i∈S(a,b). Thus (M, (F i

r(M))i∈S(a,b), rϕ,G)is an Sh-canonical lift. Therefore (M, rϕ,G) is Sh-ordinary. Thus (f) holds.

3.7.2. End of the proof of 3.3 (b). To check the “if” part of the property 3.3 (b)we can assume that g ∈ L0

G(ϕ)(W (k)) and thus we have I(M, gϕ,G) = I(M,ϕ,G), cf.Theorem 3.7.1 (e).

We check the “only if” part of the property 3.3 (b). Let g ∈ G(W (k)) be such thatwe have I(M,ϕ,G) = I(M, gϕ,G). In this paragraph we assume that G is split, adjoint,that I = S(1, n), and that Γ = (1 · · ·n) and we use the notations of Section 3.4. FromTheorem 3.6.2 (applied after shifting from the element 1 ∈ I1 to an arbitrary element ofI1) and from Corollary 3.6.3, we get that there exists an element g1 ∈ P (k) such that forall numbers i ∈ I1 we have g1

−1(P+G (gϕ)k ∩Gi,k)g1 = P+

G (ϕ)k ∩ Gi,k. From the property

3.4.2 (c) we get that for all numbers i ∈ I, the groups P+G (gϕ)k ∩Gi,k and P+

G (ϕ)k ∩Gi,k

are Gi(k)-conjugate. If i ∈ I \ I1, then Gi(k) 6 P (k). From the last three sentences weget that we can assume that the element g1 ∈ P (k) is such that for all numbers i ∈ I wehave g1

−1(P+G (gϕ)k ∩Gi,k)g1 = P+

G (ϕ)k ∩Gi,k. Therefore g1−1P+

G (gϕ)kg1 = P+G (ϕ)k.

The formation of P+G (∗) group schemes behaves well with respect to passages to adjoint

group schemes (see Subsection 2.4.5), to extensions to a finite field extension k1 of k, andto cyclic decompositions of Γ. Thus g1 exists after extension to a suitable k1 and thereforeit always exists (cf. property 2.3.1 (c)) regardless of what G and Γ are. Let g1 ∈ L be suchthat it lifts g1 and we have g−1

1 (P+G (gϕ))g1 = P+

G (ϕ), cf. Subsection 2.3.8 (e). We write

g−11 gϕg1 = rϕ, where r ∈ G(W (k)) (cf. Fact 3.1.6). As ϕ and rϕ normalize Lie(P+

G (ϕ))[ 1p ],

r also normalizes Lie(P+G (ϕ))[ 1p ]. Thus r normalizes P+

G (ϕ), cf. Lemma 2.3.6; therefore we

have r ∈ P+G (ϕ)(W (k)). Up to an inner isomorphism (i.e., up to a replacement of g1 with

g1u1, where u1 ∈ U+G (W (k)) 6 L) we can assume that r ∈ L0

G(ϕ)(W (k)), cf. Theorem3.7.1 (c). Thus gϕ = g1rϕg

−11 . This ends the proof of the property 3.3 (b).

3.8. End of the proofs of Theorems 1.4.1 and 1.4.2

In Subsections 3.8.1 to 3.8.4 we deal with the different implications required to proveTheorems 1.4.1 and 1.4.2. Until Subsection 3.8.5 we will work in the context in which Gis quasi-split, T , and B are as in Subsection 3.2.2, and (5) holds.

3.8.1. Proof of 1.4.1 (a) ⇒ 1.4.1 (c). We show that 1.4.1 (a) ⇒ 1.4.1 (c).Let g ∈ G(W (k)) be such that (M, gϕ,G) is Sh-ordinary. We have to check thatI(M, gϕ,G) = I(M,ϕ,G), cf. property 3.3 (a). For this we can assume that G is

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split, adjoint, that I = S(1, n), that Γ = (1 · · ·n), and (see Subsection 2.4.5) thatM = g. We can also assume that (M, (F i(M))i∈S(a,b), gϕ,G) is an Sh-canonical lift.

Therefore P+G (gϕ) 6 P . Thus we can choose B in Subsection 3.2.3 to be included in

P+G (gϕ). As ϕ and gϕ leave invariant Lie(B) and Lie(P+

G (gϕ)) (respectively), we havegϕ(Lie(BB(k)) = Lie(gBB(k)g

−1) ⊆ Lie(P+G (gϕ)B(k)). Thus gBB(k)g

−1 6 P+G (gϕ)B(k)

(cf. Lemma 2.3.6) and therefore gBg−1 6 P+G (gϕ). The parabolic subgroups P+

G (gϕ)B(k)

and g−1P+G (gϕ)B(k)g of GB(k) contain BB(k) and therefore (cf. property 2.3.1 (b)) they

coincide. Thus P+G (gϕ) = gP+

G (gϕ)g−1 and therefore g ∈ P+G (gϕ)(W (k)). Thus the New-

ton polygons of (M, gϕ) and (M,ϕ) coincide, cf. Lemma 3.3.1. Therefore I(M, gϕ,G) =I(M,ϕ,G), cf. Subsection 3.4.13 and the identity M = g. Thus 1.4.1 (a) ⇒ 1.4.1 (c).

3.8.2. Proof of 1.4.1 (b) ⇔ 1.4.1 (a), part I: a specialization argument. In thisand the next Subsection we will prove by induction on dim(Gk) that 1.4.1 (b) ⇔ 1.4.1 (a).This equivalence holds if dim(Gk) = 1, cf. Subsection 3.3.2. Let d ∈ N. We assume that1.4.1 (b) ⇔ 1.4.1 (a) provided dim(Gk) < d. We show that 1.4.1 (b) ⇔ 1.4.1 (a) even ifdim(Gk) = d. Let ϕ(−a) := (p−a1M )ϕ :M →M . The pair (M,ϕ(−a)) is an F -crystal.

The estimates of [Ka2, 1.4.3 or 1.5] imply that there exists a positive integer s thatdepends only on a, b, and dM and such that for all elements h ∈ G(W (k)) the Newtonpolygon of (M,hϕ(−a)) is uniquely determined by the Hodge polygons of (M, (hϕ(−a))i)’s,with i ∈ S(1, s). Let m := s(b − a + 1) ∈ N. From the property 1.3.2 (a) we get thatpb−aM ⊆ hϕ(−a)(M). This implies that for each number i ∈ S(1, s), the Hodge polygonof (M, (hϕ(−a))i) depends only on h modulo pm. To prove that 1.4.1 (b) ⇔ 1.4.1 (a) wecan make extensions to an arbitrary perfect field k1 that contains k, cf. Fact 3.2.1 (a)and the fact that (M,ϕ,G) is Sh-ordinary. Thus until Subsection 3.8.4 we can assumethat k = k and that there exists an element g1 ∈ G(W (k)) with the property that theNewton polygon N of (M, g1ϕ) is not strictly above any other Newton polygon of theform (M ⊗W (k) W (k), gk(ϕ ⊗ σk)), where k is a perfect field that contains k and where

gk ∈ G(W (k)).

Let Y be an open, affine subscheme of G through which g1 factors and such that there

exists an etale morphism eY : Y → Adim(Gk)W (k) between affine W (k)-schemes. We write

Yk = Spec(S). As Gk is a reductive group, S is an integral k-algebra. The existence of eYimplies that there exists a morphism hY : Spec(Wm(Sperf)) → YWm(k) of Wm(k)-schemes

that lifts the canonical morphism Spec(Sperf) → Spec(S) of k-schemes and such that thereexists a Teichmuller lift Tg1;m : Spec(Wm(k)) → Spec(Wm(Sperf)) whose composite withhY is g1 modulo pm. We identify hY ∈ Y (Wm(Sperf)) ⊆ G(Wm(Sperf)) with a Wm(Sperf)-linear automorphism of M ⊗W (k) Wm(Sperf). Let

CY ;m := (M ⊗W (k) Wm(Sperf), hY (ϕ(−a)⊗ ΦSperf )),

where we denote also by ϕ(−a) ⊗ ΦSperf its reduction modulo pm. We denote by Tg2;m :Spec(Wm(k)) → Spec(Wm(Sperf)) every Teichmuller lift with the property that hY Tg2;mis the reduction modulo pm of an (a priori given) element g2 ∈ Y (W (k)) ⊆ G(W (k)).

The Newton and Hodge polygons go up under specializations. From this and the abovepart referring to [Ka2], we get that there exists an open, Zariski dense subscheme Y o

k of the

77

integral scheme Yk such that a Teichmuller lift Tg2;m : Spec(Wm(k)) → Spec(Wm(Sperf))lifts a k-valued point of Sperf that maps (resp. that does not map) to Y o

k (k) if and only ifthe Newton polygon of (M, g2ϕ) isN (resp. is strictly aboveN); to be compared with [Ka2,2.3.1 and 2.3.2] (loc. cit. is stated in terms of F -crystals but its proof does not use theexistence of suitable integrable, nilpotent modulo p connections). Thus for an arbitraryelement g ∈ G(W (k)), we can choose g1 and Y such that we have g ∈ Y (W (k)) andmoreover g is not congruent to g1 modulo p. Based on the Chinese Remainder Theoremand on the existence of eY , we can also choose hY such that the Teichmuller lift Tg;m :Spec(Wm(k)) → Spec(Wm(Sperf)) does make sense. As Yk is an integral scheme, Y o

k

specializes to g modulo p and thus the Newton polygon of (M, gϕ) is above N.

Let G0 be a factor of Gad such that the simple factors of Lie(G0,B(k)) are permuted

transitively by ϕ. The restriction to Lie(G0,B(k)) of the trace form on EndB(k)(M [ 1p ]) is

non-degenerate, cf. [Bou1, Ch. I, §6, Prop. 1]. Thus we can speak about the perpendicularLie(G0,B(k))

⊥ of Lie(G0,B(k)) in EndB(k)(M [ 1p ]) with respect to this trace form. Repeatingthe specialization process of the previous two paragraphs, we get that we can assume thereexists g10 ∈ G(W (k)) (resp. g⊥10 ∈ G(W (k))) such that for each element g ∈ G(W (k)) theNewton polygon of (Lie(G0,B(k)), gϕ) (resp. of (Lie(G0,B(k))

⊥, gϕ)) is above the Newton

polygon of (Lie(G0,B(k)), g10ϕ) (resp. of (Lie(G0,B(k))⊥, g⊥10ϕ)). We can assume that g1,

g10, and g⊥10 modulo p define distinct k-valued points of Y (cf. the Zariski density in Gk

of Y 0k and of its analogues obtained working with g10 and g⊥10 instead of with g1). We can

choose hY (cf. Chinese Remainder Theorem) such that the Teichmuller lifts Tg10;m andTg⊥

10;mare also well defined. This implies that we can choose our g1 ∈ G(W (k)) such that

the Newton polygon of (Lie(G0,B(k)), g1ϕ) (resp. of (Lie(G0,B(k))⊥, g1ϕ)) is the Newton

polygon of (Lie(G0,B(k)), g10ϕ) (resp. of (Lie(G0,B(k))⊥, g⊥10ϕ)).

3.8.3. Proof of 1.4.1 (b) ⇔ 1.4.1 (a), part II: end of the argument. Let nowg ∈ G(W (k)) be such that the Newton polygon of (M, gϕ) is N. Thus the New-ton polygons of (EndW (k)(M), gϕ) and (EndW (k)(M), g1ϕ) coincide. But the New-

ton polygon of (EndW (k)(M)[ 1p], gϕ) is formed by “putting together” the Newton poly-

gons of (Lie(G0,B(k)), gϕ) and (Lie(G0,B(k))⊥, gϕ). Based on this and the last para-

graph of Subsection 3.8.2, we conclude that the Newton polygons of (Lie(G0,B(k)), g1ϕ)

and (Lie(G0,B(k))⊥, g1ϕ) coincide with the Newton polygons of (Lie(G0,B(k)), gϕ) and

(Lie(G0,B(k))⊥, gϕ) (respectively). Thus the Newton polygon of (Lie(G0,B(k)), gϕ) is be-

low the Newton polygon of (Lie(G0,B(k)), ϕ). Thus from Fact 3.4.11 and Subsection 3.4.13

we get that I(Lie(G0), ϕ, G0) ≤ I(Lie(G0), gϕ,G0). As G0 is an arbitrary factor of Gad

with the property that the simple factors of Lie(G0,B(k)) are permuted transitively by ϕ,(from the very definition of FSHW invariants) we get that I(M,ϕ,G) ≤ I(M, gϕ,G).From this and the property 3.3 (a) we get that I(M,ϕ,G) = I(M, gϕ,G). Thus up to aninner isomorphism we can assume that g ∈ L0

G(ϕ)(ϕ)(W (k)), cf. property 3.3 (b). Letr1 ∈ L0

G(ϕ)(W (k)). The Newton polygon of (M, r1ϕ) is above the Newton polygon N of(M, gϕ). We check that we have a strict inequality dim(L0

G(ϕ)k) < d. To check this we canassume that the hypotheses of Corollary 3.6.3 hold and in this case the strict inequalitydim(L0

G(ϕ)k) < d is a consequence of the fact (see Corollary 3.6.3) that P+G (ϕ) 6= G. By

our induction on dim(Gk) we get that (M, gϕ, L0G(ϕ)) is Sh-ordinary and that the Newton

78

polygon N of (M, gϕ, L0G(ϕ)) is the Newton polygon of (M,ϕ, L0

G(ϕ)). Thus (M, gϕ,G) isSh-ordinary (cf. Theorem 3.7.1 (f)) and the Newton polygon of (M,ϕ,G) is N.

Let now g ∈ G(W (k)) be such that (M, gϕ,G) is Sh-ordinary. We show that the New-ton polygon of (M, gϕ) is N. Not to introduce extra notations, (up to inner isomorphisms)we can assume that (F i(M))i∈S(a,b) is also the Sh-canonical lift of (M, gϕ,G). Thus we

have P+G (gϕ) 6 P and we recall (cf. Subsection 3.2.3) that B is a Borel subgroup scheme

of P . Let g2 ∈ P (W (k)) be such that we have B 6 g−12 P+

G (gϕ)g2, cf. Subsection 2.3.8 (d).By replacing gϕ with g−1

2 gϕg2 we can assume that g2 = 1M . Let B2 be the Borel subgroupscheme of P+

G (gϕ) whose Lie algebra contains gϕ(Lie(B)) (to be compared with Subsection3.2.3). Let h2 ∈ P+

G (gϕ)(W (k)) be such that we have h2B2h−12 = B. We get that h2gϕ

leaves invariant Lie(B). Thus b2 := h2g ∈ G(W (k)) normalizes Lie(B)[ 1p] and therefore

(cf. Lemma 2.3.6) it also normalizes BB(k); thus we have b2 ∈ B(W (k)) 6 P+G (ϕ)(W (k)).

We have P+G (b2ϕ) = P+

G (gϕ), cf. Lemma 3.3.1. Similarly, as P+G (ϕ) 6 P (see sentence

before Subsection 3.4.4) and as b2 ∈ P+G (ϕ)(W (k)), we have P+

G (b2ϕ) = P+G (ϕ). Therefore

P+G (gϕ) = P+

G (ϕ). using this, as for b2 we argue that g ∈ P+G (ϕ)(W (k)) 6 P+

G (ϕ)(W (k)).Even more, up to an inner isomorphism defined by an element of U+(ϕ)(W (k)) we canassume that g ∈ L0

G(ϕ)(W (k)) (cf. Theorem 3.7.1 (b) and (c)). Thus (M, gϕ, L0G(ϕ)) is

Sh-ordinary, cf. Theorem 3.7.1 (f). As dim(L0G(ϕ)k) < d (see previous paragraph), by in-

duction we get that the Newton polygon of (M, gϕ) is the same as of (M,ϕ) and thereforeit is N, cf. the end of the previous paragraph.

We conclude that 1.4.1 (a) ⇔ 1.4.1 (b) if G quasi-split.

3.8.4. Proof of Theorem 1.4.2. We check that if P+G (ϕ) = P+

G (ϕ), then 1.4.1 (c) ⇒1.4.1 (a). Let g ∈ G(W (k)) be such that we have I(M,ϕ,G) = I(M, gϕ,G). Let g1 ∈ Land r ∈ L0

G(ϕ)(W (k)) 6 P+G (ϕ)(W (k)) = P+

G (ϕ)(W (k)) be as in the property 3.3 (b); thusgϕ = g1rϕg

−11 . As g1 ∈ L , (g1(F

i(M)))i∈S(a,b) is a lift of (M, gϕ,G) (cf. Lemma 3.1.2).

We have P+G (rϕ) = P+

G (ϕ), cf. Lemma 3.3.1. Thus P+G (gϕ) = g1P

+G (ϕ)g−1

1 6 g1Pg−11 .

Therefore (g1(Fi(M)))i∈S(a,b) is an Sh-canonical lift of (M, gϕ,G). Thus (M, gϕ,G) is

Sh-ordinary. Therefore 1.4.1 (c) ⇒ 1.4.1 (a).

In this and the next paragraph we check that if P+G (ϕ) 6= P+

G (ϕ), then the statement

1.4.1 (c) does not imply the statement 1.4.1 (a). As P+G (ϕ) 6= P+

G (ϕ), L0G(ϕ) is not a

subgroup scheme of P+G (ϕ). Thus not all Newton polygon slopes of (Lie(L0

G(ϕ)), ϕ) are

non-negative and therefore L0G(ϕ) is not a subgroup scheme of P . Thus P ∩ L0

G(ϕ) 6=L0G(ϕ) and moreover L0

G(ϕ) is not a torus (i.e., it is not T ). As P ∩ L0G(ϕ) 6= L0

G(ϕ),we can assume that P ∩ L0

G(ϕ) ∩ G1 6= L0G(ϕ) ∩ G1. For every direct factor L0

G(ϕ)0 ofL0G(ϕ)

ad such that ϕ permutes transitively the simple factors of Lie(L0G(ϕ)0)[

1p ], we have

I(Lie(L0G(ϕ)0), ϕ, L

0G(ϕ)0) > 0 (cf. Subsection 3.2.2 as well as Subsections 3.4.12 and 3.4.13

applied to (Lie(L0G(ϕ)0), ϕ, L

0G(ϕ)0)). Let w ∈ L0

G(ϕ)(W (k)) be such that it normalizesT and w(B ∩ L0

G(ϕ))w−1 is the opposite of B ∩ L0

G(ϕ) with respect to T , cf. property2.3.8 (d). We check that the fact that P ∩ L0

G(ϕ) 6= L0G(ϕ) implies that (M,wϕ, L0

G(ϕ))is not Sh-ordinary. We choose L0

G(ϕ)0 such that it contains a simple factor of L0G(ϕ)

ad inwhich µ has a non-trivial image. It is easy to see that the choices of w and L0

G(ϕ)0 implythat I(Lie(L0

G(ϕ)0), wϕ, L0G(ϕ)0) = 0. Not to introduce extra notations, we will detail here

79

the analogue of this last equality for the context of (M,wϕ,G) itself; thus wBw−1 is theopposite of B with respect to T . We can assume that G is split, adjoint, that I = S(1, n),that Γ = (1 · · ·n), and (to fix notations) that µ has a non-trivial image in [G1∩L0

G(ϕ)0)]ad.

The choice of w implies that that we have (wΨ)n(g1) ⊆ pg1 + b1, cf. formulas (6a) and(6b) and the equality (wΨ)n = pnδ(wϕ)n (see Subsection 3.4.13; see also Subsection 3.4.3for our standard notations). Thus (wΨ)n+1(g1) ⊆ pg2 and this implies (wΨ)2n(g) ⊆ pg.Therefore I(M,wϕ,G) = 0.

As I(Lie(L0G(ϕ)0), wϕ, L

0G(ϕ)0) = 0 < I(Lie(L0

G(ϕ)0), ϕ, L0G(ϕ)0) and as 1.4.1 (a) ⇒

1.4.1 (c), the triple (Lie(L0G(ϕ)0), wϕ, L

0G(ϕ)0) is not Sh-ordinary; therefore the triple

(Lie(L0G(ϕ)

ad), wϕ, L0G(ϕ)

ad) is also not Sh-ordinary. Thus (M,wϕ, L0G(ϕ)) is not Sh-

ordinary, cf. Corollary 3.1.5. Therefore (M,wϕ,G) is not Sh-ordinary (cf. Theorem 3.7.1(f)) but we have I(M,wϕ,G) = I(M,ϕ,G) (cf. Theorem 3.7.1 (e)). Thus 1.4.1 (c) doesnot imply 1.4.1 (a) if P+

G (ϕ) 6= P+G (ϕ).

To end the proof of Theorem 1.4.2 we only have to show that we have P+G (ϕ) = P+

G (ϕ)if and only if for each element i ∈ I such that the simple factors of Gi,W (k) are

PGLℓ+1 group schemes with ℓ ≥ 3 and for every element h0 ∈ G(W (k)) such that(M ⊗W (k) W (k), h0(ϕ ⊗ σk), GW (k)) is Sh-ordinary, the image of the unipotent radical

of P+G

W (k)(h0(ϕ⊗ σk)) in Gi,W (k) is of nilpotent class at most 2. To prove this equivalence

we can assume that G is split, adjoint (cf. Corollary 3.1.5), that I = S(1, n), and thatΓ = (1 · · ·n).

We first check the “if” part. Thus we first assume that the long statement on an atmost 2 nilpotent class holds. By taking h0 = 1M , from the nilpotent class part of Corollary3.6.3 we get that in the case (a) (of Subsection 3.4.6) we have |E| ≤ 2. Thus for all cases (ofSubsection 3.4.6) we have E = E . Therefore P+

G (ϕ)∩G1 = P+G (ϕ)∩G1, cf. Corollary 3.6.3.

Thus P+G (ϕ) = P+

G (ϕ), cf. property 3.4.2 (b) (applied with g = 1M ). To check the “only

if” part we can assume that k = k. We assume that P+G (ϕ) = P+

G (ϕ). Let h0 ∈ G(W (k))be such that (M,h0ϕ,G) is Sh-ordinary. Based on the implication 1.4.1 (a) ⇒ 1.4.1 (c)and on the property 3.3 (a), we have I(M,h0ϕ,G) = I(M,ϕ,G). Thus up to an innerisomorphism we can assume that h0 ∈ P+

G (ϕ)(W (k)) = P+G (ϕ)(W (k)), cf. property 3.3

(b). Therefore P+G (h0ϕ) = P+

G (ϕ) = P+G (ϕ) (cf. Lemma 3.3.1 for the first equality) and

thus the long statement on an at most 2 nilpotent class follows from Corollary 3.6.3 andthe definitions of E and E in Subsection 3.4.8. This ends the proof of Theorem 1.4.2.

3.8.5. The non quasi-split case. We know that Theorem 1.4.1 holds if G is quasi-split,cf. Subsections 3.8.1 and 3.8.3. We now show that Theorem 1.4.1 holds even if G is notquasi-split. Thus Gk is not quasi-split (cf. Subsection 2.3.8 (a)) and therefore the field kis infinite (cf. [Bo, Ch. V, Prop. 16.6]). Let k1 be a Galois extension of k such that Gk1

is split. As Theorem 1.4.1 holds in the context of (M ⊗W (k) W (k1), h(ϕ⊗ σk1), GW (k1))’s

with h ∈ G(W (k1)), to prove Theorem 1.4.1 we only have to show that there exists anelement g ∈ G(W (k)) such that (M, gϕ,G) is Sh-ordinary. As in Subsection 3.8.2, wecheck this by induction on dim(Gk).

The case dim(Gk) = 1 is trivial (cf. Subsection 3.3.2) and the inductive step goes asfollows. As the perfect field k is infinite, the set G(k) is Zariski dense in Gk (cf. [Bo, Ch. V,

80

Cor. 18.3]). Thus there exists an element g1 ∈ G(W (k)) that maps to a k-valued point ofthe open, Zariski dense subscheme Z of Gad

k introduced in Fact 3.2.1 (b). Therefore for eachelement g2 ∈ G(W (k1)) we have I(M ⊗W (k) W (k1), g2(ϕ⊗ σk1

), GW (k1)) ≤ I(M, g1ϕ,G),

cf. Fact 3.2.1 (b). An L-conjugate of µ factors through a Levi subgroup scheme L0(g1) ofP+G (g1ϕ), cf. Lemma 3.1.11. In the remaining part of this paragraph we will check that

every such Levi subgroup scheme L0(g1)′ of P+

G (g1ϕ) is L ∩ P+G (g1ϕ)(W (k))-conjugate

to L0(g1). The argument for this goes as follows. We can assume µk factors throughboth L0(g1)k and L0(g1)

′k. Let Ck be the centralizer of µk in Gk. Thus Ck is a reduc-

tive subgroup (cf. Subsection 2.3.13) of Pk. Moreover the intersections L0(g1)k ∩ Ck

and L0(g1)′k ∩ Ck are Levi subgroups of the parabolic subgroup P+

G (ϕ)k ∩ Ck of Ck.Therefore these two intersections are (P+

G (ϕ)k ∩ Ck)(k)-conjugate, cf. [Bo, Ch. V, Prop.

20.5]. Thus, as (P+G (ϕ)k ∩ Ck)(k) 6 P (k), we can assume that L0(g1)k and L0(g1)

′k

have a common maximal torus. Therefore we have L0(g1)k = L0(g1)′k, cf. the unique-

ness part of Subsection 2.3.8 (c). As in Subsection 2.3.8 (d) we argue that a suitableKer(P+

G (g1ϕ)(W (k)) → P+G (g1ϕ)(k))-conjugate of a maximal torus of L0(g1) is a maximal

torus of L0(g1)′. Thus a suitable Ker(P+

G (g1ϕ)(W (k)) → P+G (g1ϕ)(k))-conjugate of L

0(g1)

is L0(g1)′, cf. Subsection 2.3.8 (c).

Not to introduce extra notations, we will assume that µ itself factors through L0(g1).This implies that the intersection g1ϕ(Lie(L

0(g1))[1p])∩Lie(P+

G (g1ϕ)) is the Lie algebra of

a Levi subgroup scheme of P+G (g1ϕ) which (cf. Subsection 2.3.8 (f)) is U+

G (g1ϕ)(W (k))-

conjugate to L0(g1). Thus there exists an element u ∈ U+G (g1ϕ)(W (k)) such that ug1ϕ

normalizes Lie(L0(g1))[1p ]. Therefore (M,ug1ϕ, L

0(g1)) is a Shimura p-divisible object. We

have dim(L0(g1)k) < dim(Gk) (as otherwise statements 3.2.2 (a) to (d) would hold). Byinduction we know that we can choose u such that (M,ug1ϕ, L

0(g1)) is Sh-ordinary (herewe might have to replace u by a L0(g1)(W (k))-conjugate of it and to replace g1 by a rightL0(g1)(W (k))-translate of it). Let g := ug1. We claim that (M, gϕ,G) is Sh-ordinary.

After extensions to k1 (cf. Fact 3.2.1 (a)) and up to inner isomorphisms, to checkthis claim we can assume that (5) holds, that (cf. property 3.3 (a) and the choice of g1)I(M,ϕ,G) = I(M, g1ϕ,G), and (cf. property 3.3 (b) and Theorem 3.7.1 (d)) that P+

G (ϕ) =

P+G (g1ϕ). From the last equality and the above part on L ∩ P+

G (g1ϕ)(W (k))-conjugation,

we get that up to inner isomorphisms we can also assume that L0G(ϕ) = L0(g1). Thus the

fact that (M, gϕ,G) is Sh-ordinary follows from Theorem 3.7.1 (f). This ends the proof ofTheorem 1.4.1.

3.9. On Hasse–Witt invariants

We consider the case when a = 0. We refer to the multiplicity of the Newton polygonslope 0 of (M, gϕ) as the Hasse–Witt invariant HW (M, gϕ) of (M, gϕ).

In many cases the fact that (M, gϕ,G) is Sh-ordinary or not depends only on thevalue of HW (M, gϕ) ∈ N ∪ 0 (we emphasize that for this, one often might have toperform first a shift as in Subsection 1.3.4). For instance, this applies if b = 1 (resp.if b = 2) and we have a direct sum decomposition M = ⊕i∈IMi, where Mi is a freeW (k)-module of rank 2ℓ (resp. 2ℓ + 1), such that Mi[

1p ]’s are permuted transitively by ϕ

81

and G =∏

i∈I GSp(Mi, ψi) (resp. and G =∏

i∈I GSO(Mi, ψi)), where ψi is a perfectalternating (resp. perfect symmetric) bilinear form on Mi.

If b = 2 we do not know if the reduction modulo p of ϕ can be used to define sometype of truncation modulo p of (M, gϕ,G) which could substitute the FSHW map of(M, gϕ,G). This, the fact that (i) and (ii) below exclude about half of the cases coveredby Theorem 1.4.2 (i.e., of the cases when 1.4.1 (a) ⇔ 1.4.1 (c)), and the need of fullgenerality and of entirely intrinsic and practical methods, are the main reasons why in thismonograph we always work (mainly) in an adjoint context. To reach a context in which wecan use HW (M, gϕ) to decide if (M, gϕ,G) is Sh-ordinary or not, we often have to passfirst to an adjoint context and then to “lift” the things to contexts modeled on the twoexamples of the previous paragraph; thus it seems that it is more practical to avoid these“lifts”. With the notations of Subsections 3.4.6 and 3.4.8, the adjoint contexts to whichsuch “lifts” are possible are divided into two types:

(i) we are in cases (a), (b), (c), or (d) and we have |E| = 1;

(ii) we are in the case (d) and we have E = ℓ− 1, ℓ.

82

4 Abstract theory, II: proofs of Theorems 1.4.3 and 1.4.4

We recall that we use the notations listed in Section 2.1. In Sections 4.1 and 4.2 weprove Theorems 1.4.3 and 1.4.4 (respectively). In Section 4.3 we include five complements.

4.1. Proof of Theorem 1.4.3

See Subsection 3.1.8 for the implication 1.4.3 (a) ⇒ 1.4.3 (b). Obviously 1.4.3 (b) ⇒1.4.3 (a). Thus 1.4.3 (a) ⇔ 1.4.3 (b). We now deal with the other implications required toprove Theorem 1.4.3.

4.1.1. Proof of 1.4.3 (b) ⇒ 1.4.3 (c). We assume that the statement 1.4.3 (b) holdsand we prove that the statement 1.4.3 (c) holds. For this we can assume that k = k,that for all i ∈ S(a, b) we have F i

0(M) = F i(M), and that µ : Gm → G is the inverseof the canonical split cocharacter of (M, (F i(M))i∈S(a,b), ϕ). Thus µB(k) factors throughZ(L0

G(ϕ)B(k)), cf. Subsection 3.1.8. Let σϕ and MZpbe as in Section 2.4. See Example

2.4.2 for the reductive, closed subgroup scheme GZpof GLMZp

. For each non-negative

integer m let σmϕ (µ) be the cocharacter of G that is the σm

ϕ -conjugate of µ. Let T (0)

be the smallest subtorus of G through which all σmϕ (µ)’s factor, cf. Corollary 3.1.9. We

have T(0)B(k) 6 Z0(L0

G(ϕ)B(k)) and the triple (M,ϕ, T (0)) is a Shimura p-divisible object, cf.

Corollary 3.1.9. We have a direct sum decomposition M = ⊕γ∈QW (γ)(M,ϕ), cf. Lemma2.6.3 applied to (M,ϕ, T (0)). Thus L0

G(ϕ) is a Levi subgroup scheme of P+G (ϕ), cf. Lemma

2.6.3. As ϕ normalizes Lie(T (0)), from Lemma 2.4.1 we get the existence of a torus

T(0)Zp

of GLMZpwhose extensions to W (k) is T (0). Obviously T

(0)Zp

is a torus of GZp.

Let q be the smallest positive integer such that µ : Gm → T (0) is the extension to

W (k) of a cocharacter of T(0)W (Fpq )

. Let k0 := Fpq . We get that σmϕ (µ) is the extension

to W (k) of a cocharacter of T(0)W (k0)

to be denoted also by σmϕ (µ). We have σq

ϕ(µ) = µ.

For each pair (i, j) ∈ S(1, q)× ∈ S(a, b), let iF j be the maximal free W (k0)-submodule ofMZp

⊗ZpW (k0) on which σi

ϕ(µ) acts via the −j power of the identity character of Gm.

For each function f : S(1, q) → S(a, b), let

Ff :=⋂

i∈S(1,q)

iF f(i).

83

Let L be the set of such functions f for which we have Ff 6= 0. As σiϕ(µ)’s commute,

we have a direct sum decomposition MZp⊗Zp

W (k0) = ⊕f∈LFf . Let S : L→L be the

bijection defined by the rule S(f)(i) := f(i− 1), where f(0) := f(q). We have:

(20) ϕ(Ff ) = pf(0)FS(f).

Let I0S:= f ∈ L|S(f) = f. Let S =

∏l∈I

SS l be written as a product of disjoint

cyclic permutations. We allow trivial cyclic permutations i.e., we take the index set IS to

contain I0S. Each function f ∈ I0

Sis said to be associated to Sf . Let I1

S:= IS \ I0

S. If

l ∈ I1S, then each function f ∈ L such that we have S l(f) 6= f is said to be associated

to S l. As Sq= 1L, the order ql of S l divides q. As σs

ϕ(µ) 6= µ if s ∈ S(1, q − 1) and asσqϕ(µ) = µ, we have q = l.c.m.ql|l ∈ IS. For each element l ∈ IS we pick up an element

f l ∈ L associated to S l. We have

ϕql(Ffl) = p

qlq ℓ(f l)Ffl

, where ℓ(f l) := f l(1) + f l(2) + · · ·+ f l(q).

Moreover ϕql acts on Ff l

as pqlq ℓ(f l)σql

ϕ does, cf. (20). Let pFf l:= x ∈ F

f l|σql

ϕ (x) = x;

it is a free W (Fpql )-submodule of Ff lof the same rank as Ff l

(the rank part follows from

[Bo, Ch. AG, Sect. 14.2, Prop.] applied over B(Fpql ) to Ff l[ 1p]). From this and the fact

that σqlϕ (Ff l

) = Ff l, we get that Ffl

= pFf l⊗W (Fpql )

W (k0). For l ∈ IS we choose a

W (Fpql )-basis ℓs|s ∈ Jl for pFf land we also view it as a W (k0)-basis for Ff l

. If l ∈ I1ϕ,

then for each function f ∈ L associated to Sl but different from f l, let j ∈ N be the

smallest number such that we have f = Sj(f l). We get a W (k0)-basis

1pnj,l ϕ

j(ℓs)|s ∈ Jl

for Ff , where nj,l :=∑j

i=1 f l(q + 1− i). This expression of nj,l is a consequence of either

formula (20) or of the following formula

(21) ϕj = (

j∏

m=1

σmϕ (µ))(

1

p)σj

ϕ = σjϕ(

j∏

m=1

σm−jϕ (µ))(

1

p).

Let l ∈ IS and let f be associated to S l. As ql is the order of S l, the function

θl : Z/qlZ → S(a, b) that takes [j] ∈ Z/qlZ to f(j) is not periodic.

Let B0 = e(0)1 , . . . , e

(0)dM

be the W (k0)-basis for MZp⊗Zp

W (k0) obtained by putting

together the chosen W (k0)-bases for Ff , with f ∈ L. We view B0 also as a W (k)-basis

for M . Let π be the permutation of S(1, dM) such that for each number j ∈ S(1, dM)

we have ϕ(e(0)j ) = pnje

(0)π(j), where nj ∈ S(a, b) is such that e

(0)j ∈ Fnj (M). We conclude

that (M,ϕ) has the desired cyclic form with respect to B0. Moreover Im(µ) normalizes

W (k)e(0)j for all numbers j ∈ S(1, dM).

Let µ0 : Gm → G, π0 : S(1, dM)→S(1, dM), and e1, . . . , edM ⊆ M be as in the

statement 1.4.3 (c). We check that µ0 = µ. Let T big be the maximal torus of GLM

84

that normalizes W (k)ej for all numbers j ∈ S(1, dM). We have ϕ(Lie(T big)) = Lie(T big),cf. proof of Lemma 2.8.2. As µ0 : Gm → G factors through T big we have an inclusionIm(dµ0,B(k)) ⊆ Lie(T big

B(k))∩Lie(GB(k)) ⊆ Lie(L0G(ϕ)B(k)). Thus µ0 factors through L0

G(ϕ)

(cf. end of Subsection 2.3.8 (b)) and therefore it commutes with Z(L0G(ϕ)) as well as with

µ. Thus µ0 = µ (cf. Lemma 3.1.1) and therefore the implication 1.4.3 (b) ⇒ 1.4.3 (c)holds.

4.1.2. Proofs of 1.4.3 (c) ⇒ 1.4.3 (d) and 1.4.3 (d) ⇒ 1.4.3 (b). Due to the uniquenessparts of statements 1.4.3 (b) and 1.4.3 (d) and due to the existence of descent for lifts (seeLemma 3.1.13), to prove the two implications 1.4.3 (c) ⇒ 1.4.3 (d) and 1.4.3 (d) ⇒ 1.4.3 (b)we can assume that k = k. We first show that the statement 1.4.3 (c) implies the statement1.4.3 (d). We use the notations of the statement 1.4.3 (c). Let (F i

0(M))i∈S(a,b) be the lift of(M,ϕ,G) defined by µ0. Let τ0 := (j1 · · · jq) be a cycle of π0, where q, j1, . . . , jq ∈ S(1, dM).The F -crystal over k

(M(τ0), p−aϕ) := (⊕q

s=1W (k)ejs , p−aϕ)

has only one Newton polygon slope. As (M, p−aϕ) is a direct sum of such F -crystals(M(τ0), p

−aϕ), it is a direct sum ⊕γ∈Q(Mγ, p−aϕ) as in the statement 1.4.3 (d) with

Mγ := W (γ)(M, p−aϕ). As each W (k)ej is normalized by Im(µ0), for each element i ∈S(a, b) there exists a subset Ji of S(1, dM) such that we have F i

0(M) = ⊕j∈JiW (k)ej . This

implies that for each element i ∈ S(a, b) we have F i0(M) = ⊕γ∈QF

i0(M) ∩Mγ .

Let now (F i1(M))i∈S(a,b) be another lift of (M,ϕ,G) such that for each element

i ∈ S(a, b) we have F i1(M) = ⊕γ∈QF

i1(M) ∩ Mγ . Let µ1 : Gm → G be the (factor-

ization of the) inverse of the canonical split cocharacter of (M, (F i1(M))i∈S(a,b), ϕ), cf.

Subsection 3.1.7. It normalizes Mγ for all γ ∈ Q, cf. the functorial aspects of [Wi,p. 513]. As W (0)(EndB(k)(M [ 1p ]), ϕ) = ⊕γ∈QEndB(k)(Mγ [

1p ]) and as Lie(L0

G(ϕ)B(k)) =

W (0)(EndB(k)(M [ 1p ]), ϕ)∩ g[ 1p ], the image of dµ1,B(k) belongs to Lie(L0G(ϕ)B(k)) and thus

(cf. end of Subsection 2.3.8 (b)) µ1 factors through L0G(ϕ). Also µ0 factors through L0

G(ϕ),cf. end of Subsection 4.1.1.

Let L0,scQp

be the simply connected semisimple group over Qp whose semisimple Lie al-

gebra over Qp is x ∈ Lie(L0G(ϕ)

derB(k))|ϕ(x) = x. The adjoint group of L0,sc

Qpis the identity

component of the group of Lie automorphisms of x ∈ Lie(L0G(ϕ)

derB(k))|ϕ(x) = x. This

implies that L0,scB(k) is the simply connected semisimple group cover of L0

G(ϕ)derB(k). Thus we

can speak about the image Im(L0,scQp

(Qp) → L0G(ϕ)B(k)(B(k))). Let h ∈ Im(L0,sc

Qp(Qp) →

L0G(ϕ)B(k)(B(k))) ∩ L; we have hϕ = ϕh. The Zariski closure of such h’s in L0

G(ϕ)derB(k)

is L0G(ϕ)

derB(k) itself. Due to the uniqueness of µ0 we have hµ0h

−1 = µ0. Thus µ0,B(k)

is a cocharacter of L0G(ϕ)B(k) that commutes with L0

G(ϕ)derB(k). Therefore µ0,B(k) factors

through Z0(L0G(ϕ)B(k)). Thus µ0 and µ1 are two cocharacters of L0

G(ϕ) that commuteand therefore (cf. Lemma 3.1.1) they coincide. Thus for each number i ∈ S(a, b) we haveF i0(M) = F i

1(M). This proves the existence (cf. first paragraph of Subsection 4.1.2) andthe uniqueness (cf. this paragraph) of a lift (F i

1(M))i∈S(a,b) of (M,ϕ,G) such that for

85

each number i ∈ S(a, b) we have a direct sum decomposition F i1(M) = ⊕γ∈QF

i1(M)∩Mγ.

Therefore 1.4.3 (c) ⇒ 1.4.3 (d).

We show that 1.4.3 (d) ⇒ 1.4.3 (b). As the statement 1.4.3 (d) holds, L0G(ϕ) is a Levi

subgroup scheme of P+G (ϕ) (cf. Lemma 2.6.3). Let µ1 : Gm → G be the (factorization

of the) inverse of the canonical split cocharacter of (M, (F i1(M))i∈S(a,b), ϕ), cf. Subsection

3.1.7. As the statement 1.4.3 (d) holds, from its uniqueness part we get that every inner au-tomorphism of (M,ϕ,G) defined by an element h ∈ Im(L0,sc

Qp(Qp) → L0

G(ϕ)B(k)(B(k)))∩L

normalizes (F i1(M))i∈S(a,b). Thus we have hµ1h

−1 = µ1. As in the previous paragraph,we argue that the identities hµ1h

−1 = µ1 imply that µ factors through Z0(L0G(ϕ)). There-

fore L0G(ϕ) centralizes Im(µ1) and thus it normalizes the filtration (F i

1(M))i∈S(a,b) ofM . Therefore (F i

1(M))i∈S(a,b) is a U-canonical lift. Thus 1.4.3 (d) ⇒ 1.4.3 (a). As1.4.3 (a) ⇔ 1.4.3 (b), we get that 1.4.3 (d) ⇒ 1.4.3 (b).

4.1.3. End of the proof of Theorem 1.4.3. The four statements of Theorem 1.4.3 areequivalent, cf. Section 4.1 and Subsections 4.1.1 and 4.1.2. As µ0 = µ1 (cf. Subsection4.1.2), for each number i ∈ S(a, b) we have F i

0(M) = F i1(M).

We consider the case when k = k and for all numbers j ∈ S(1, dM) we have ej = e(0)j .

Thus π0 = π. For each cycle τ0 = (j1 · · · jq) of π0 = π, the function f(τ0) : Z/qZ → S(a, b)that takes [s] ∈ Z/qZ to the number njs ∈ S(a, b) of the statement 1.4.3 (c), is not periodic(cf. the non-periodicity of the functions θl’s introduced in Subsection 4.1.1). Thus the triple(M(τ0), (F

i(M)∩M(τ0))i∈S(a,b), ϕ) is circular indecomposable. As (M, (F i(M))i∈S(a,b), ϕ)is the direct sum of (M(τ0), (F

i(M) ∩M(τ0))i∈S(a,b), ϕ)’s, it is cyclic.

We point out that Lemma 2.8.3 implies that this type of direct sum decomposition of(M, (F i(M))i∈S(a,b), ϕ) can not be refined.

4.1.4. Canonical cocharacters. Let (M, (F i(M))i∈S(a,b), ϕ, G) be a U-canonical lift.

As pointed out in Subsection 4.1.1, L0G(ϕ) is a Levi subgroup scheme of P+

G (ϕ). As thenatural homomorphism L0

G(ϕ) → P−G (ϕ)/U−

G (ϕ) is an isomorphism (cf. Lemma 2.3.9),L0G(ϕ) is also a Levi subgroup scheme of P−

G (ϕ).

We take µ : Gm → G to be the inverse of the canonical split cocharacter of(M, (F i(M))i∈S(a,b), ϕ). We refer to such a µ as the canonical cocharacter of (M,ϕ,G). Itsgeneric fibre factors through Z0(L0

G(ϕ)B(k)), cf. Subsection 3.1.8. Thus µ factors throughZ0(L0

G(ϕ)). The cocharacter µW (k) : Gm → GW (k) is µ0 : Gm → GW (k) of the statement

1.4.3 (c), cf. end of Subsection 4.1.1.

4.1.5. Automorphisms. Let (M,ϕ,G) be a Shimura p-divisible object over k. Byan automorphism (resp. an inner automorphism) of it we mean an automorphism h :(M,ϕ)→(M,ϕ) that takes G onto G (resp. that is defined by an element h ∈ G(W (k))).

If (M,ϕ,G) is U-ordinary and (F i(M))i∈S(a,b) is its U-canonical lift, then each suchh normalizes (F i(M))i∈S(a,b) (cf. the uniqueness part of 1.4.3 (b)). Conversely we have:

4.1.6. Lemma. We assume that k = k. If there exists a lift (F i(M))i∈S(a,b) of(M,ϕ,G) that is normalized by every inner automorphism of (M,ϕ,G), then the quadruple(M, (F i(M))i∈S(a,b), ϕ, G) is a U-canonical lift.

86

Proof: We take µ : Gm → G to be the canonical split cocharacter of the triple(M, (F i(M))i∈S(a,b), ϕ). Let L0,sc

Qpbe as in Subsection 4.1.2. Each element h of the in-

tersection Im(L0,scQp

(Qp) → L0G(ϕ)B(k)(B(k))) ∩ L is an inner automorphism of (M,ϕ,G).

Thus the lift (F i(M))i∈S(a,b) of (M,ϕ,G) is normalized by such an h and therefore (dueto the functorial property of canonical split cocharacters) we have an identity hµh−1 = µ.As in Subsection 4.1.2 we argue that the identities hµh−1 = µ imply that µB(k) fac-tors through Z0(L0

G(ϕ)B(k)). Using this, as in the end of Subsection 4.1.2 we get that(M, (F i(M))i∈S(a,b), ϕ, G) is a U-canonical lift.

4.1.7. Automorphisms as Zp-valued points. We consider the case when k = k and

(M,ϕ,G) is U-ordinary. Let µ : Gm → G, σϕ,MZp, GZp

, T (0), and T(0)Zp

be as in Subsection

4.1.1. Let µ0 : Gm → G be as in the statement 1.4.3 (c). We have µ0 = µ, cf. Subsection

4.1.4. Let C1,Zp(resp. C0,Zp

) be the centralizer of T(0)Zp

in GLMZp(resp. in GZp

); it is a

reductive group scheme over Zp (cf. Subsection 2.3.13). If h is an automorphism (resp.an inner automorphism) of (M,ϕ,G), then from the uniqueness property of µ0 in 1.4.3 (c)we get that hµ0h

−1 = hµh−1 = µ0 = µ. Thus h is fixed by σϕ = ϕµ(p) and therefore wehave h ∈ GLMZp

(Zp). As T (0) is generated by σmϕ (µ)’s with m ∈ N ∪ 0, we also get

that h is fixed by T (0). Thus h is defined by an element of C1,Zp(Zp) (resp. of C0,Zp

(Zp)).The converse of this holds as well and therefore C1,Zp

(Zp) (resp. C0,Zp(Zp)) is the group

of automorphisms (resp. of inner automorphisms) of (M,ϕ,G).

As ϕ fixes each element of Lie(C0,Qp), we have Lie(C0,B(k)) ⊆ Lie(L0

G(ϕ)B(k)). As

T(0)B(k) 6 Z0(L0

G(ϕ)B(k)), the group L0G(ϕ)B(k) commutes with T

(0)B(k) and therefore we have

Lie(L0G(ϕ)B(k)) ⊆ Lie(C0,B(k)). Thus Lie(L0

G(ϕ)B(k)) = Lie(C0,B(k)). As both groupsL0G(ϕ)B(k) and C0,B(k) are connected, we get L0

G(ϕ)B(k) = C0,B(k) (cf. Lemma 2.3.6).

Therefore L0G(ϕ) = C0,W (k) is the centralizer of T (0) in G.

4.2. Proofs of Theorem 1.4.4 and Corollary 1.4.5

Until Subsection 4.2.1 we assume that k = k. To prove Theorem 1.4.4 we can assumethat (M, (F i(M))i∈S(a,b), ϕ, G) is an Sh-canonical lift. Let µ : Gm → G, σϕ, MZp

, GZp,

T (0), and T(0)Zp

be as in Subsection 4.1.1. We fix a family (tα)α∈J of tensors of T (MZp)

as in Subsection 3.1.7; we think of it as a family of tensors of T (M) fixed by G andϕ and given to us before considering MZp

. As Lie(P+G (ϕ)B(k)) is normalized by µ and

ϕ, it is also normalized by σϕ. Thus σϕ normalizes Lie(P+G (ϕ)) = Lie(P+

G (ϕ)B(k)) ∩ g

and therefore P+G (ϕ) is the extension to W (k) of a subgroup scheme P+

Zpof GLMZp

, cf.

Lemma 2.4.1. Obviously P+Zp

is a parabolic subgroup scheme of GZpthat contains T

(0)Zp

.

The centralizer C0,Zpof T

(0)Zp

in GZpis a Levi subgroup of P+

Zp(overW (k), this follows from

Subsection 4.1.7). Thus there exists a Borel subgroup scheme BZpof P+

Zpthat contains

T(0)Zp

. Let TZpbe a maximal torus of the Borel subgroup scheme BZp

∩ C0,Zpof C0,Zp

, cf.

Subsection 2.3.8 (b); it is a maximal torus of BZpthat contains T

(0)Zp

. Let T := TW (k) and

B := BW (k). The notations match with Subsection 3.2.3 i.e., ϕ normalizes t := Lie(T ) and

87

leaves invariant the Lie subalgebra b := Lie(B) of Lie(P ). But the cocharacter µ : Gm → Gis uniquely determined by its G(W (k))-conjugacy class [µ] and by the facts that: (i) itfactors through T , and (ii) it acts via inner conjugation on b through non-positive powersof the identity character of Gm (this makes sense as b ⊆ Lie(Lie(P+

G (ϕ))) ⊆ Lie(P )).Thus, as ϕ = σϕµ(

1p ), the inner isomorphism class of (M,ϕ,G) is uniquely determined by

[µ] and by the quintuple

(22) (MZp, GZp

, BZp, TZp

, (tα)α∈J ).

Obviously the statement 1.4.4 (a) implies both statements 1.4.1 (a) and 1.4.4 (b). Wecheck that 1.4.1 (a) ⇒ 1.4.4 (a). Let (M1,Zp

, G1,Zp, B1,Zp

, T1,Zp, (tα)α∈J ) be the quintuple

similar to (22) but obtained starting from another Sh-canonical lift (M, (F ig(M))i∈S(a,b), gϕ,G)

and from the canonical cocharacter µg : Gm → G of (M, gϕ,G). As µ is also a Hodgecocharacter of (M, gϕ,G), we have [µ] = [µg] (cf. Lemma 3.1.3). Thus to prove that(M,ϕ,G) and (M, gϕ,G) are inner isomorphic, it suffices to show there exists an isomor-phismS : (MZp

, (tα)α∈J )→(M1,Zp, (tα)α∈J ) that takesBZp

ontoB1,Zpand takes TZp

ontoT1,Zp

. But the pairs (T,B) and (T1,W (k), B1,W (k)) are G(W (k))-conjugate (cf. Subsection2.3.8 (d)) and thus such an isomorphism S exists after tensorization withW (k) and there-fore (cf. a standard application of Artin’s approximation theorem) also after tensorizationwith the ring of Witt vectors with coefficients in some finite field kg of characteristic p.Therefore the existence of S is measured by a class in H1(Gal(W (kg)/Zp), TZp

). Buteach such class is trivial, cf. Lang theorem of [Se2, Ch. III, §2, case 2.3 a)] and thefact that Zp is henselian. Thus S exists. Therefore 1.4.1 (a) ⇒ 1.4.4 (a) and thus also1.4.1 (a) ⇒ 1.4.4 (b).

We check by induction on dim(Gk) that 1.4.4 (b) ⇒ 1.4.1 (a) (the argument extendsuntil Subsection 4.2.1). We can assume that G is adjoint and that the cocharacter µ :Gm → G is non-trivial, cf. Corollary 3.1.5 and Subsection 3.3.2. We can also assumethat I = S(1, n) and Γ = (1 · · ·n). Let N be the closed subgroup scheme of G definedbefore Lemma 3.1.1. We recall that σϕ(N) is the commutative, connected, unipotent,smooth, closed subgroup scheme of G whose Lie algebra is σϕ(Lie(N)) = Ψ(Lie(N)), cf.Subsections 2.4.7 and 2.5.5. Let g ∈ G(k) be g modulo p. Let h ∈ G(k) be such that we havehgΨ = gΨh. Thus h normalizes Ker(Ψ) = Ker(gΨ). We have Ker(Ψ) = ⊕i∈I1Lie(Pk∩Gi,k)(see (10a) and see Subsection 3.4.7 for the subset I1 of I). Thus for each element i ∈ I1,the component of h in Gi,k(k) belongs to (Pk ∩Gi,k)(k) (cf. Lemma 2.3.4). Thus we haveh ∈ P (k). By replacing gϕ with hgϕh−1, where h ∈ P (W (k)) lifts h, we can assumegΨ = Ψ. Thus I(M, gϕ,G) = I(M,ϕ,G) = dim(I), where I := Im((gΨ)dim(Gk)) =

Im(Ψdim(Gk)

). If δ := |I1|n , then we have I = W (−δ)(g, ϕ)⊗W (k) k (cf. Subsections 3.4.12

and 3.4.13 applied with g = 1M ). Thus the reduced scheme of the normalizer of I in Gk

is P−G (ϕ)k, cf. Corollary 3.6.4 (see Subsections 2.6.5 and 2.6.6 for P+

G (ϕ), P−G (ϕ), P+

G (gϕ),

and P−G (gϕ)). As P−

G (gϕ)k also normalizes I, we have P−G (gϕ)k 6 P−

G (ϕ)k. We check that

the groups P−G (ϕ)k, P

+G (ϕ)k, P

−G (gϕ)k, and P

+G (gϕ)k have equal dimensions. Based on the

ranks part of Subsection 2.7.1, it suffices to show that dim(P+G (ϕ)k) = dim(P+

G (gϕ)k). To

check this last equality, up to an inner isomorphism we can assume that g ∈ P+G (ϕ) (cf.

property 3.3 (b)), and thus the last equality follows from Theorem 3.7.1 (d).

88

By reasons of dimensions we get that P−G (gϕ)k = P−

G (ϕ)k. Thus by replacing gϕ

with g1gϕg−11 , where g1 ∈ Ker(G(W (k)) → G(k)), we can assume that P−

G (gϕ) = P−G (ϕ)

(cf. Subsection 2.3.8 (e)). We reached the following context: g ∈ P−G (ϕ)(W (k)) and g

centralizes Im(Ψ).

In this paragraph we check that the fact that g centralizes Im(Ψ) implies that we haveg ∈ σϕ(N)(k). If i ∈ I1 (resp. if i ∈ I \ I1), then the component of g in Gi+1(k) centralizesthe Lie algebra of a non-trivial and commutative unipotent radical σϕ(N)k ∩ Gi+1,k of aparabolic subgroup of Gi+1,k (resp. centralizes Lie(Gi+1,k)); thus this component belongsto (σϕ(N)k ∩ Gi+1,k)(k) (resp. is the identity element), cf. Lemma 2.3.5 (resp. cf. thefact that Gi+1,k is adjoint). Here we identify the indices n+ 1 = 1.

As N and σϕ(N) are products of Ga group schemes (cf. Subsection 2.5.5 for N), wecan identify them with the vector group schemes defined by their Lie algebras. Thus, asg ∈ σϕ(N)(k), there exists an element x ∈ Lie(N) such that under the last identificationthe element g corresponds to the reduction modulo p of Ψ(x) ∈ Lie(σϕ(N)); in other words

we have exp(Ψ(x)) := 1M +∑

i≥1Ψ(x)i

i! ∈ G(W (k)) and its reduction modulo p is g (to

be compared with Subsection 2.5.4). Let g1 ∈ Ker(N(W (k)) → N(k)) 6 P−G (ϕ)(W (k))

be such that under the last identifications we have g1 = exp(−px) := 1M +∑

i≥1(−px)i

i!.

As ϕg−11 ϕ−1 = exp(Ψ(x)) ∈ G(W (k)) modulo p is g, by replacing gϕ with g1gϕg

−11 ,

we can assume g is the identity element. Let L0G(ϕ) and U−

G (ϕ) be as in Section 2.6.

We write g = ur, where u ∈ U−G (ϕ)(W (k)) and r ∈ L0

G(ϕ)(W (k)) are congruent to1M modulo p. We consider a filtration M0,+ ⊆ M1,+ ⊆ . . . ⊆ M s,+ = M of Mby direct summands that is normalized by P+

G (ϕ) and such that U+G (ϕ) acts trivially

on all the factors M j+1,+/M j,+ (here s ∈ N ∪ 0 and j ∈ S(1, s − 1)). There-fore we have (M j+1,+/M j,+, gϕ) = (M j+1,+/M j,+, rϕ) and thus the Newton polygonsof (M j+1,+/M j,+, gϕ) and (M j+1,+/M j,+, rϕ) are equal. Therefore the Newton poly-gons of (M, gϕ) and (M, rϕ) are also equal. Thus, as 1.4.1 (a) ⇔ 1.4.1 (b), to provethat 1.4.4 (b) ⇒ 1.4.1 (a) we can assume that g = r. As g = r and 1M are con-gruent modulo p, the FSHW maps of (M,ϕ, L0

G(ϕ)) and (M, gϕ, L0G(ϕ)) coincide. As

dim(L0G(ϕ)k) < dim(Gk) (see first paragraph of Subsection 3.8.3), by induction we get

that (M, gϕ, L0G(ϕ)) is Sh-ordinary. Therefore (M, gϕ,G) is Sh-ordinary, cf. Theorem

3.7.1 (f). This ends the induction. Thus 1.4.4 (b) ⇒ 1.4.1 (a).

This ends the proof of Theorem 1.4.4.

4.2.1. Proof of 1.4.5. To prove Corollary 1.4.5 we can assume that G is adjoint, cf.Corollary 3.1.5. We first assume that k = k. Let Y and Y o

k be as in Subsection 3.8.2. As1.4.1 (a) ⇔ 1.4.4 (b), the fact that (M, gϕ,G) is Sh-ordinary depends only on g = gad; theanalogue of this holds for the gk’s of Corollary 1.4.5. Thus, as 1.4.1 (a) ⇔ 1.4.1 (b), wecan take the open subscheme U of Gk = Gad

k to be defined by the property that for eachaffine, open subscheme Y of G as in Subsection 3.8.2, we have U ∩ Yk = Y o

k .

We come back to the general case when k is not necessarily k. As Uk exists, it isautomatically invariant under the action of Gal(k) on Gk = Gad

k. Thus we have a canonical

Galois descent datum on the open subscheme of Uk of Gk. It is effective (cf. [BLR, Ch.6, Sect. 6.1, Thm. 6]) and thus Uk is the natural pull back of a unique open subscheme U

89

of Gk. From Fact 3.2.1 (a) we get that U has the desired property. This proves Corollary1.4.5.

4.2.2. Remark. The fact that 1.4.1 (a) ⇒ 1.4.1 (b) follows also from the existence ofthe quintuple (22) and from [RR, Thm. 4.2]. Also the fact that 1.4.1 (b) ⇒ 1.4.1 (c)can be obtained by combining loc. cit. with Subsections 3.2.3 and 3.4.11. But Theorem1.4.2 and the implications 1.4.1 (b) ⇒ 1.4.1 (a) and 1.4.4 (b) ⇒ 1.4.4 (a) do not followfrom [RR]. Also, we think that the elementary arguments of Subsections 3.8.1 to 3.8.3, aresimpler than the ones of [RR] that led to the proof of [RR, Thm. 4.2]. Moreover, the partof Subsection 3.8.2 that involves the existence of a smallest Newton polygon N works inthe general context of an arbitrary triple (M,ϕ,G) for which the property 1.3.2 (c) holds,where G is a smooth, closed subgroup scheme of GLM whose special fibre Gk is connected.

4.3. Complements

We end this Chapter with five complements.

4.3.1. Proposition. Let (M, (F i(M))i∈S(a,b), ϕ, G) be an Sh-canonical lift.

(a) We have Lie(P+G (ϕ)) ⊆ F 0(g) = F 0(g)⊕F 1(g) and Lie(P−

G (ϕ)) ⊆ F 0(g)⊕F−1(g).Thus P+

G (ϕ) 6 P and P−G (ϕ) 6 Q, where P and Q are as in Subsections 2.5.3 and 2.5.5.

(b) Let h ∈ G(B(k)) be such that the quadruple (h(M), (h(F i(M)))i∈S(a,b), ϕ, G(h)) isa Shimura filtered p-divisible object, where G(h) is the Zariski closure of GB(k) in GLh(M).

Then (h(M), (F i(M)[ 1p ] ∩ h(M))i∈S(a,b), ϕ, G(h)) is an Sh-canonical lift.

Proof: To prove (a) we can assume that k = k, that G is adjoint, that I = S(1, n), andthat Γ = (1 · · ·n). Based on the first paragraph of Section 4.2, we can also assume that(5) holds. Thus the first part (a) follows from the end of Subsection 3.4.3. The secondpart follows from the first part and from Lemma 2.3.6.

We now prove (b). Let g := h−1ϕhϕ−1 ∈ G(B(k)), cf. Lemma 2.4.6. The element h−1

defines an isomorphism (h(M), (h(F i(M)))i∈S(a,b), ϕ, G(h))→(M, (F i(M))i∈S(a,b), gϕ,G).Thus we have ϕ−1(M) = (gϕ)−1(M) (cf. property 1.3.2 (a)) and therefore g(M) = M .Thus g ∈ G(W (k)). Moreover (M,ϕ) and (M, gϕ) have the same Newton polygon andtherefore (cf. Theorem 1.4.1) (M, gϕ,G) is Sh-ordinary. Let µ and µh be the canonicalcocharacters of (M,ϕ,G) and (h(M), ϕ, G(h)) (respectively). Let (F i

h(M))i∈S(a,b) be theSh-canonical lift of (h(M), ϕ, G(h)). We know that h−1µhh is G(W (k))-conjugate to µ (cf.Lemma 3.1.3) and thus the cocharacters of Gab = G(h)ab defined by µ and µh coincide.

Let Ph,B(k) be the normalizer of (F ih(M)[ 1

p])i∈S(a,b) in GB(k). The groups Ph,B(k) and

PB(k) are G(B(k))-conjugate (as µh,B(k) and µB(k) are G(B(k))-conjugate) and contain

P+G (ϕ)B(k) (cf. the very definition of Sh-ordinariness); thus we have Ph,B(k) = PB(k) (cf.

property 2.3.1 (b)). As µB(k) and µh,B(k) are two cocharacters of Z0(L0G(ϕ)B(k)) (cf.

Subsection 4.1.4), they commute and are cocharacters of PB(k) = Ph,B(k). Moreover, they

act as the inverse of the identity character of Gm on F 1(g)[ 1p] (i.e., on the Lie algebra of

the unipotent radical of Ph,B(k) = PB(k)) and act trivially on F 0(g)[ 1p ]/F1(g)[ 1p ]. Thus

the cocharacters of GadB(k) defined by µB(k) and µh,B(k) coincide. From this and the above

90

part pertaining to Gab = G(h)ab, we get that µB(k) = µh,B(k). This implies that for all

i ∈ S(a, b) we have F ih(M)[ 1

p] = F i(M)[ 1

p] and therefore also F i

h(M) = F i(M)[ 1p]∩h(M).

4.3.2. Inheritance properties. We consider the case when k = k and (M,ϕ,G) is

Sh-ordinary (resp. U-ordinary). Let GZpand T

(0)Zp

be as in Subsection 4.1.1. Let GZp

be a reductive, closed subgroup scheme of GZpthat contains T

(0)Zp

. Let G := GW (k).

As P+

G(ϕ) (resp. L0

G(ϕ)) is contained in P+

G (ϕ) (resp. in L0G(ϕ)), the quadruple

(M, (F i(M))i∈S(a,b), ϕ, G) is an Sh-canonical (resp. a U-canonical) lift. Due to the unique-

ness part of 1.4.3 (c), the canonical cocharacters of (M,ϕ, G) and (M,ϕ,G) coincide (i.e.,define the same cocharacter of G).

Let TZpbe a maximal torus of the centralizer of T

(0)Zp

in GZp, cf. Subsections

2.3.13 and 2.3.8 (b). Similarly, let T bigZp

be a maximal torus of the centralizer of TZp

in GLMZp. Thus T big

Zpis a maximal torus of GLMZp

that contains TZp. Let T := TW (k)

and T big := T bigW (k); we have T 6 T big 6 GLM . The quadruples (M, (F i(M))i∈S(a,b), ϕ, T )

and (M, (F i(M))i∈S(a,b), ϕ, Tbig) are Sh-canonical lifts.

4.3.3. U-ordinariness via Sh-ordinariness. We assume that (M, (F i(M))i∈S(a,b), ϕ, G)is an Sh-canonical lift. Let G′ be a reductive, closed subgroup scheme of GLM that con-tains G and such that (M,ϕ,G′) is a Shimura p-divisible object. As we have a direct sumdecomposition M = ⊕γ∈QW (γ)(M,ϕ) (cf. statement 1.4.3 (d)), from Lemma 2.6.3 we getthat L0

G′(ϕ) is a Levi subgroup scheme of P+G′(ϕ).

In this paragraph we check that if Z0(L0G(ϕ)) is a subtorus of Z0(L0

G′(ϕ)) (e.g., thisholds if L0

G(ϕ)der = L0

G′(ϕ)der), then (M, (F i(M))i∈S(a,b), ϕ, G′) is a U-canonical lift. The

canonical split cocharacter µ of (M,ϕ,G) factors through Z0(L0G(ϕ)) (cf. Subsection 4.1.4)

and therefore also through Z0(L0G′(ϕ)). Thus L0

G′(ϕ) commutes with Im(µ) and thereforeit normalizes (F i(M))i∈S(a,b). Thus (M, (F i(M))i∈S(a,b), ϕ, G

′) is a U-canonical lift.

Example: if L0G′(ϕ) is a torus, then (M, (F i(M))i∈S(a,b), ϕ, G

′) is a U-canonical lift.

4.3.4. Finiteness of U-ordinary classes. We consider the case when k = k and(M,ϕ,G) is U-ordinary. Referring to Subsection 4.3.2, we choose the group scheme GZp

to be a maximal torus TZpof GZp

. Up to an inner isomorphism, we can assume G = TW (k)

is a fixed maximal torus T of G whose special fibre Tk is a subgroup of Pk. As in Subsection3.2.3 we argue that there exists an element w0 ∈ G(W (k)) that normalizes T and suchthat (M,w0ϕ,G) is Sh-ordinary. If t ∈ T (W (k)), then (M,ϕ, T ) and (M, tϕ, T ) are innerisomorphic (cf. Theorem 1.4.4) and therefore also (M,ϕ,G) and (M, tϕ,G) are innerisomorphic. Thus the number NU of inner isomorphism classes of U-ordinary Shimurap-divisible objects of the form (M, gϕ,G), where g ∈ G(W (k)), is at most equal to theorder |WG| of the Weyl group scheme WG of G.

4.3.5. Example for NU . In this Subsection we assume that G is split, adjoint andwe use the notations of Section 3.4. We also assume that k = k, that I = S(1, n), thatΓ = (1 · · ·n), that X = B1 = A1, that I1 = I, and that (M, (F i(M))i∈S(a,b), ϕ, G) is anSh-canonical lift. Let T be a maximal torus of G whose Lie algebra is normalized by ϕ,

91

cf. Subsection 4.3.2. As X = B1 = A1 and as I1 = I, the normalizer P of (F i(M))i∈S(a,b)

in G is a Borel subgroup scheme of G; moreover G is isomorphic to PGLn2 . Let N

be the normalizer of T in G. The Weyl group scheme WG := N/T is (Z/2Z)n. Forw ∈ WG(W (k)), we denote also by w an arbitrary element of N(W (k)) that lifts w.The triple (M,wϕ, T ) is Sh-ordinary and (see Subsection 4.3.4) its isomorphism classdoes not depend on the choice of the lift (of w). We have a direct sum decompositionM = ⊕γ∈QW (γ)(M,wϕ) (cf. statement 1.4.3 (d)) and therefore from Lemma 2.6.3 we getthat L0

G(wϕ) is a Levi subgroup scheme of P+G (wϕ). As wϕ normalizes Lie(T ), we have

Lie(T ) 6 W (0)(g, wϕ) = Lie(L0G(wϕ)). Thus T 6 L0

G(wϕ), cf. Lemma 2.3.6.

From the property 3.4.2 (a) we get that either P+G (wϕ) = G or for each element i ∈ I

the intersection P+G (wϕ) ∩Gi is a Borel subgroup scheme of Gi. If P

+G (wϕ) = G, then all

Newton polygon slopes of (g[ 1p ], wϕ) are 0 and thus we have L0G(wϕ) = G 66 P ; therefore

(M,wφ,G) is not U-ordinary. If each intersection P+G (wϕ)∩Gi is a Borel subgroup scheme

of Gi, then L0G(wϕ) is a maximal torus of G and thus it is T ; therefore (M,wφ,G) is U-

ordinary. Let NU be the number of elements w ∈WG such that (M,wφ,G) is U-ordinary.

We check that if n is odd (resp. is even), then NU = 2n−1 (resp. NU = 2n−1−(n−1

n2

)).

Let NZ := 2n −NU . Based on the previous paragraph, it suffices to show that there existprecisely NZ = 2n−1 (resp. NZ = 2n−1 +

(n−1

n2

)) elements w ∈ WG such that (g[ 1p ], wϕ)

has all Newton polygon slopes 0. Let gα1be the Lie algebra of the unipotent radical of

P ∩G1 and let g−α1be its opposite with respect to T (see Subsections 3.4.3 to 3.4.7). We

write w =∏n

i=1wi, where wi ∈ (N ∩Gi)(W (k)).

We assume that n is odd. If (wϕ)n(gα1[ 1p ]) is gα1

[ 1p ] (resp. is g−α1[ 1p ]), then gα1

6⊆

W (0)(g, wϕ) (resp. then all Newton polygon slopes of (g[ 1p ], wϕ) are 0). Thus, if w2, . . . , wn

are fixed, then there exists a unique choice for w1 in order that all Newton polygon slopesof (g[ 1p ], wϕ) are 0. Therefore we have NZ = 2n−1 = NU .

We assume that n is even. If (wϕ)n(gα1[ 1p ]) = g−α1

[ 1p ], then all Newton polygon slopes

of (g[ 1p ], wϕ) are 0. As in the previous paragraph we argue that for precisely 2n−1 elements

w ofWG we have (wϕ)n(gα1[ 1p ]) = g−α1

[ 1p ]. We now count the number of elements w ∈WG

such that (wϕ)n(gα1[ 1p ]) = gα1

[ 1p ] and all Newton polygon slopes of (g[ 1p ], wϕ) are 0. For i ∈

S(1, n+1) let θ(w)i ∈ −1, 1 be defined inductively by the rule: p

∑i

j=1−θ(w)j (wϕ)i(gα1

)is a direct summand of g. This makes sense as I = I1. We always have θ(w)1 = 1. The setNG(W (k)) is in bijection to the set of n+1-tuples (θ(w)1, . . . , θ(w)n+1) ⊆ −1, 1n+1 thatsatisfy θ(w)1 = 1. We have (wϕ)n(gα1

) = gα1and all Newton polygon slopes of (g[ 1p ], wϕ)

are 0 if and only if θ(w)n+1 = 1 and∑n

i=1 θ(w)i = 0. The number of n + 1-tuples(θ(w)1, . . . , θ(w)n+1) ∈ −1, 1n+1 such that

∑ni=1 θ(w)i = 0 and θ(w)1 = θ(w)n+1 = 1 is

equal to the number of n−1-tuples (θ(w)2, . . . , θ(w)n) ∈ −1, 1n−1 that contain preciselyn2 entries −1 and thus it is

(n−1

n2

). We conclude that NZ = 2n−1 +

(n−1

n2

).

Let w be w modulo p. For i ∈ I we have Ker((wΨ)2) ∩ Lie(Gi,k) = Lie(Gi,k) if andonly if wi+1 is not the identity element (here wn+1 := w1). This implies that the innerisomorphism class of (M,wϕ,G) is uniquely determined by w. Thus the number NU wehave computed is exactly the number NU introduced in Subsection 4.3.4.

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5 Abstract theory, III: formal Lie groups

In this Chapter we assume (M, (F i(M))i∈S(a,b), ϕ, G) is an Sh-canonical lift. Except

for Lemma 5.3.3, in this Chapter we also assume that k = k. We emphasize that in thisChapter we take

∗ ∈ −,+

and non-−∗ means non-negative if ∗ = + and non-positive if ∗ = −; thus the upper index∗ will not refer to duals of modules or to pull backs of morphisms.

We recall that we use the notations listed in Section 2.1. In particular, we haveg = Lie(G). Following the pattern of the p-divisible groups D+(0), D+

W (k), D−(0), and

D−W (k) of Subsection 1.2.4, in Section 5.1 we introduce sign p-divisible groups of (M,ϕ,G).

In Section 5.2 we get minimal decompositions for some of them. In Sections 5.3 and 5.4we introduce the first and the second formal Lie groups of (M,ϕ,G).

5.1. Sign p-divisible groups

Let µ : Gm → G, σϕ = ϕµ(p), MZp, GZp

, T (0), and T(0)Zp

be as in Subsection 4.1.1.Let BZp

, TZp, B = BW (k), and T = TW (k) be as in the first paragraph of Section 4.2.

Thus TZpis a maximal torus of a Borel subgroup scheme BZp

of GZp, we have T

(0)Zp

6 TZp,

and the Lie algebras of T and B are normalized and left invariant (respectively) by ϕ. We

recall that we view µ as a cocharacter of T (0) = T(0)W (k), that the group schemes P ∗

G(ϕ)

and L0G(ϕ) were defined in Section 2.6, that U∗

G(ϕ) is the unipotent radical of P ∗G(ϕ), and

(cf. Subsection 4.1.4) that L0G(ϕ) is a Levi subgroup scheme of P ∗

G(ϕ). Let S∗G(ϕ) be the

solvable, smooth, closed subgroup scheme of P ∗G(ϕ) generated by U∗

G(ϕ) and T(0); as T (0)

normalizes U∗G(ϕ) we have a short exact sequence 1 → U∗

G(ϕ) → S∗G(ϕ) → T (0) → 1. Let

t(0) := Lie(T (0)), u∗ := Lie(U∗G(ϕ)), and s∗ := t(0) ⊕ u∗ = Lie(S∗

G(ϕ)).

Let ϕ(−) be the σ-linear endomorphism of g that takes x ∈ g to pϕ(x). Let ϕ(+) := ϕ.We know that t(0) is normalized by ϕ and (being fixed by µ) is contained in F 0(g). FromProposition 4.3.1 (a) and the equivalence of statements 2.3.3 (b) and (d), we get thatF ∗1(g) ⊆ u∗ ⊆ F ∗1(g) ⊕ F 0(g). From the last two sentences we get that the triples(s+, F 1(g), ϕ(+)) and (s−, s− ∩ F 0(g), ϕ(−)) are filtered Dieudonne modules.

5.1.1. Definitions. We call the p-divisible group

D∗0

93

over k whose Dieudonne module is(s∗, ϕ(∗)),

as the standard non-−∗ p-divisible group of (M,ϕ,G). If t(0) is replaced by an arbitrarydirect summand E(0) of Lie(L0

G(ϕ)) normalized by ϕ(∗), we get the notion of a non-−∗p-divisible group D∗0(E(0)) of (M,ϕ,G); its Dieudonne module is (E(0)⊕u∗, ϕ(∗)). If E(0)

has rank 1 (resp. if E(0) is Lie(L0G(ϕ))), then we refer to D∗0(E(0)) as a minimal (resp. as

the maximal) non-−∗ p-divisible group of (M,ϕ,G).

By the positive (resp. the negative) p-divisible group

D+W (k) (resp. D−

W (k))

overW (k) of (M,ϕ,G) we mean the p-divisible group overW (k) whose filtered Dieudonnemodule is

(u+, F 1(g), ϕ) (resp. is (u−, u− ∩ F 0(g), ϕ(−))).

The uniqueness of D∗W (k) follows from properties 1.2.1 (b) and (c) once we recall that: (i)

all Newton polygon slopes of (u+, ϕ(+)) are positive (see formula for u+ = Lie(U+G (ϕ)) in

Subsection 2.6.2) and thus belong to the interval (0, 1], and (ii) all Newton polygon slopesof (u−, ϕ) belong to the interval [−1, 0) (cf. the oppositions of Section 2.7) and thereforeall Newton polygon slopes of (u−, ϕ(−)) belong to the interval [0, 1).

5.1.2. Lemma. The W (k)-submodule u− of g is the smallest direct summand of g thatcontains F−1(g) and is left invariant by ϕ(−). We have u− = F−1(g) if and only if allNewton polygon slopes of (g, ϕ) are integral.

Proof: We can assume that G is adjoint, that I = S(1, n), and that Γ = (1 · · ·n). Basedon the first paragraph of Section 4.2 (or on the equivalence 1.4.1 (a) ⇔ 1.4.4 (a)), we canuse the notations of Section 3.4. Thus the first part follows from (7). With the notationsof Subsection 3.4.7, both statements of the second part of the Lemma are equivalent tothe identity (I1, |E|) = (I, 1) (cf. (7)).

5.1.3. Lemma. The p-divisible group D∗W (k) is cyclic.

Proof: As L0G(ϕ) is the centralizer of T

(0) in G (cf. Subsection 4.1.7), the inner conjugationrepresentation T (0) → GLu∗ has a trivial kernel. Thus we can identify T (0) with a torusof GLu∗ . The quadruples (u−, u− ∩ F 0(g), ϕ(−), T (0)) and (u+, F 1(g), ϕ, T (0)) are Sh-canonical lifts. Thus (u+, F 1(g), ϕ) and (u−, u− ∩ F 0(g), ϕ(−)) are cyclic, cf. end ofTheorem 1.4.3. Thus D∗

W (k) is cyclic, cf. Fact 2.8.1 (b).

5.1.4. Groups of automorphisms. Each automorphism of (M,ϕ,G) normalizes(F i(M))i∈S(a,b) and u∗ and therefore it defines naturally automorphisms of both triples

(u+, F 1(g), ϕ) and (u−, u− ∩ F 0(g), ϕ(−)) and therefore also (cf. properties 1.2.1 (b) and(c)) of both D+

W (k) and D−W (k). Let A∗

1 (resp. A∗0) be the image of the group of auto-

morphisms (resp. of inner automorphisms) of (M,ϕ,G) in the group A∗2 := Aut(D∗

W (k)).

For i ∈ 0, 1 we refer to A∗i as the i-th group of automorphisms of D∗

W (k) with respect to

(M,ϕ,G). We have A∗0 6 A∗

1 6 A∗2.

94

5.2. Minimal decompositions

Until Section 5.4 we will assume that G is adjoint, that I = S(1, n), and that Γ =(1 · · ·n). We use the notations of Section 3.4 introduced before Subsection 3.4.11. Inparticular, e ∈ 1, 2, 3 (see Subsection 3.4.4), ℓ is the rank of the Lie type X of everysimple factor Gi of G, for each i ∈ I1 ⊆ I = S(1, en) we have a semisimple element li ∈ gi(see Section 3.4 and Subsection 3.4.4), for each i ∈ I the Lie algebra of Ti := Gi ∩ Tis denoted as ti, for each i ∈ I we have a root decomposition gi = ti ⊕α∈Φi

gi(α) (seeSubsection 3.4.3), and for each i ∈ I we have a standard base ∆i = α1(i), . . . , αℓ(i) ofΦi (see Subsection 3.4.5). Among the possible values of the pair (X , e), we single out theset of exceptional cases

E := (Aℓ, 2)|ℓ ∈ 2N.

5.2.1. Proposition. There exists a unique direct sum decomposition

(23) (u∗, ϕ(∗)) = ⊕j∈J(u∗j , ϕ(∗))

such that the following six properties hold:

(i) if (X , e) /∈ E, then for all elements j ∈ J the direct summand u∗j of u∗ is the Liealgebra of a closed subgroup scheme Uj of U∗

G(ϕ) that is a product of Ga subgroup schemesof G normalized by T ; if (X , e) ∈ E, then for all j ∈ J the direct sum u∗j ⊕ [u∗j , u

∗j ] is the

Lie algebra of a unipotent, smooth, closed subgroup scheme Uj of U∗G(ϕ) normalized by T

and of nilpotent class at most 2 and moreover, if this class is 2 (i.e., if [u∗j , u∗j ] 6= 0), then

there exists an element j′ ∈ J with the property that u∗j′ = [u∗j , u∗j ];

(ii) for all elements j1, j2 ∈ J with the property that [u∗j1 , u∗j2] 6= 0, there exists a

unique element j3 ∈ J \ j1, j2 such that we have [u∗j1 , u∗j2] ⊆ u∗j3;

(iii) all Newton polygon slopes of (u∗j , ϕ(∗)) are equal and belong to the interval (0, 1]if ∗ = + and to the interval [0, 1) if ∗ = −;

(iv) for each element j ∈ J there exists an element tj ∈ Lie(T(0)Zp

) whose action on

every intersection u∗j ∩ gi with i ∈ I, has only one eigenvalue that belongs to Gm(W (k));

if e = 1, then there exists a Zp-basis for Lie(T(0)Zp

) formed by elements whose actions on

every u∗j ∩ gi have only one eigenvalue, for all pairs (j, i) ∈ J × I;

(v) there exists a total ordering (to be denoted as ≤) on the set J of indices, withthe property that for each element j1 ∈ J the direct sum ⊕j≤j1u

∗j is the Lie algebra of a

connected, smooth, unipotent, normal, closed subgroup scheme of U∗G(ϕ);

(vi) if (u∗, ϕ(∗)) = ⊕j′∈J ′(u∗j′ , ϕ(∗)) is another decomposition such that the abovecondition (i) holds, then for each element j′ ∈ J ′ there exists an element j ∈ J such thatwe have u∗j ⊆ u∗j′.

Proof: For each root α ∈ Φ∗i we write α = ∗

∑ℓs=1 cs(α)αs(i), where cs(α) ∈ N ∪ 0. Let

Φ∗i (E) := α ∈ Φ∗

i |there is x ∈ E with cx(α) 6= 0.

95

We have (Φ∗i (E) + Φ∗

i (E)) ∩ Φi ⊆ Φ∗i (E) i.e., Φ

∗i (E) is a closed subset of Φi. From (7) we

get thatu∗ ∩ gi = ⊕α∈Φ∗

i(E)gi(α).

To define a direct summand u∗j of u∗ that is a T -module, it is enough to describe foreach i ∈ I the subset Φ∗

i (j) of Φ∗i (E) such that we have

u∗j ∩ gi = ⊕α∈Φ∗i(j)gi(α).

Let J∗ be the set of orbits of the permutation of ∪ni=1Φ

∗i (E) defined via restriction by

the permutation ΓΦ of Subsection 3.4.3. We identify J∗ with the set J∗of orbits of the

permutation of Φ∗1(E) defined via restriction by the permutation ΓΦ1

of Subsection 3.4.4.

The multiplication of roots by −1 defines bijections J+→J− and J+→J

−and therefore

for simplicity we will take J := J+ and J := J+. Let j ∈ J be the image of an arbitrary

element j ∈ J under the natural bijection J→J . For j ∈ J let Φ+i (j) := α ∈ Φ+

i (E)|α ∈j, let Φ−

i (j) := −Φ+i (j), and let u∗j := ⊕i∈I ⊕α∈Φ∗

i(j) gi(α).

If (X , e) /∈ E, then we have (Φ∗i (j) + Φ∗

i (j)) ∩ Φ∗i = ∅. If (X , e) ∈ E, then either

(Φ∗i (j) + Φ∗

i (j)) ∩ Φ∗i = ∅ or there exists an element j′ ∈ J such that we have identities

(Φ∗i (j) + Φ∗

i (j)) ∩ Φ∗i = Φ∗

i (j′) and (Φ∗

i (j′) + Φ∗

i (j)) ∩ Φ∗i = (Φ∗

i (j′) + Φ∗

i (j′)) ∩ Φ∗

i = ∅.To check all these properties, we can assume that i = 1 and thus we can work only withu∗j ∩ g1 and u∗j′ ∩ g1; thus all these properties follow easily from [Bou2, plates I to VI]. Wedetail here only the most complicated case when (X , e) = (Aℓ, 2) ∈ E. We will work withi = 1 and the set of orbits J . We recall that we drop the lateral right index (1) from thenotation of the roots in Φ1, cf. Subsection 3.4.5. The orbit j has one or two elements. Ifj has only one element, then obviously (Φ∗

1(j) + Φ∗1(j)) ∩ Φ∗

1 = ∅ as 2Φ∗1 ∩ Φ∗

1 = ∅. If j

has two elements, then we can write j = ∑i2

s=i1αs,

∑ℓ+1−i1s=ℓ+1−i2

αs, where 1 ≤ i1 ≤ i2 ≤ ℓand i1 < ℓ + 1 − i2. If 2i2 6= ℓ, then (Φ∗

1(j) + Φ∗1(j)) ∩ Φ∗

1 = ∅. If 2i2 = ℓ, then

(Φ∗1(j) + Φ∗

1(j)) ∩ Φ∗1 =

∑s=ℓ+1−i1s=i1

αs; as j′ we can take ∑s=ℓ+1−i1

s=i1αs and obviously

both sets (Φ∗i (j

′) + Φ∗i (j)) ∩ Φ∗

i and (Φ∗i (j

′) + Φ∗i (j

′)) ∩ Φ∗i are empty.

At the level of Lie algebras the previous paragraph gets translated as follows. Thedirect summand u∗j of u∗ is a commutative Lie subalgebra if (X , e) /∈ E. If (X , e) ∈ Eand u∗j is not a commutative Lie subalgebra of u∗, then there exists an element j′ ∈ Jsuch that we have an inclusion [u∗j , u

∗j ] ⊆ u∗j′ ; this inclusion is a W (k)-linear isomorphism

(cf. Chevalley rule; see the property 3.6.1 (b) for the case i = 1). If u∗j is (resp. is not)a commutative Lie algebra, then Φ∗

i (j) (resp. then the union Φ∗i (j) ∪ Φ∗

i (j′)) is a closed

subset of Φ∗i . Thus the existence of Uj follows from [DG, Vol. III, Exp. XXII, Cor. 5.6.5].

Therefore (i) holds.

It is easy to check based on [Bou2, plates I to VI] that (ii) holds. For instance, if e = 2and X = E6, then (ii) is a consequence of the fact that there exists no root α ∈ Φ∗

1 suchthat we have α+ΓΦ1

(α) ∈ Φ∗1; this is so as there exists no root α ∈ Φ∗

1 such that we havec4(α), c2(α) ∈ 0, 2 and c4(α) + c2(α) > 0.

Formula (6a) implies that all Newton polygon slopes of (u∗j , ϕ(∗)) are equal. The partof (iii) involving intervals was already explained before Lemma 5.1.2. Thus (iii) holds.As T (0) is generated by the images of σm

ϕ (µ)’s with m ∈ Z, the projection of t(0) on

96

g1 has B01 := ϕ1−i(li)|i ∈ I1 ⊆ g1 as a W (k)-basis. We take tj to be fixed by ϕand such that its component tj,1 in g1 is a Gm(W (Fpen))-linear combination of elementsof B01. We can choose these coefficients such that (iv) holds. We detail here only thecases e ∈ 1, 3 as the case e = 2 is very much the same. If e = 1 we take tj,1 to

be an arbitrary element of B01 that acts identically on u−j . Every Zp-basis for Lie(T(0)Zp

)formed by elements whose components in g1 are Zp-linear combinations of elements of B01,satisfies the extra requirement of (iv) for e = 1. If e = 3, we take tj such that we havetj,1 = cl1 + σn(c)ϕn(l1) + σ2n(c)ϕ2n(l1) for some c ∈ Gm(W (Fp3n)). We can take c = 1except when p ∈ 2, 3 and l1 + ϕn(l1) + ϕ2n(l1) acts on u−j via the multiplication with p.

If p = 2 we require an element c such that both c + σn(c) and c + σ2n(c) (and thereforeimplicitly also σn(c)+σ2n(c)) belong to Gm(W (Fp3n)). If p = 3 we require an c such thatc+ σn(c) + σ2n(c) ∈ Gm(W (Fp3n)). Obviously such elements c ∈ Gm(W (Fpen)) do exist.

Let t01 be the sum of the elements of B01. Let aj ∈ N be the absolute value of theonly eigenvalue of the action of t01 on u∗j ∩ g1. This makes sense i.e., we do have only one

such eigenvalue which is positive, as for each root α ∈ j the sum∑

x∈E cx(α) depends only

on j and not on α (cf. constructions) and as the semisimple element li acts identicallyon F−1(gi) (see paragraph before Subsection 3.4.1). We choose a total ordering of J suchthat for two elements j1, j2 ∈ J , the strict inequality aj1 < aj2 implies j1 > j2. If j1, j2,and j3 are as in (ii), then aj3 = aj1 +aj2 . Thus for each element j1 ∈ J , the set ∪j≤j1Φ

∗i (j)

is a closed subset of Φi. Therefore (v) follows from [DG, Vol. III, Exp. XXII, Cor. 5.6.5].

The minimality part of (vi) is implied by the fact that for each element j ∈ J the set∪ni=1Φ

∗i (j) is permuted transitively by ΓΦ.

5.2.2. Simple properties. (a) If the group scheme Uj is (resp. is not) commutative,then we say j is of type 1 (resp. of type 2). The element j ∈ J is of type 2 if and onlyif (X , e) ∈ E and we have j =

∑s∈S(i1,

ℓ2 )αs,

∑s∈S( ℓ

2+1,ℓ+1−i1)αs for some number

i1 ∈ S(1, ℓ2), cf. proof of Proposition 5.2.1. If j1 and j2 have the same type, then either

[u∗j1 , u∗j2] = 0 or the element j3 ∈ J as in the property 5.2.1 (ii) is of type 1. If j1 and j2

have distinct types and if [u∗j1 , u∗j2] 6= 0, then the element j3 ∈ J as in the property 5.2.1

(ii) can be of either type 1 or type 2. For instance, if X = A4, e = 2, j1 = α1, α4, andj2 = α2, α3, then j3 := α1 + α2, α3 + α4, j1 is of type 1 and j2 and j3 are of type 2.

(b) For i ∈ I the image of T (0) in Gi has rank |E|. To check this we can assumethat i = 1. But this case follows from the fact that the W (k)-basis B01 of the proof ofProposition 5.2.1 has |E| elements. However, often the rank of T (0) is less than n |E|.

If |E| = 1 and if t ∈ Lie(T(0)Zp

) generates a direct summand of Lie(T(0)Zp

), then the

action of t on u− is a W (k)-linear automorphism. To check this, we first remark thatthe component of t in g1 is a Gm(Zp)-multiple of l1 and moreover we have [l1, x] = x

if x ∈ u− ∩ g1 = F−1(g) ∩ g1 (as |E| = 1 this last identity follows from (7)). Thus[t, u− ∩ g1] = u− ∩ g1 and therefore (as t is fixed by ϕ) we have [t, u− ∩ gi] = u− ∩ gi forall i ∈ I.

(c) Let D+j,W (k) and D−

j,W (k) be the p-divisible groups over W (k) whose filtered

Dieudonne modules are (u+j , u+j ∩ F 1(g), ϕ) and (u−j , u

−j ∩ F 0(g), ϕ(−)) (respectively). To

97

the decomposition (23) corresponds naturally a product decomposition

D∗W (k) =

∏

j∈J

D∗j,W (k).

5.2.3. Remark. The direct summands u∗j of u∗ are normalized by every inner auto-morphism of (M,ϕ,G) defined by an element of TZp

(Zp). The torus TZpof GZp

andtherefore also the decomposition (23) is uniquely determined up to inner automorphismsof (M,ϕ,G). If e ∈ 1, 3, then we have a variant of Proposition 5.2.1 in which properties5.2.1 (i) to (v) hold and the property 5.2.1 (vi) gets replaced by the following maximalproperty:

(vi’) if (u∗, ϕ(∗)) = ⊕j′∈J ′(u∗j′ , ϕ(∗)) is another decomposition such that properties5.2.1 (i) and (iv) hold, then for each element j′ ∈ J ′ there exists an element j ∈ J suchthat we have u∗j′ ⊆ u∗j .

For e = 3 we refer to Example 5.3.6 below. If e = 1, then due to the property 5.2.1(iv), we can take J ′ to be the set of all functions c : E → 0, 1 for which there exists aroot α ∈ Φ+

1 (E) such that we have c(x) = cx(α) for all elements x ∈ E . Accordingly, wecan take Φ+

i (c) := α ∈ Φ+i |cx(α) = c(x) ∀x ∈ E.

Such maximal decompositions have the advantages that involve sets J with a smallernumber of elements and that they are normalized by each inner automorphism of (M,ϕ,G).But from the point of view of refinements and of treating all cases at once, it is moreadequate to work with minimal decompositions instead of maximal decompositions.

5.2.4. Duality. The non-degenerate Killing form K on g induces a bilinear form

bj : u−j ⊗ u+j →W (k).

If α, β ∈ Φi and α + β 6= 0, then we have K(gi(α), gi(β)) = 0 (cf. [Hu, Ch. II, Subsect.8.1, Prop.] applied over B(k)). Thus bj factors through a bilinear form

⊕i∈I ⊕α∈Φ+i(j) gi(−α)⊗W (k) gi(α) →W (k).

All roots in Φ∗i (j) are permuted by Aut(Φi) and thus have equal length. Thus all roots in

Φ∗i (j) are permuted transitively by the Weyl subgroup of Aut(Φi), cf. [Hu, Ch. II, Subsect.

10.4, Lemma C]. This implies that there exists a non-negative integer nj such that each bi-linear form gi(−α)⊗W (k)gi(α) →W (k) with α ∈ Φ+

i (j), is pnj times a perfect bilinear form

(i.e., it gives birth naturally to a W (k)-monomorphism gi(−α) → HomW (k)(gi(α),W (k))whose image is pnjHomW (k)(gi(α),W (k))). Therefore we can write bj = pnj b0j , where

b0j : u−j ⊗ u+j → W (k) is a perfect bilinear form. Let s+j : u+j →HomW (k)(u−j ,W (k)) and

s−j : u−j →HomW (k)(u+j ,W (k)) be the isomorphisms defined naturally by b0j .

For all elements x ∈ u−j and y ∈ u+j we have b0j(ϕ(−)(x), ϕ(y)) = pσ(b0j(x, y)). Thus, asK is fixed by G and therefore by Im(µ), the perfect bilinear morphism of filtered Dieudonnemodules

b0j : (u−j , u−j ∩ F 0(g), ϕ(−))⊗ (u+j , u

+j ∩ F 1(g), ϕ) → (W (k),W (k), pσ)

98

defines naturally (via s−j ) an isomorphism between (D+j,W (k))

t and D−j,W (k).

5.2.5. Commutating maps. Let

J3 := (j1, j2, j3) ∈ J3|0 6= [u−j1 , u−j2] ⊆ u−j3.

We can have an element (j1, j1, j3) ∈ J3 if and only if (X , e) ∈ E. For (j1, j2, j3) ∈ J3 weconsider the Lie bracket bilinear form u−j1 ⊗W (k) u

−j2

→ u−j3 ; it is non-zero (cf. property

3.6.1 (a)). Passing to duals and using the isomorphisms s+ji ’s, we get a cobilinear (non-zero)morphism

(24) cj1,j2 : (u+j3 , u+j3∩ F 1(g), ϕ) → (u+j1 , u

+j1∩ F 1(g), ϕ)⊗W (k) (u

+j2, u+j2 ∩ F

1(g), ϕ)

of filtered Dieudonne modules.

5.2.6. Standard reductive subgroup schemes. Let j ∈ J . It is easy to check that[Lie(U+

j ),Lie(U−j )] ⊆ Lie(T ). Thus the direct sum

Lie(T )⊕ Lie(U+j )⊕ Lie(U−

j )

is a Lie subalgebra of g which contains Lie(T ) and therefore which is normalized by T . LetGj be the unique reductive, closed subgroup scheme of G that contains T and whose Liealgebra is this last direct sum, cf. [DG, Vol. III, Exp. XXII, Def. 5.4.2, Thm. 5.4.7, andProp. 5.10.1]. The triple

(M,ϕ,Gj)

is an Sh-ordinary p-divisible object (cf. Subsection 4.3.2).

For j ∈ J and i ∈ I we have |Φ∗i (j)| ≤ e. Moreover, if (X , e) /∈ E (resp. if (X , e) ∈ E),

then for each number i ∈ S(1, n) the adjoint group scheme of Gj ∩ Gi is a PGLej2 group

scheme with ej ∈ S(1, e) (resp. is either a PGL2 or a PGL22 or a PGL3 group scheme).

As j is an orbit of ΓΦ, the simple factors of Lie(Gadj,B(k)) are permuted transitively by ϕ.

For the rest of this Section we present four examples.

5.2.7. Example. Let (e1, . . . , en) be the standard ordered W (k)-basis for W (k)n. Weconsider the case when E has only one element x. Thus e = 1 and we have ΓΦ(αs(i)) =αs(i+ 1) for all pairs (i, s) ∈ I × S(1, ℓ) (see (6a) and Subsection 3.4.5). Let f : Z/nZ →0, 1 be the function that takes the modulo n class [i] defined by i ∈ I to 1 if and only ifηi 6= 0 (i.e., if and only if ηi = x). Let d ∈ S(1, n) be such that the modulo n class [d] is theperiod of f ; it is a divisor of n. Let D1 be the circular p-divisible group over W (k) whosefiltered Dieudonne module is (W (k)n, F 1

1 , ϕn), where ϕn takes ei to pf([i])ei+1 with en+1 :=

e1 and where F 11 := ⊕i∈S(1,n), f([i])=1W (k)ei. Let D0 be the circular indecomposable p-

divisible group over W (k) whose filtered Dieudonne module is (W (k)d, F 10 , ϕd), where

ϕd takes the d-tuple (e1, . . . , ed) to the d-tuple (pf([1])e2, . . . , pf([d])e1) and where F 1

0 :=⊕i∈S(1,d), f([i])=1W (k)ei.

From (6a) we get that D+W (k) is a direct sum of rkW (k)(F

1(g1)) copies of D1. More

precisely, each α ∈ Φ+1 that contains αx in its expression defines an orbit α(1), . . . , α(n)

99

of J to which corresponds uniquely one such copy of D1. Let Tn be the maximaltorus of GLW (k)n that normalizes all W (k)ei’s with i ∈ S(1, n). We know that

(W (k)n, F 11 , ϕn) is a direct sum of n

dcopies of (W (k)d, F 1

0 , ϕd), cf. Subsection 4.1.1 appliedto (W (k)n, F 1

1 , ϕn, Tn). Thus D1 is a direct sum of nd copies of D0, cf. properties 1.2.1 (b)

and (c). Thus

D+W (k) = D

nrkW (k)(F1(g1))

d0 .

If (f [1], . . . , f [d]) is up to a cyclic permutation (0, . . . , 0, 1) (resp. (1, . . . , 1, 0)), thenD0 is the unique circular indecomposable p-divisible group over W (k) of height d andNewton polygon slope 1

d (resp. and Newton polygon slope d−1d ) and we denote it by D 1

d

(resp. by D d−1d). If d = 4 and (f [1], . . . , f [4]) is up to a cyclic permutation (1, 1, 0, 0),

then D0 is the unique circular indecomposable p-divisible group over W (k) of height 4 andNewton polygon slope 1

2.

5.2.8. Example. We consider the case when n = 1, e = 2, X = Dℓ with ℓ ≥ 4, andE = ℓ− 1, ℓ. For what follows, it is irrelevant if η1 is ℓ− 1 or ℓ. The number of roots in

Φ+1 that contain the sum αℓ−1+αℓ in their expressions, is (ℓ−2)(ℓ−1)

2. The number of roots

in Φ+1 that contain αℓ−1 but do not contain αℓ in their expressions, is ℓ − 1. See [Bou2,

plate IV] for the last two sentences. Thus there exist (ℓ−2)(ℓ−1)2

(resp. ℓ− 1) orbits of ΓΦ1

(i.e., elements of J = J) that have only one element (resp. have two elements).

By the standard trace form on g we mean the form on g induced by the trace formon g[ 1p ] associated to a standard simple g[ 1p ]-module W of dimension 2ℓ (if ℓ = 4 we have

three such standard simple g[ 1p ]-modules; all of them give birth to the same standard trace

form on g). If s is an sl2 Lie subalgebra of g[ 1p ] normalized by TB(k), then the s-module Wis a direct sum of a trivial s-module of dimension 2ℓ− 4 and of two copies of the standardirreducible s-module of dimension 2 (to check this, one can allow the case when ℓ is 2 andcan use the fact that the D2 Lie type is in fact the A1 + A1 Lie type). This implies thatthe standard trace form on g induces a bilinear form u−j ⊗ u+j → W (k) that is 2 times

a perfect bilinear form. Let n′j ∈ N ∪ 0 be the greatest number such that pn

′j divides

ℓ− 2. The Killing form K is 2(ℓ− 2) times the standard trace form on g (for the analogueof this over C see [He, Ch. III, §8, (11)]). We conclude that if p ≥ 3 (resp. if p = 2), thennj = n′

j (resp. nj = 2 + n′j).

Let D 12be as in the end of Example 5.2.7. We have a direct sum decomposition

D+W (k) = µµµ

(ℓ−2)(ℓ−1)2

p∞ ⊕Dℓ−112

,

where to each j ∈ J with |j| = 1 (resp. |j| = 2) corresponds a copy of µµµp∞ (resp. of D 12).

5.2.9. Example. We consider the case when e = 1, X = Aℓ, ℓ ≥ 3, and E = S(1, ℓ).Thus n ≥ ℓ. We have P+

G (ϕ) ∩ G1 = ∩ℓx=1Px = B ∩ G1, cf. Corollary 3.6.3 for the first

identity. As e = 1, each set Φ∗i (j) has only one root. Thus the derived group scheme of the

intersection Gj∩Gi is an SL2 group scheme, for all j ∈ J and all i ∈ S(1, n). The standardtrace form on g induces a perfect bilinear form u−j ⊗ u+j → W (k). Moreover, the Killing

100

form K is 2(ℓ+ 1) times the standard trace form on g (for the analogue of this over C see[He, Ch. III, §8, (5)]). Thus the number nj ∈ N ∪ 0 of Subsection 5.2.4, is the greatest

number such that pnj |2(ℓ+ 1). The number of positive roots in Φi isℓ(ℓ+1)

2and therefore

the height of D∗W (k) is nℓ(ℓ+1)

2. The dimension of D+

k is the rank of F 1(g) and thereforedepends on the values of η1, . . . , ηn computed with multiplicities. If n = ℓ ≥ 4, then theisomorphism class of D+

W (k) depends on the η-type (η1, . . . , ηn) of (M,ϕ,G). Thus for

simplicity we will consider here only the cases when n = ℓ ∈ 3, 4. To fix the notations,we can assume that η1 = 1.

We first consider the case when n = ℓ = 3. Thus |J | =∣∣J∣∣ = rkW (k)(u

∗ ∩ g1) = 6.If p = 2, then nj = 3. If p > 2, then nj = 0. We have two possibilities for the η-type:(1, 2, 3) and (1, 3, 2) (the first one relates to Example 1.4.6).

See end of Example 5.2.7 for D 13and D 2

3. We have a direct sum decomposition

D+W (k) = µµµ3

p∞ ⊕D223⊕D3

13

(and therefore the dimension ofD+k is 10). This decomposition corresponds to the following

direct sum decomposition

u+ ∩ g1 = (gα1+α2+α3)⊕ (gα1+α2

⊕ gα2+α3)⊕ (gα1

⊕ gα2⊕ gα3

)

left invariant by ϕ3.

We now consider the case when n = ℓ = 4. Thus |J | =∣∣J∣∣ = rkW (k)(u

∗ ∩ g1) = 10. Ifp divides 10, then nj = 1. If p > 5 or if p = 3, then nj = 0. We have six possibilities for theη-type (η1, η2, η3, η4) = (1, η1, η3, η4) of (M,ϕ,G). We have a direct sum decomposition

D+W (k) = µµµ4

p∞ ⊕D234⊕Dm1

2,4 ⊕Dm212

⊕D414,

where m1, m2 ∈ 0, 1, 2, 3, 4 and where D 12, D 1

4, D2,4, and D 3

4are as in the end of

Example 5.2.7. If (η1, . . . , η4) is (1, 2, 3, 4) or (1, 4, 3, 2), then (m1, m2) = (3, 0) and theD3

2,4 factor of D+W (k) corresponds to the elements α1 + α2, α2 + α3, α3 + α4 ∈ J .

If (η1, . . . , η4) is (1, 2, 4, 3) or (1, 3, 4, 2), then (m1, m2) = (2, 2), the D22,4 factor of D+

W (k)

corresponds to the elements α1 + α2, α3 + α4 ∈ J , and the D212

factor of D+W (k)

corresponds to the element α2 + α3 ∈ J . If (η1, . . . , η4) is (1, 3, 2, 4) or (1, 4, 2, 3), then(m1, m2) = (1, 4), the D2,4 factor of D+

W (k) corresponds to the element α2 + α3 ∈ J ,

and the D412

factor of D+W (k) corresponds to the elements α1 + α2, α3 + α4 ∈ J . The

dimension of D+k is 20.

5.2.10. Example. We consider the case when n = 1, e = 2, and X = A2. Thus GZp

is a PGU3 group scheme. Let j ∈ J = J . Thus j is either α1, α2 or α1 + α2. Wecan assume that the η-type of (M,ϕ,G) is (1, 2). Thus ϕ takes gα1

, gα2, and gα1+α2

topgα2

, gα1, and pgα1+α2

(respectively). Thus we have a direct sum decomposition D+W (k) =

µµµp∞ ⊕D 12. Moreover if j = α1, α2, then Gj = G is a PGL3 group scheme.

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5.3. Formal Lie groups: the adjoint, cyclic case

Let CA (resp. CN) be the category of abstract abelian (resp. of abstract nilpotent)groups. Let CS be the category of sets. Let IA : CA → CS and IN : CN → CS be thefunctors that forget the group structures. Let Art(W (k)) be the category of local, artinianW (k)-schemes that have k as their residue fields. We denote by Z an arbitrary object ofArt(W (k)). We will use the setting of Section 5.2. We emphasize that in what follows allproducts indexed by elements j ∈ J are ordered in accordance with the ordering of J ofthe property 5.2.1 (v).

5.3.1. The first formal Lie group. By the first formal Lie group F1 of (M,ϕ,G)we mean the formal Lie group over Spf(W (k)) of D+

W (k). Thus the contravariant functor

defined by F1 (and denoted also by F1)

F1 : Art(W (k)) → CA

is such that we have F1(Z) := ∪m∈NHomZ(Z,D+Z [p

m]). The dimension of F1 is thedimension of D+

k and therefore it is the rank of F 1(g).

For j ∈ J let F1,j be the formal Lie group over Spf(W (k)) of the p-divisible groupD+

j,W (k) of the property 5.2.2 (c). We have a product decomposition

F1 =∏

j∈J

F1,j.

If U−j is (resp. is not) commutative, then F1,j (resp. F1,j×F1,j′ with j

′ as in the property

5.2.1 (i)) is the first formal Lie group of (Lie(Gadj ), ϕ, Gad

j ).

5.3.2. Bilinear morphisms. Let (j1, j2, j3) ∈ J3. It is known that to (24) correspondsnaturally a (non-zero) bilinear commutating morphism

(25) Cj1,j2 : D+j1,W (k) ×W (k) D

+j2,W (k) → D+

j3,W (k).

As we do not know a reference for this result, for the rest of this paragraph we will presentthe very essence of its proof. Let cj1,j2 be the morphism of F -crystals over k definednaturally by (24) by forgetting the filtrations. For s ∈ S(1, 3) and m ∈ N, we writeD+

js,W (k)[pm] = Spec(Rjs). Let Ijs be the ideal of Rjs that defines the identity section

of D+js,W (k)[p

m]. We consider the homomorphism hj1,m : Z/pmZ → D+j1,W (k)[p

m]Rj1/pRj1

over Spec(Rj1/pRj1) defined naturally by 1Rj1/pRj1

. By composing the extension of (24)(viewed without filtrations but with Verschiebung maps and connections) to Rj1/p

mRj1

with D(hj1,m)⊗D(1D+j2,W (k)

[pm]Rj1/pRj1

), we get a morphism

hj1,j2,j3,m : D(D+j3,W (k)[p

m]Rj1/pRj1

) → D(D+j2,W (k)[p

m]Rj1/pRj1

).

As all Newton polygon slopes of the special fibre of D+j1,W (k) are positive, the finite flat

group scheme D+j1,W (k)[p

m]k is connected. From this and [Wa, 14.4] we get that Rj1/pRj1 is

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a k-algebra of the form k[x1, . . . , xuj1]/(x

n1,j11 , . . . , x

nuj1,j1

uj1), where uj1 , n1,j1 , . . . , nuj1

,j1 ∈N∪0. Thus the Dieudonne functor D is fully faithful on finite, flat, commutative groupschemes over Spec(Rj1/pRj1) = D+

j1,W (k)[pm]k, cf. [BM, Rm. 4.3.2 (i)]. Therefore we can

writehj1,j2,j3,m = D(cj1,j2,j3,m),

where cj1,j2,j3,m : D+j2,W (k)[p

m]Rj1/pRj1

→ D+j3,W (k)[p

m]Rj1/pRj1

is a homomorphism over

Spec(Rj1/pRj1). It is defined naturally by a k-homomorphism Rj3/pRj3 → Rj1/pRj1 ⊗k

Rj2/pRj2 which in fact also defines the special fibre of

Cj1,j2 [pm] : D+

j1,W (k)[pm]×W (k) D

+j2,W (k)[p

m] → D+j3,W (k)[p

m].

As Ij1 modulo p is a nilpotent ideal of Rj1/pRj1 , the natural divided power structure ofthe ideal pIj1 of Rj1 is nilpotent modulo pm. Thus the Grothendieck–Messing deformationtheory (applied in the context of cj1,j2,j3,m) allows us to construct the fibre of Cj1,j2 [p

m]over Wm(k). By taking m→ ∞ we get Cj1,j2 .

We denote also by Cj1,j2 : F1,j1 × F1,j2 → F1,j3 the bilinear morphism of formal Liegroups over Spf(W (k)) that corresponds naturally to (25).

5.3.3. Lemma. For this Lemma only, we will take k to be an arbitrary perfect fieldof characteristic p. Let DW (k) be an arbitrary p-divisible group over W (k) whose specialfibre D does not have slope 0. Let r be the rank of D. Let d be the dimension of D. LetEDW (k)

: Art(W (k)) → CA be the functor that takes Z ∈ Art(W (k)) to the abelian groupof isomorphism classes of extensions of Qp/Zp (over Z) through DZ (i.e., of isomorphism

classes of short exact sequences of the form 0 → DZ → D → Qp/Zp → 0). Then EDW (k)is

represented by the formal Lie group FDW (k)of DW (k) (i.e., there exists a natural equivalence

hDW (k): FDW (k)

→ EDW (k)) and therefore it has dimension d.

Proof: Each morphism mZ : Z → DW (k) over W (k) defines naturally a homomor-phism nZ : Zp → DZ over Z. By pushing forward the short exact sequence 0 →Zp → Qp → Qp/Zp → 0 (viewed over Z) via nZ , we get a short exact sequence

0 → DZ → D → Qp/Zp → 0 over Z to be denoted as hDW (k)(Z)(mZ). The associa-

tion mZ → hDW (k)(Z)(mZ) defines a natural transformation hDW (k)

: FDW (k)→ EDW (k)

.If hDW (k)

(mZ) splits, then nZ extends to a homomorphism oZ : Qp → DZ . But asZ ∈ Art(W (k)), each such homomorphism oZ is trivial and this implies that mZ factorsthrough the identity section of DW (k). Thus hDW (k)

is a monomorphism natural transfor-mation i.e., each homomorphism hDW (k)

(Z) : FDW (k)(Z) → EDW (k)

(Z) is a monomorphism.

It is well known that FDW (k)is a formal Lie group over Spf(W (k)) of dimension d. For

the rest of this paragraph we check that EDW (k)is also a formal Lie group over Spf(W (k))

of dimension d. Let D(D) : Art(W (k)) → CS and D(D ⊕Qp/Zp) : Art(W (k)) → CS bethe functors that parameterize isomorphism classes of deformations of D and D ⊕Qp/Zp

(respectively). Each deformation of D⊕Qp/Zp over Z is naturally the extension of Qp/Zp

(over Z) through a deformation of D over Z. This gives birth naturally to an epimorphismnatural transformation D(D ⊕Qp/Zp) ։ D(D). The pull back back of this last naturaltransformation via the trivial subfunctor D(D)triv of D(D) that defines trivial deformations

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of D, is canonically identified with a natural transformation IA EDW (k)→ D(D)triv. The

last two sentences imply that EDW (k)is representable by a formal Lie group over Spf(W (k)).

The dimensions of D(D) and D(D ⊕Qp/Zp) are d(r − d) and d(r + 1− d) (respectively),cf. [Il, Thm. 4.8]. Thus the dimension of EDW (k)

is d(r + 1− d)− d(r − d) = d.

As hDW (k)(Spec(k[x]/x2)) is an injective k-linear map between k-vector spaces of di-

mension d, it is a k-linear isomorphism. This implies that hDW (k)is a natural equivalence.

5.3.4. The second formal Lie group. Let s1 < s2 < . . . < s|J| be the elements of Jlisted increasingly. We now construct a second contravariant functor

F2 : Art(W (k)) → CN.

For j ∈ J let ej,Z be the identity element of F1,j(Z) and let Bj(Z) := ∪j1∈J, j1≤jF1,j1(Z),the union being disjoint. Let F2(Z) be the quotient of the free group F (Bs|J|

(Z)) on theset Bs|J|

(Z) by the following three types of relations:

(a) x1x2 = x3, if there exists an element j ∈ J such that u∗j is commutative and x1,x2, x3 ∈ F1,j(Z) satisfy the relation x1x2 = x3;

(b) e2j1,Z = ej1,Z = ej2,Z for all j1, j2 ∈ J ;

(c) x1x2x−11 x−1

2 = x3, if there exists a triple (j1, j2, j3) ∈ J3 such that we havexi ∈ F1,ji(Z) for i ∈ S(1, 3) and Cj1,j2(Z)(x1, x2) = x3.

If Z → Z is a morphism of Art(W (k)), then the homomorphisms F1,j(Z) → F1,j(Z)are compatible with the above type of relations and therefore define a homomorphismF2(Z) → F2(Z). Let x = (xs1 , . . . , xs|J|

) ∈ F1(Z), where xsi ∈ F1,si(Z). The map of sets

pZ : F1(Z) → F2(Z)

that takes x to xs1xs2 · · ·xs|J|, is a natural transformation between IAF1 and INF2 that

in fact is an isomorphism. Thus F2 is representable by a formal Lie group over Spf(W (k))denoted also by F2 and called the second formal Lie group of (M,ϕ,G).

Let F2,j(Z) be the subgroup of F2(Z) generated by Bj(Z). We get a subfunctor

F2,j : Art(W (k)) → CN

of F2. As pZ(∏

j1≤j F1,j1(Z)) = F2,j(Z), this subfunctor is defined by a formal Lie sub-group of F2 to be denoted also by F2,j. We get a filtration

(26) 0 → F2,s1 → F2,s2 → . . . → F2,s|J|= F2

by normal formal Lie subgroups. We have natural identifications F2,s1 = F1,s1 andF2,si/F2,si−1

= F1,si , with i ∈ S(2, |J |).

5.3.5. Remarks. (a) The nilpotent class of F2(Z) is at most |E|. If |E| ≥ 2, then it iseasy to see that the identities of the property 3.6.1 (b) imply that there exist objects Z ofArt(W (k)) such that the nilpotent class of F2(Z) is |E|.

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(b) The filtration (26) depends on the ordering of J . The isomorphism class of thep-divisible group D+

W (k) as well as of F1 and F2 depend only on the type of (M,ϕ,G) as

defined in Subsection 3.4.9.

In the next two examples, the group law of F1 (resp. of F2) will be denoted additively(resp. multiplicatively). If [uj1 , uj2] = 0, let Cj1,j2 : F1,j1 × F1,j2 → F1,s1 be the zerobilinear morphism.

5.3.6. Example. Let e = 3. The set J has five elements s1, . . . , s5. They can bedescribed via J = s1, . . . , s5 as follows: s5 := α1, α3, α4, s4 := α1 +α2, α2 +α3, α2 +α4, s3 := α1 + α2 + α3, α1 + α2 + α4, α2 + α3 + α4, s2 := α1 + α2 + α3 + α4, ands1 := α1 + 2α2 + α3 + α4. The set J has four possible orderings such that the property5.2.1 (v) holds: they are defined by choosing which one among s4 and s5 and respectivelyamong s1 and s2 is greater. Thus filtration (26) is not uniquely determined. The maximaldecomposition mentioned in Remark 5.2.3 is defined naturally by the set with four elementsJ ′ := s1, s2, s3, s4 ∪ s5.

For m ∈ S(1, 5) let xm, ym ∈ F1,sm(Z). Let zm := xm + ym ∈ F1,sm(Z). LetCi1,i2 := Csi1 ,si2

. The multiplication in F2(Z) is defined by the rule: the productpZ(x1, x2, x3, x4, x5)pZ(y1, y2, y3, y4, y5) is

pZ(z1 + C4,3(Z)(x4, y3) + C4,3(Z)(x4, C5,4(Z)(x5, y4)), z2 + C5,3(Z)(x5, y3), z3, z4, z5).

5.3.7. Example. We assume that |E| = 2. Let xj , yj ∈ F1,j(Z). Let zj := xj + yj. LetJ(2) := j ∈ J |aj = 2, where aj is as in the end of the proof of Proposition 5.2.1. Wehave J(2) = si|i ∈ S(1, |J(2)|). The formal Lie group F2,s|J(2)|

is commutative; thus therules xJ(2) :=

∑j∈J(2) xj , yJ(2) :=

∑j∈J(2) yj, and zJ(2) := xJ(2) + yJ(2) define naturally

elements of F2,s|J(2)|(Z). Let u∗J(2) := ⊕j∈J(2)u

∗j . As |E| = 2, the Lie algebra u∗/u∗J(2) is

commutative. If p ≥ 3, we introduce a new product formula on the set underlying F1(Z)by the rule

(27) (x1, x2, . . . , x|J|)× (y1, y2, . . . , y|J|) = (z1, . . . , z|J(2)|, z1+|J(2)|, . . . , z|J|),

where the zj ’s are defined by the following summation property

∑

j∈J(2)

zj =∑

j∈J(2)

zj +

|J|∑

j1, j2=1+|J(2)|

1

2Cj1,j2(Z)(xj1 , yj2).

This new product is independent on the possible orderings of J and is preserved by innerautomorphisms. It satisfies the relations of Subsection 5.3.4 and thus the resulting formalLie group is isomorphic to F2. Formulas similar to (27) can be established for |E| > 2 andp ≥ 3. Not to make this monograph too long, they will not be included here.

5.4. Formal Lie groups: the general case

We do not assume any more that G is adjoint or that Γ is a cycle. All Subsectionsfrom 5.2 to 5.3.5 make sense in this generality: we only have to work with direct sums of

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decompositions as in Proposition 5.2.1 (to each cycle of the permutation Γ of the set Icorresponds a direct sum decomposition as in Proposition 5.2.1). In particular, we speakalso about the first and second formal Lie groups F1 and F2 of (M,ϕ,G). They dependonly on the adjoint Shimura Lie F -crystal (Lie(Gad), ϕ).

5.4.1. Functorial aspects. Let G be a reductive, closed subgroup scheme of G obtainedas in Subsection 4.3.2. Thus (M, (F i(M))i∈S(a,b), ϕ, G) is an Sh-canonical lift. Let g,

F 0(g), F 1(g), u∗, D∗W (k), K, and F1 be the analogues of g, F 0(g), F 1(g), u∗, D∗

W (k),

K, and F1 (respectively) but obtained working with (M, (F i(M))i∈S(a,b), ϕ, G) instead of(M, (F i(M))i∈S(a,b), ϕ, G). The natural inclusions

(u−, u− ∩ F 0(g), ϕ(−)) → (u−, u− ∩ F 0(g), ϕ(−)) and (u+, F 1(g), ϕ) → (u+, F 1(g), ϕ)

define respectively homomorphisms of p-divisible groups over W (k) (cf. properties 1.2.1(b) and (c))

f− : D−W (k) → D−

W (k) and f+ : D+W (k) → D+

W (k).

Both f+ and f− are epimorphisms and therefore their kernels are p-divisible groups overW (k).

Let f+t : D+

W (k) → D+W (k) be the closed embedding defined naturally by the Cartier

dual of f− and by the duality properties of Subsection 5.2.4. We emphasize that theendomorphism

f+ f+t : D+

W (k) → D+W (k)

is not always an automorphism (and in particular it is not always 1D+W (k)

). This is so

as in Subsection 5.2.4 we used Killing forms to identify positive p-divisible groups overW (k) with Cartier duals of negative p-divisible groups over W (k) and the restriction of

K to u− ⊗W (k) u+ can be non-perfect even if K and K are perfect. Both f− and f+ are

functorial but (due to the same reason) f+t is not functorial in general. Thus we will not

stop here to get general functorial properties for second formal Lie groups. To f+ (resp.to f+

t ) corresponds an epimorphism (resp. a closed monomorphism) denoted in the sameway f+ : F1 ։ F1 (resp. f+

t : F1 → F1).

5.4.2. Remarks. (a) For each cyclic p-divisible group D0 over W (k) there exists anSh-ordinary p-divisible object (M,ϕ,G) over k such that F1 is the formal Lie group ofD0 and all simple factors of Gad are of B1 = A1 Lie type. Using direct sums, to checkthis we can assume that D0 is circular indecomposable. Thus we can assume D0 is as inExample 5.2.7. But if in Example 5.2.7 we take X = B1 = A1 and d = n, then we haveD+

W (k) = D0.

We point out that this remark in fact classifies all possible first formal Lie groups F1

over Spf(W (k)) (of Sh-ordinary triples (M,ϕ,G) over k), cf. Lemma 5.1.3.

(b) We have variants of Subsections 5.2.4 and 5.4.1 where instead of Killing formswe work with restrictions of the trace form on EndW (k)(M). The disadvantage (resp. theadvantage) of this approach is that it depends on M and the numbers nj of Subsection

106

5.2.4 tend to grow and often are hard to be computed (resp. is that the endomorphismf+ f+

t of D+W (k) becomes a power of p times 1D+

W (k)).

(c) We could have worked out Section 5.3 using the slope decomposition of (u∗, ϕ(∗)).The advantage of this approach is that this decomposition is normalized by all automor-phisms of (M,ϕ,G) and the resulting ascending slope filtration of F2 is canonical. Thedisadvantage of this approach is that we lose properties listed in properties 5.2.1 (i) and(iv) and Subsection 5.2.6 and it is harder to formalize Subsection 5.2.4 as it can happenthat there exist elements j1, j2 ∈ J such that nj1 6= nj2 but the slopes of (u∗j1 , ϕ(∗)) and(u∗j2 , ϕ(∗)) are equal.

(d) If k 6= k, then in general we can not get decompositions as in Section 5.2. However,we can still speak about the positive and negative p-divisible groups overW (k) of (M,ϕ,G)and about the first formal Lie group of (M,ϕ,G). If all numbers nj of Subsection 5.2.4are equal or if there exists a maximal torus T of G whose Lie algebra is normalized by ϕ,then we can also speak about the second formal Lie group of (M,ϕ,G).

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6 Geometric theory: proofs of the Theorems 1.6.1 to 1.6.3

We recall that p ∈ N is an arbitrary prime. The passage from the abstract theory ofSection 1.4 to the geometric theory of Section 1.6 will involve two extra tools. The firsttool is Fontaine comparison theory that relates the p-adic etale context of Subsection 1.5.2to the crystalline context of Subsection 1.5.4 (see Section 6.1). The second tool is Faltingsdeformation theory of [Fa2, §7] that keeps track of how the Shimura filtered F -crystals(M,F 1, ϕ, G) of Subsection 1.5.4 vary in terms of lifts z ∈ N (W (k)) of points y ∈ N (k)(see Section 6.2). Its main role is to substitute the abstract “pseudo-deformations” CY ;m

used in Subsection 3.8.2. We also use Faltings deformation theory to show that the triple(M,ϕ, G) of Subsection 1.5.4 depends only on y ∈ N (k) and not on its lift z ∈ N (W (k))(see Proposition 6.2.7 (d)). In Subsection 6.2.3 we extend Faltings deformation theory toa more general and axiomatized context of the so called GD rings. In Subsection 6.3 weput together Sections 1.4, 6.1, and 6.2 to prove Theorem 1.6.1. In Section 6.4 we introducethe language of PD hulls in order to prove a result (see Proposition 6.4.6) on the existenceof lifts of (M,ϕ, G) attached to W (k)-valued lifts of y ∈ N (k); this result is also the veryessence of the proof of Theorem 1.6.2. In Section 6.5 we prove Theorems 1.6.2 and 1.6.3;see Subsection 6.5.5 for the definition of the condition W.

In Subsections 6.1.1 to 6.1.4 we work with a potential standard Hodge situation(f, L, v). From Subsection 6.1.5 until the end of the monograph we will assume that(f, L, v) is a standard Hodge situation in the sense of Definition 1.5.3.

The notations f : (GQ, X) → (GSp(W,ψ), S), Sh(GQ, X), E(GQ, X), v, k(v), O(v),L, L(p) = L ⊗Z Z(p), GZ(p)

, Kp = GSp(L, ψ)(Zp), Hp = Kp ∩ GQ(Qp) = GZ(p)(Zp),

r ∈ N, M, N , (A,ΛA), (vα)α∈J , (wAα )α∈J , Lp = L(p) ⊗Z(p)

Zp = L ⊗Z Zp, ψ∗, T0, B0,

µ0 : Gm → T0,W (k(v)), M0 = L∗p ⊗Zp

W (k(v)) = F 10 ⊕ F 0

0 , and C0 = (M0, ϕ0, GW (k(v)))will be as in Subsections 1.5.1 and 1.5.2. We recall that GZ(p)

is a reductive group schemeover Z(p), cf. assumption 1.5.1 (*).

We refer to [Me], [Ka1], [De4], [BBM], [BM], and [Fa2, §7] for (principally quasi-polarized) filtered F -crystals over R/pR of (principally polarized) abelian schemes overSpec(R). Here R is as in Subsection 2.8.4.

6.1. Fontaine comparison theory

Let k be a perfect field of characteristic p. We fix an algebraic closure B(k) of B(k).

Let W (k)∧

be the p-adic completion of the normalization of W (k) in B(k). The group

Gal(B(k)) := Gal(B(k)/B(k)) acts naturally on W (k)∧. If p ≥ 3 (resp. if p = 2) let

B+(W (k)) be the Fontaine ring used in [Fa2, §4] (resp. used in [Fa2, §8] and obtained

108

using the 2-adic completion). The reason we do not use the divided power topology forp = 2 is that (despite the claims of [Fa2, §8]) we could not convince ourselves that theresulting Fontaine ring has a natural Frobenius lift (to be compared with [Fa2, §4, p.125] where the arguments refer only to odd primes). We recall that B+(W (k)) is a local,integral, commutativeW (k)-algebra, endowed with a decreasing, exhaustive, and separatedfiltration (F i(B+(W (k))))i∈N∪0, with a Frobenius lift L, and with a natural Gal(B(k))-action (see also [Fo2, Ch. 2]). We also have a Gal(B(k))-invariant W (k)-epimorphism

sW (k) : B+(W (k)) ։ W (k)

∧

whose kernel has a natural divided power structure. Let V be a totally ramified discretevaluation ring extension of W (k). Let eV ∈ N be the index of ramification of V .

We start with a point y : Spec(k) → N . Let z : Spec(W (k)) → N be a lift of y.Let A = z∗(A), Dz , (M,F 1, ϕ, G), and (tα)α∈J be as in Subsection 1.5.3 but for the timebeing we do not assume that conditions 1.5.3 (a) and (b) hold. Let λM (resp. λDz

) be theprincipal quasi-polarization of (M,ϕ) (resp. of Dz) defined by λA := z∗(ΛA). As A has alevel s symplectic similitude structure for all positive integers s relatively prime to p, thefield k contains all s-th roots of unity and thus contains F. Let

H1 := H1et(AB(k)

,Zp).

Let λH1 : H1 ⊗ZpH1 → Zp be the perfect form that is the Zp-etale realization of λA. For

Fontaine comparison theory we refer to [Fo1] to [Fo3], [Ts], and [Fa2]. From this theoryapplied to A we get the existence of a B+(W (k))-monomorphism

iA :M ⊗W (k) B+(W (k)) → H1 ⊗Zp

B+(W (k))

that has the following four properties:

(a) It respects the tensor product filtrations (the filtration of H1 is defined by:F 1(H1) = 0 and F 0(H1) = H1).

(b) It respects the Galois actions (the Galois action on M ⊗W (k)B+(W (k)) is defined

naturally via sW (k) and the fact that Ker(sW (k)) has a divided power structure).

(c) It respects the tensor product Frobenius (the Frobenius of H1 being 1H1).

(d) There exists an element β0 ∈ F 1(B+(W (k))) such that we have L(β0) = pβ0and iA becomes an isomorphism iA[

1β0] after inverting β0. Moreover iA[

1β0] takes λM to a

Gm(B+(W (k))[ 1β0])-multiple of λH1 .

The existence of iA is a particular case of [Fa2, Thm. 7], cf. end of [Fa2, §8]. Fromthe way [Fa2, §8] is stated, it is clear that loc. cit. also treats the prime 2. We pointout that the ring RV used in [Fa2, §8] for p = 2 is not definable as in [Fa2, §2] but itis the ring ReV of Subsection 2.8.7. As B+(W (k)) is a W (k)-algebra, we can define thecrystalline cohomology group of the p-divisible group Dz of A relative to B+(W (k)) usingthe site CRIS(Spec(W (k))/Spec(W (k))) (cf. end of [Fa2, §8]). Working with this site, the

109

homology and the cohomology of the p-divisible group Qp/Zp overW (k) can be computedas in [Fa2, §6] (modulo powers of p, this is also completely detailed in [BM, Ch. 2]). Thusone constructs the B+(W (k))-monomorphism iA as in [Fa2, §6] even for p = 2.

We recall few complements on de Rham cycles as defined in [Bl]. For de Rham ringBdR(V ) and the de Rham conjecture we refer to [Fa1], [Fa2], [Fo2], [Fo3], and [Bl]. In whatfollows we only use the facts that B+(W (k)) is a W (k)-subalgebra of the commutative,integral ring BdR(V ) and that β0 is an invertible element of BdR(V ). We have:

6.1.1. Theorem. Let K := V [ 1p ]. Let z ∈ N (V ). Let A := z∗(A). Then each Hodge cycle

on AK that involves no etale Tate twist is a de Rham cycle (i.e., under the isomorphismiA ⊗ BdR(V ) of the de Rham conjecture, its de Rham component is mapped to the p-component of its etale component).

Proof: Based on the refinement [Va3, Principle 5.2.16] of the Principle B of [Bl, Thm.(3.1)], we can assume that AK is definable over a number field contained in K (strictlyspeaking, in [Va3, Subsect. 5.2] an odd prime is used; however the proof of [Va3, Principle5.2.16] applies to all primes). But in this case the Theorem is well known (for instance,see [Bl, Thm. (0.3)]).

6.1.2. Corollary. For all α ∈ J we have ϕ(tα) = tα.

Proof: The cycle wAα involves no etale Tate twist. Thus iA[

1β0] takes tα to vα, cf. Theorem

6.1.1. Thus, as vα is fixed by 1H1 ⊗ L, the Corollary follows.

6.1.3. Remark. We have ϕ(Lie(GB(k))) = Lie(GB(k)), cf. Corollary 6.1.2. As inSubsection 3.1.7 we argue that the inverse of the canonical split cocharacter of (M,F 1, ϕ)factors through G. Thus the essence of the condition 1.5.3 (b) is that G is a reductivegroup scheme over W (k).

6.1.4. The group schemes G0Z(p)

and G0. Let G0Q be the subgroup of GQ that fixes ψ.

Let G0Z(p)

be the Zariski closure of G0Q in GZp

. As the finite group Z(Sp(W ⊗QR, ψ))→µµµ2

is contained in the maximal compact subtorus of the image of every homomorphismResC/RGm → GR that is an element of X , the group G0

Q is connected. Therefore

Im(G0Q → Gab

Q ) is a subtorus of GabQ whose Zariski closure G0,ab

Z(p)in Gab

Z(p)is a torus over Z(p)

(cf. Fact 2.3.12 applied over Zp). As G0Z(p)

is the pull back of G0,abZ(p)

via the natural epimor-

phism GZ(p)։ Gab

Z(p), we have a short exact sequence 1 → Gder

Z(p)→ G0

Z(p)→ G0,ab

Z(p)→ 1.

This implies that G0Z(p)

is a reductive group scheme whose abelianization is exactly G0,abZ(p)

.

We have a second short exact sequence 1 → G0Z(p)

→ GZ(p)→ Gm → 1 and GZ(p)

acts on

the Z(p)-span of ψ via this Gm quotient of it. Thus G0Z(p)

is the closed subgroup scheme

of GZ(p)that fixes ψ.

Let G0 be the closed subgroup scheme of G that fixes λM . Arguments similar to theones of the previous paragraph show that if G is a reductive group scheme overW (k), thenG0 is a reductive group scheme over W (k).

110

6.1.5. Simple facts. Until the end we assume that (f, L, v) is a standard Hodge situation.Thus G is a reductive group scheme over W (k). There exist non-canonical identificationsH1 = L∗

p under which (see Subsection 1.5.1):

– the perfect form λH1 on H1 is the perfect form ψ∗ on L∗p (defined by ψ) and for all

α ∈ J the p-component of the etale component of wα := z∗E(GQ,X)(wAα ) is vα.

Strictly speaking this holds if there exists an O(v)-monomorphism eW (k) :W (k) → C;but to define H1 we can assume that k is of countable transcendental degree and thereforethat eW (k) exists. Also Subsection 1.5.1 mentions a Gm(Zp)-multiple β(p) of ψ

∗; but as thecomplex 1 → G0

Zp(Zp) → GZp

(Zp) → Gm(Zp) → 1 is exact, we can assume that β(p) = 1.Let

G := GW (k).

We fix a Hodge cocharacter of (M,ϕ, G) to be denoted as

µ : Gm → G.

Until Section 6.2 we assume that k = k. We know that iA ⊗BdR(W (k)) is an isomor-phism that takes tα to vα (cf. Corollary 6.1.2) and that takes λM to a Gm(BdR(W (k)))-multiple of λH1 = ψ∗ (cf. property 6.1 (d)). By composing iA ⊗ BdR(W (k)) witha BdR(W (k))-linear automorphism of M ⊗W (k) BdR(W (k)) defined by an element ofIm(µ)(BdR(W (k))), we get the existence of an isomorphism

(M ⊗W (k) BdR(W (k)), λM , (tα)α∈J )→(M0 ⊗W (k(v)) BdR(W (k)), ψ∗, (vα)α∈J ).

But the set H1(B(k), G0B(k)) contains only the trivial class, cf. [Se2, Ch. II, §3, 3.3, c)

and Ch. III, §2, case 2.3 c)]. Thus from the last two sentences we get the existence of anisomorphism

ρ : (M [1

p], (tα)α∈J )→(M0 ⊗W (k(v)) B(k), (vα)α∈J )

under which λM is mapped to ψ∗. In what follows, it is convenient to allow ρ to take λMto some Gm(B(k))-multiple of ψ∗.

Let gy := ρϕρ−1(ϕ0 ⊗ σ)−1. It is a B(k)-linear automorphism of M0 ⊗W (k(v)) B(k)that fixes vα for all α ∈ J . Thus we have

gy ∈ G(B(k)).

Due to the property 1.5.2 (a), the cocharacters ρµB(k)ρ−1 and µ0,B(k) of GB(k) become

G(C)-conjugate over C. As GB(k) is split, we get that ρµB(k)ρ−1 and µ0,B(k) are G(B(k))-

conjugate. Thus by replacing ρ with ρg, where g ∈ G(B(k)), we can assume that we have

(28) ρµB(k)ρ−1 = µ0,B(k) : Gm → GB(k).

Therefore we have ρ(F 1[ 1p ]) = F 10 ⊗W (k(v)) B(k).

111

6.1.6. The adjoint context. Let ρad : Lie(GadB(k))→Lie(Gad

B(k)) be the Lie isomorphism

over B(k) that is defined by ρ. We have two hyperspecial subgroups of G(B(k)). Thefirst one is ρG(W (k))ρ−1 6 G(B(k)) and therefore it normalizes ρ(M). The second oneis G(W (k)) and therefore it normalizes M0 ⊗W (k(v)) W (k). They are conjugate under

an element iy ∈ Gad(B(k)), cf. [Ti2, end of Subsect. 2.5]. Thus we have an identity

iyρad(Lie(Gad))i−1

y = Lie(Gad). Let µad0,W (k) : Gm → Gad and µad

y : Gm → Gad be the

cocharacters defined by µ0,W (k) and by the inner conjugate under iy of ρµρ−1 : Gm →

GLρ(M) (respectively). As the generic fibres of µad0,W (k) and µ

ady are Gad(B(k))-conjugate

and as Gad is split, the cocharacters µad0,W (k) and µad

y are Gad(W (k))-conjugate. Thus

we can choose iy such that it fixes µ0,B(k) i.e., such that µad0 = µad

y . The σ-linear Lie

automorphism iyρadϕ(ρad)−1i−1

y of Lie(Gad)[ 1p ] is of the form

gy(ϕ0 ⊗ σ),

where gy ∈ Gad(B(k)). But as µad0 = µad

y , from property 1.3.2 (a) we get that we have an

identity (gy(ϕ0 ⊗ σ))−1(Lie(Gad)) = (ϕ0 ⊗ σ)−1(Lie(Gad)). Thus gy normalizes Lie(Gad).But Gad(W (k)) is a maximal bounded subgroup of Gad(B(k)) (cf. [Ti2, Subsect. 3.2])which is a subgroup of the normalizer of Lie(Gad) in Gad(B(k)). Thus Gad(W (k)) is thislast normalizer and therefore we have

gy ∈ Gad(W (k)).

6.1.7. New tensors. Until Subsection 6.1.9 we assume that the following two conditionshold:

(i) there exists a maximal torus T of G such that the quadruple (M,F 1, ϕ, T ) is aShimura filtered F -crystal, and

(ii) either p ≥ 3 or p = 2 and (M,ϕ) has no integral slopes.

Let (tα)α∈J0be the family of elements of Lie(T ) fixed by ϕ. As k = k, their W (k)-

span is Lie(T ) itself. Each tα is the crystalline realization of an endomorphism of Dz (forinstance, cf. (ii) and properties 1.2.1 (b) and (c)). For each α ∈ J0, let vα ∈ EndZp

(H1)be the element that corresponds to tα via Fontaine comparison theory for A (or for Dz).

6.1.8. Lemma. Let κ be the Zp-span of vα|α ∈ J0. Then κ is the Lie algebra of a

maximal torus Tet of the reductive, closed subgroup scheme GZpof GLH1 = GLL∗

p.

Proof: Let α ∈ J0 be such that tα is a regular element of Lie(GB(k)). Thus vα is a regular

element of Lie(GQp). Thus κ[ 1p ] is the Lie algebra of a maximal torus Tet,Qp

of GQp. Let

Tet be the Zariski closure of Tet,Qpin GZp

. Let T big be a maximal torus of GLM that

contains T and such that (M,F 1, ϕ, T big) is an Sh-canonical lift, cf. Subsection 4.3.2. Let

κbig and T biget be the analogues of κ and Tet but obtained working with T big instead of T .

We know that Lie(T big) isW (k)-generated by projectors ofM on rank 1 direct summandsof it. Thus using W (k)-linear combinations of elements of κbig, we can “capture” the rank

112

1 direct summands of L∗p ⊗Zp

W (k) that are normalized by T biget,W (k). Thus T big

et,W (k) is a

maximal torus of GLL∗p⊗ZpW (k) and therefore T big

et is a torus. Thus Tet is a torus over Zp,cf. Fact 2.3.12.

Obviously κ ⊆ Lie(Tet). Every element v ∈ Lie(Tet) defines naturally an endomor-phism of Dz and therefore it of the form tα for some α ∈ J0. Thus Lie(Tet) ⊆ κ. Thereforewe have Lie(Tet) = κ.

The Zariski closure in GLH1 (resp. in GLM ) of the subgroup of GLH1[ 1p ](resp. of

GLM [ 1p ]) that fixes vα (resp. tα) for all α ∈ J := J ∪ J0, is Tet (resp. is T ).

6.1.9. Two variants. (a) If (M,F 1, ϕ, G) is an Sh-canonical lift, then condition 6.1.7 (i)holds (cf. Subsection 4.3.2). Moreover, even if the condition 6.1.7 (ii) does not hold, thesubgroup Tet,Qp

of GLH1[ 1p ]is well defined (as Grothendieck–Messing deformation theory

implies that for p = 2 each p2tα with α ∈ J0 is the crystalline realization of a well definedendomorphism of Dz) and (cf. the proof of Lemma 6.1.8) it is a torus over Qp.

(b) Let x ∈ X . Let Tx be a maximal torus of GR such that x : ResC/R → GR

factors through Tx. It is well known that Tx is the extension of a compact torus by Gm =Z(GSp(W,ψ)), cf. [De3, axiom (2.1.1.2)] applied to the Shimura pair (GSp(W,ψ), S).Let TZ(p)

be a maximal torus of GZ(p)whose extension to R is GQ(R)-conjugate to Tx, cf.

[Ha, Lem. 5.5.3]. Thus there exists an element x0 ∈ X such that x0 factors through TR.We now assume that: (i) k = F, and (ii) z ∈ N (W (F)) is such that via an O(v)-embedding

W (F) → C, it defines the complex point [x0, 1] ∈ N (C) = GZ(p)(Z(p))\(X × GQ(A

(p)f ))

(see Subsection 1.5.1). Let T be the flat, closed subgroup scheme of G that corresponds toTZp

via Fontaine comparison theory for A. The arguments of Lemma 6.1.8 can be reversed(i.e., we can pass from the Zp-etale context to the crystalline context instead of the other

way round) to show that T is a torus of G. This works even for p = 2 as each element ofLie(TZp

) does define naturally an endomorphism of Dz.

6.2. Deformation theory

We start with some notations that describe the concrete principally quasi-polarizedF -crystals of the formal deformations of (A, λA) = z∗(A,ΛA) associated naturally to Nand M. In Subsection 6.2.1 we introduce some abstract principally quasi-polarized F -crystals and then in Subsection 6.2.3 we recall a theorem of Faltings that allows us (seeSubsection 6.2.6) to “relate” them to the concrete principally quasi-polarized F -crystals wehave mentioned. Some conclusions are gathered in Propositions 6.2.7 to 6.2.9. For futurereferences, in Subsection 6.2.3 we work in the axiomatized context of the class of the socalled GD (good deformation) rings which “contains” the ring R of Subsection 2.8.4.

Let

d := dimC(X) and e := dimC(S).

We denote also by z the two closed embeddings Spec(W (k)) → NW (k) and Spec(W (k)) →MW (k) defined naturally by z : Spec(W (k)) → N . Let R (resp. R1) be the completionof the local ring of the k-valued point of NW (k) (resp. of MW (k)) defined by y ∈ N (k).

113

We identify R (resp. R1) with W (k)[[x1, . . . , xd]] (resp. with W (k)[[w1, . . . , we]]) in sucha way that the ideal

I := (x1, . . . , xd) ⊆ R (resp. I1 := (w1, . . . , we) ⊆ R1)

defines z. Let ΦR (resp. ΦR1) be the Frobenius lift of R (resp. of R1) that is compatible

with σ and takes xi to xpi (resp. wi to w

pi ) for i ∈ S(1, d) (resp. for i ∈ S(1, e)). Let

i1 : Spec(R) → Spec(R1), mR : Spec(R) → N , and mR1: Spec(R1) → M

be the natural morphisms. Thus mR1 i1 is the composite of mR with the morphism

N → M. Let mR[ 1p ]: Spec(R[ 1p ]) → NE(GQ,X) be the morphism defined naturally by mR.

LetCR = (MR, F

1R, ϕR,∇R, λMR

)

be the principally quasi-polarized filtered F -crystal over R/pR of m∗R(A,ΛA). For each

α ∈ J let

tRα ∈ T (MR[1

p])

be de Rham component of m∗R[ 1p ]

(wAα ).

We recall that ϕR acts naturally on (MR[1p ])

∗ via the rule: if x ∈ (MR[1p ])

∗ and if

y ∈MR[1p ], then ϕR(x) ∈ (MR[

1p ])

∗ is such that ϕR(x)(ϕR(y)) = ΦR(x(y)); here we denote

also by ΦR the endomorphism of R[ 1p ] defined by ΦR. Also ϕR acts naturally in the usual

tensor way on T (MR[1p]). We apply Corollary 6.1.2 to the composite of a Teichmuller lift

zgen : Spec(W (k1)) → Spec(R)

with mR, where k1 is an algebraically closed field and where the special fibre of zgen isdominant. We get that the tensor tRα ∈ T (MR[

1p ]) is fixed by ϕR. Let

CR1= (MR1

, F 1R1, ϕR1

,∇R1, λMR1

)

be the principally quasi-polarized filtered F -crystal over R1/pR1 of the pull back throughmR1

: Spec(R1) → M of the universal principally polarized abelian scheme over M.

6.2.1. An explicit deformation of (M,F 1, ϕ, λM , (tα)α∈J ). Let M = F 1 ⊕ F 0 be thedirect sum decomposition normalized by µ. Let N1 be the unipotent radical of the maximalparabolic subgroup scheme of GSp(M,λM ) that is the normalizer of F 0 in GSp(M,λM ).In other words, N1 is the maximal connected, unipotent, commutative, smooth, closedsubgroup scheme of GSp(M,λM) such that the composite cocharacter µ : Gm → G →GSp(M,λM ) acts on Lie(N1) via the identity character of Gm. The relative dimension ofN1 is e and N1 acts trivially on both F 0 and M/F 0. The intersection

N0 := N1 ∩ G

114

is the unipotent radical of the parabolic subgroup scheme Q of G that is the normalizer ofF 0 in G (see Subsection 2.5.5). We identify the spectrum of the local ring of the completionof N1 along its identity section with Spec(R1) in such a way that the spectrum of the localring of the completion of N0 along its identity section becomes the closed subscheme

Spec(R0)

of Spec(R1) defined by the ideal (wd+1, . . . , we); we also identifyR0 withW (k)[[w1, . . . , wd]].Let

I0 := (w1, . . . , wd) ⊆ R0 =W (k)[[w1, . . . , wd]].

Let ΦR0be the Frobenius lift of R0 defined by ΦR1

(via passage to quotients or viarestriction toR0 viewed as aW (k)-subalgebra ofR1); for i ∈ S(1, d) we have ΦR0

(wi) = wpi .

Following [Fa2, §7], let

C1 := (M ⊗W (k) R1, F1 ⊗W (k) R1, gR1

(ϕ⊗ ΦR1),∇1, λM )

be the principally quasi-polarized filtered F -crystal over R1/pR1 defined by the naturaluniversal morphism gR1

: Spec(R1) → N1 (that identifies Spec(R1) with the spectrum ofthe local ring of the completion of N1 along its identity section). The connection ∇1 onM⊗W (k)R1 is uniquely determined (cf. Fact 2.8.5) by the requirement that gR1

(ϕ⊗ΦR1) is

horizontal with respect to∇1 (in the sense of Subsection 2.8.4) and this explains why indeedλM defines a principal quasi-polarization of (M⊗W (k)R1, F

1⊗W (k)R1, gR1(ϕ⊗ΦR1

),∇1).We recall that the existence of ∇1 is guaranteed by [Fa2, Thm. 10]. Let

C0 := (M ⊗W (k) R0, F1 ⊗W (k) R0, gR0

(ϕ⊗ ΦR0),∇0, λM )

be the pull back of C1 to R0/pR0. Thus gR0: Spec(R0) → N0 is the natural universal

morphism (that identifies Spec(R0) with the spectrum of the local ring of the completion ofN0 along its identity section). We denote also by ∇0 the connection on T (M ⊗W (k)R0[

1p ])

induced naturally by ∇0.

6.2.2. Proposition. The following three things hold:

(a) for each element α ∈ J we have ∇0(tα) = 0;

(b) each tensor tα belongs to the F 0-filtration of T (M ⊗W (k) R0[1p]) defined by

F 1 ⊗W (k) R0[1p];

(c) the Kodaira–Spencer map

K0 : (⊕di=1R0dwi)

∗ → HomR0(F 1⊗W (F)R0, (M/F 1)⊗W (F)R0) = HomW (k)(F

1, F 0)⊗W (F)R0

of ∇0 is injective and its image is the direct summand of HomW (k)(F1, F 0)⊗W (F) R0 that

is (naturally identified with) the R0-module Lie(N0)⊗W (k) R0.

Proof: The tensor tα ∈ T (M ⊗W (k) R0[1p]) is fixed by gR0

and ϕ ⊗ ΦR0and thus also

by gR0(ϕ ⊗ ΦR0

). Therefore we have ∇0(tα) = (gR0(ϕ ⊗ ΦR0

) ⊗ dΦR0)(∇0(tα)). As

115

dΦR0(dwi) = pwp−1

i dwi, by induction on q ∈ N we get that the following belongnessrelation ∇0(tα) ∈ (M ⊗W (k) R0[

1p]) ⊗R0[

1p ]

⊕di=1I

q0 [

1p]dwi holds. As R0 is complete with

respect to the I0-topology, we get that ∇0(tα) = 0. Thus (a) holds. Part (b) is obvious.

We check (c). Let the flat connection ∇R0triv on M ⊗W (K)R0 be as in Subsection 2.8.4.

As ΦR0(I0) ⊆ I2

0 , the connection ∇0 is modulo Ip−10 of the form ∇R0

triv − g−1R0dgR0

. Thus,as gR0

is a universal morphism, we get that K0 modulo I0 is injective. As for all α ∈ Jwe have ∇0(tα) = 0, Im(K0) is included in the image of Lie(G)⊗W (k) R0 in

HomW (k)(F1, F 0)⊗W (F) R0 = (EndW (k)(M)/F 0(EndW (k)(M)))⊗W (k) R0,

where F 0(EndW (k)(M)) := x ∈ EndW (k)(M)|x(F 1) ⊆ F 1. As we have a direct sum

decomposition Lie(G) = Lie(N0) ⊕ F 0(EndW (k)(M)) ∩ Lie(G) (cf. property 1.3.2 (b)),Im(K0) is included in the direct summand of HomW (k)(F

1, F 0)⊗W (F)R0 that is (naturallyidentified with) the R0-module Lie(N0)⊗W (k)R0. As K0 modulo I0 is injective, by reasonsof ranks we get that Im(K0) is in fact the last mentioned direct summand. Thus (c) holds.

6.2.3. GD rings. By a GD ring over W (k) we mean a triple

(C,ΦC , (Iq)q∈N),

where C is a commutative W (k)-algebra, ΦC is a Frobenius lift of C, and (Iq)q∈N is adecreasing filtration of C by ideals such that the following four axioms hold:

(a) we have C = proj.lim.q→∞C/Iq and C/I1 = W (k);

(b) for all positive integers q we have ΦC(Iq) ∪ I2q ⊆ Iq+1;

(c) for all positive integers q the W (k)-module Iq/Iq+1 is either 0 or torsion free;

(d) all W (k)-algebras C and C/Iq’s are p-adically complete; here q is an arbitrarypositive integer.

Above GD stands for good deformation and this terminology is justified by Theorem6.2.4 below. We consider a pair (CC , (t

Cα )α∈J ), where CC = (MC , F

1C , ϕC ,∇C , λC) and

where (tCα )α∈J is a family of tensors of T (MC [1p ]), such that the following five things hold:

(i) MC is a free C-module of finite rank, ∇C is a connection on MC , F1C is a direct

summand of MC , ϕC is a ΦC-linear endomorphism of MC with the property that MC isC-generated by ϕC(MC + 1

pF1C), and λC is a perfect alternating form on MC with the

property that for all elements x, y ∈MC we have λC(ϕC(x)⊗ ϕC(y)) = pΦC(λC(x, y));

(ii) each tensor tCα is fixed by the ΦC-linear endomorphism of T (MC [1p]) defined by

ϕC and belongs to the F 0-filtration of T (MC [1p]) defined by F 1

C [1p];

(iii) the reduction modulo I1 of (MC , F1C , ϕC , λC , (t

Cα )α∈J ) is (M,F 1, ϕ, λM , (tα)α∈J );

(iv) the quadruple (MC , ϕC ,∇C , λC) is the evaluation at the thickening associatednaturally to the closed embedding Spec(C/pC) → Spec(C) of a principally quasi-polarizedF -crystal over C/pC.

116

Let (ΩC/W (k))∧ be as in Subsection 2.8.4. We recall that (iv) implies that the con-

nection ∇C : MC → MC ⊗C (ΩC/W (k))∧ is integrable, nilpotent modulo p and that ΦC

is horizontal with respect to ∇C (in the sense of Subsection 2.8.4). We now include arefinement of a weaker version of a result of Faltings (see [Fa2, §7]: in [Fa2, §7] it is consid-ered the particular case when one has Iq = Cq

1 and C→R, with R as in Subsection 2.8.4;moreover loc. cit. also proves the existence and the uniqueness of ∇C).

6.2.4. Theorem. Let the triple (C,ΦC, (Iq)q∈N) be a GD ring. Let the pair (CC , (tCα )α∈J )

be such that conditions 6.2.3 (i) to (iv) hold. Then there exists a unique morphism eC :Spec(C) → Spec(R0) of W (k)-schemes that has the following two properties:

(a) at the level of rings, the ideal I0 is mapped to I1;

(b) there exists an isomorphism e∗C(C0, (tα)α∈J )→(CC , (tCα )α∈J ) which modulo I1 is

the identity automorphism of (M,F 1, ϕ, λM , (tα)α∈J ).

Proof: Due to axioms 6.2.3 (a) and (b), it is enough to show by induction on q ∈ Nthat there exists a unique morphism eC,q : Spec(C/Iq) → Spec(R0) of W (k)-schemes suchthat at the level of rings I0 is mapped to I1/Iq and the analogue of (b) holds for eC,q.This is so as the axiom 6.2.3 (b) implies that there exists no non-trivial automorphismof e∗C,q(C0, (tα)α∈J ) which modulo I1/Iq is defined by 1M . Obviously eC,1 exists and isunique. The inductive passage from q to q + 1 goes as follows. We can assume thatIq 6= Iq+1. We first choose eC,q+1 : Spec(C/Iq+1) → Spec(R0) to be an arbitrary lift ofeC,q. Let Jq := Iq/Iq+1; it is a torsion free W (k)-module (cf. axiom 6.2.3 (c)). We haveJ2q = 0, cf. axiom 6.2.3 (b). We endow Jq with the trivial divided power structure; thus

J[2]q = 0. It is known that (M ⊗W (k) R0, F

1 ⊗W (k) R0, gR0(ϕ⊗ΦR0

),∇0) is the filtered F -crystal of a p-divisible group over Spf(R0) that lifts the p-divisible group over Spf(W (k))defined by Dz (see [Fa2, Thm. 10]). Thus, as the ideals In

0 /In+10 with n ∈ N are

equipped naturally with the trivial nilpotent divided power structure, from Grothendieck–Messing deformation theory we get that C0 is the principally quasi-polarized filtered F -crystal of a principally quasi-polarized p-divisible group over Spf(R0) which modulo I0is the principally quasi-polarized p-divisible group over Spf(W (k)) defined by (Dz, λDz

).Therefore from the property 6.2.3 (iv) and from Grothendieck–Messing deformation theorywe get that e∗C,q+1(C0) is naturally identified with

(MC/Iq+1MC , F1C;q+1, ϕC ,∇C , λC),

where we denote also by (ϕC ,∇C , λC) its reduction modulo Iq+1 and where F 1C;q+1 is a

direct summand of MC/Iq+1MC which modulo Jq is F 1C/IqF

1C . Let Gq+1, N0;q+1, and

N1;q+1 be the closed subgroup schemes of GLMC/Iq+1MCthat correspond to GR0/I

q+10

,

N0,R0/Iq+10

, and N1,R0/Iq+10

(respectively) through this last identification Jq+1. Let tCα,q+1

be the reduction modulo Iq+1 of tCα . Under Jq+1, the tensor tα,q+1 := (eC,q+1)∗(tα) is

tCα,q+1 for each α ∈ J . This is so as (cf. the inclusion ΦC(Iq) ⊆ Iq+1 of the axiom 6.2.3(b)) we have:

tα,q+1 − tCα,q+1 = ϕC(tα,q+1 − tCα,q+1) ∈ ϕC(IqMC [1

p]/Iq+1MC [

1

p]) = 0.

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As F 1C;q+1 and F 1

C/Iq+1F1C are maximal isotropic C/Iq+1-submodules of MC/Iq1MC

with respect to λC that are congruent modulo Jq, there exists a unique element

h ∈ Ker(N1;q+1(C/Iq+1) → N1;q+1(C/Iq))

such that we have h(F 1C;q+1) = F 1

C/Iq+1F1C . We check using purely group theoretical

properties that in fact we have

(29) h ∈ Ker(N0;q+1(C/Iq+1) → N0;q+1(C/Iq)).

Let T (M) = ⊕i∈ZFi(T (M)) be the direct sum decomposition such that β ∈

Gm(W (k)) acts on F i(T (M)) via µ as the multiplication with β−i. Let F i(T (MC/Iq+1MC))

be the direct summand of T (MC/Iq+1MC) that corresponds naturally to F i(T (M)) via

Jq+1. As tα ∈ F 0(T (M))[ 1p], we have tα,q+1 ∈ F 0(T (MC/Iq+1MC))[

1p]. For each

α ∈ J the tensor tα,q+1 belongs to the F 0-filtration of T (MC/Iq+1MC [1p]) defined by

F 1C/Iq+1F

1C [

1p], cf. the property 6.2.3 (i). As h−1(F 1

C/Iq+1F1C) = F 1

C;q+1, we get that

h−1(tα,q+1) ∈ ⊕i≥0Fi(MC/Iq+1MC)[

1

p].

As h−1(tα,q+1) has a zero component in F−1(T (MC/Iq+1MC))[1p ], the element

h−1 − 1MC/Iq+1MC∈ Lie(N1;q+1) ⊆ F−1(T (MC/Iq+1MC))

annihilates tα,q+1 ∈ F 0(T (MC/Iq+1MC))[1p] for all α ∈ J ; this implies that we have

h−1 − 1MC/Iq+1MC∈ Lie(N1;q+1) ∩ Lie(Gq+1) and thus (as Jq is a torsion free W (k)-

module) that h−1 − 1MC/Iq+1MC∈ Jq(Lie(N1;q+1) ∩Lie(Gq+1)). As N0 = N1 ∩ G, we also

have N0;q+1 = N1;q+1∩Gq+1. This implies that Lie(N0;q+1) = Lie(N1;q+1)∩Lie(Gq+1) and

therefore that JqLie(N0;q+1) = Jq(Lie(N1;q+1) ∩ Lie(Gq+1)). Thus we have the relation

h−1 − 1MC/Iq+1MC∈ JqLie(N0;q+1). This implies that property (29) holds.

From the description of Im(K0) (see Proposition 6.2.2 (c)), we get that for each el-ement h ∈ Ker(N0;q+1(C/Iq+1) → N0;q+1(C/Iq)), there exists a unique lift eC,q+1,h :

Spec(C/Iq+1) → Spec(R0) of eC,q such that the role of h gets replaced by the one of hh.Thus our searched for lift eC,q+1 is eC,q+1,h−1 and it is unique. This ends the inductionand therefore also the proof.

6.2.5. Complements to 6.2.3 and 6.2.4. (a) If Iq/Iq+1 is a free W (k)-module of finiterank, then property (29) can be also argued as in [Fa2, §7, Rm. iii), p. 136] at the level ofstrictness of filtrations between morphisms of the category MF(W (k)) of [Fa1]. Moreover,the proof of [Fa2, Thm. 10] can be used to show that ∇C is unique and that Theorem6.2.4 holds without assuming that the property 6.2.3 (iv) holds.

(b)We list few standard examples of GD rings. We first remark that if (C,ΦC , (Iq)q∈N)is a GD ring, then for each positive integer l the triple (C/Il,ΦC/Il , (Iq+ Il/Il)q∈N) is also

118

a GD ring; here ΦC/Il is the Frobenius lift of C/Il induced naturally by ΦC (cf. axiom

6.2.3 (b)). Let now R, ΦR, RPD, ΦR, I, and I [l] (with l ∈ N) be as in Subsection 2.8.6. If

ΦR(I) ⊆ I2, then the triples (R,ΦR, (Iq)q∈N) and (RPD,ΦRPD , (I [q])q∈N) are the most

basic examples of GD rings.

(c) In this monograph, without any extra comment, we will apply Theorem 6.2.4 toGD rings (C,ΦC , (Iq)q∈N) of the form (R,ΦR, (I

q)q∈N) but we will always mention only

(R,ΦR, I).

6.2.6. First applications of 6.2.4. From Theorem 6.2.4 applied to (R,ΦR, I), we getthe existence of a unique morphism

e0 : Spec(R) → Spec(R0)

of W (k)-schemes that has the following two properties:

(a) At the level of rings the ideal I0 is mapped to I.

(b) We have an isomorphism e∗0(C0, (tα)α∈J )→(CR, (tRα )α∈J ) which modulo I is the

identity automorphism of (M,F 1, ϕ, λM , (tα)α∈J ).

A similar argument shows the existence of a unique endomorphism e1 : Spec(R1) →Spec(R1) of W (k)-schemes that at the level of rings maps I1 to I1 and we have an iso-morphism e∗1(C1)→CR1

which modulo I1 is the identity automorphism of (M,F 1, ϕ, λM ).As the Kodaira–Spencer map of ∇R1

is injective, e1 is a W (k)-linear automorphism. Dueto the uniqueness part of Theorem 6.2.4, the following diagram is commutative

Spec(R)e0−→ Spec(R0)yi1

yi01

Spec(R1)e1−→ Spec(R1),

where i01 is the natural closed embedding and where i1 is the natural morphism (see thesecond paragraph of Section 6.2).

6.2.7. Proposition. The following four things hold:

(a) the morphism e0 : Spec(R) → Spec(R0) of W (k)-schemes is an isomorphism;

(b) the Kodaira–Spencer map of ∇R is injective;

(c) the natural morphism i1 : Spec(R) → Spec(R1) of W (k)-schemes is a closedembedding;

(d) the tensors tα ∈ T (M [ 1p ]) (α ∈ J ) and therefore also (M,ϕ, G) depend only on

the point y ∈ N (k) and not on the lift z ∈ N (W (k)) of y.

Proof: We recall that N is the normalization of the Zariski closure of Sh(GQ, X)/Hp inMO(v)

. The morphism N → MO(v)of O(v)-schemes is finite, cf. [Va3, proof of Prop.

3.4.1]. Thus NW (k) is the normalization of the Zariski closure of (Sh(GQ, X)/Hp)B(k) inMW (k) and the morphism NW (k) → MW (k) of W (k)-schemes is finite. Therefore i1 is a

119

finite morphism of W (k)-schemes and its fibre over B(k) is a closed embedding. As e1 isan automorphism of W (k)-schemes and as i01 is a closed embedding, we get that e0 is afinite morphism of W (k)-schemes and that e0,B(k) is a closed embedding. As R0 and Rare regular local rings of equal dimension, we get that e0 is a birational, finite morphismof W (k)-schemes. As R0 is normal, we conclude that in fact e0 is an isomorphism ofW (k)-schemes. Thus (a) holds. As e0 is an isomorphism of W (k)-schemes and as theKodaira–Spencer map K0 of Proposition 6.2.2 (c) is injective, we get that (b) holds. Asi01 is a closed embedding and as i1 = e−1

1 i01 e0, we get that i1 is a closed embedding.Thus (c) holds.

As e0 is an isomorphism of W (k)-schemes and due to the property 6.2.6 (b), thepull back of (CR, (t

Rα )α∈J ) via each W (k)-valued point z of Spec(R) is of the form

(M, F 1, ϕ, λM , (tα)α∈J ). Thus (d) also holds.

Let (Mgen, F1gen, ϕgen, λMgen

, (tgenα )α∈J ) := z∗gen(MR, F1R, ϕR, λMR

, (tRα )α∈J ), wherethe Teichmuller lift zgen : Spec(W (k1)) → Spec(R) is as in Subsection 6.2.3. See Sub-

section 6.1.4 for the reductive, closed subgroup scheme G0 of G.

6.2.8. Proposition. We assume that k = k. Let y1 ∈ N (k) be such that it factors throughthe same connected component of Nk(v) as y ∈ N (k) does. Let z1 ∈ N (W (k)) be a lift of y1.

We consider the analogue (M1, F11 , ϕ1, G1, λM1

, (t1,α)α∈J ) of (M,F 1, ϕ, G, λM , (tα)α∈J )obtained by working with z1 instead of z. Then there exists an isomorphism

jz,z1 : (M,λM , (tα)α∈J )→(M1, λM1, (t1,α)α∈J )

that takes F 1 onto F 11 . In particular, we can identify naturally:

(i) G = G1, uniquely up to G(W (k))-conjugation, and

(ii) (M1, ϕ1, G1, λM1, (t1,α)α∈J ) = (M, g1ϕ, G, λM , (tα)α∈J ) for some element g1 ∈

G0(W (k)).

Proof: There exists an isomorphism

jz : (M ⊗W (k) W (k1), λM , (tα)α∈J )→(Mgen, λMgen, (tgenα )α∈J )

that takes F 1 ⊗W (k) W (k1) onto F1gen, cf. the definition of C0 and the property 6.2.6 (b).

Working with z1, let z1,gen, (M1,gen, F11,gen, ϕ1,gen, λM1,gen

, (tgen1,α)α∈J ), and

jz1 : (M1 ⊗W (k) W (k1), λM1, (t1,α)α∈J )→(M1,gen, λM1,gen

, (tgen1,α)α∈J )

be the analogues of zgen, (Mgen, F1gen, ϕgen, λMgen

, (tgenα )α∈J , λMgen), and jz (respectively).

We choose zgen and z1,gen such that their special fibres coincide. Thus we have a naturalisomorphism jgen : (Mgen, ϕgen, λMgen

, (tgenα )α∈J )→(M1,gen, ϕ1,gen, λM1,gen, (tgen1,α)α∈J ), cf.

also Proposition 6.2.7 (d). Thus F 1gen and j−1

gen(F11,gen) are two lifts of the Shimura F -crystal

(Mgen, ϕgen, Ggen), where Ggen := jzGW (k1)j−1z 6 GLMgen

. Thus F 1gen and j−1

gen(F11,gen)

are Ggen(W (k1))-conjugate (see Lemma 3.1.3) and therefore also G0gen(W (k1))-conjugate,

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where G0gen is the reductive, closed subgroup scheme of Ggen that fixes λMgen

. Thus wecan choose jgen such that F 1

gen = j−1gen(F

11,gen). Thus we have an isomorphism

j−1z1

jgen jz : (M ⊗W (k) W (k1), λM , (tα)α∈J )→(M1 ⊗W (k) W (k1), λM1, (t1,α)α∈J )

that takes F 1 ⊗W (k) W (k1) onto F11 ⊗W (k) W (k1). Therefore jz,z1 exists in the faithfully

flat topology of W (k).

By multiplying each vα with pa with a ∈ N and a >> 0, we can assume that we havetα ∈ T (M) and t1,α ∈ T (M1) for all α ∈ J . Let P be theW (k)-scheme that parameterizesisomorphisms between (M,λM , (tα)α∈J ) and (M1, λM1

, (t1,α)α∈J ) that take F 1 onto F 11 .

The normalizer P 0 of F 1 in G0 is a parabolic subgroup scheme of G0 and thus it hasconnected fibres. The group scheme P 0 acts from the left naturally on the Zariski closureP0 of PB(k) in P. From the last sentence of the previous paragraph, we get that P0 is a

right torsor of P 0. As k = k, this torsor is trivial and therefore has W (k)-valued points.

Thus jz,z1 exists. As jz,z1 is unique up to its left composites with elements of

P 0(W (k)), we get that (i) holds. As jz,z1(F1) = F 1

1 , g1 := j−1z,z1

ϕ1jz,z1ϕ−1 is an ele-

ment of GLM (W (k)). But g1 fixes tα for all α ∈ J . Therefore we have g1 ∈ G(W (k)).Moreover, as λM and λM1

are principal quasi-polarizations of (M,ϕ) and (M1, ϕ1) (re-spectively) and as jz,z1 takes λM to λM1

, we get that g1 fixes λM . Thus g1 ∈ G0(W (k))and therefore (ii) holds.

6.2.9. Proposition. We assume that the Newton polygon of (M,ϕ) is the same as theNewton polygon of (Mgen, ϕgen). Then (M,ϕ, G) is Sh-ordinary.

Proof: To prove this, we can assume that k = k (cf. Fact 3.2.1 (a)). Due to the existenceof jz (see proof of Proposition 6.2.8), we can identify (Mgen, ϕgen, Ggen) with (M ⊗W (k)

W (k1), ggen(ϕ⊗ σk1), GW (k1)) for some element ggen ∈ G(W (k1)). To show that (M,ϕ, G)

is Sh-ordinary it suffices to show that (Mgen, ϕgen, Ggen) is Sh-ordinary, cf. the equivalence

1.4.1 (a) ⇔ 1.4.1 (b) (see Theorem 1.4.1). Let d be the relative dimension of the reductivegroup scheme G0. Let R2 be the local ring of the completion of G0 along the identitysection. As N0 is a closed subgroup scheme of G0, we have a natural closed embeddingi02 : Spec(R0) → Spec(R2). We identify R2 with W (k)[[w1, . . . , wd]] in such a way thatthe ideal (wd+1, . . . , wd) of R2 defines the closed subscheme Spec(R0) of Spec(R2). Thus

I2 := (w1, . . . , wd) is the ideal of R2 that defines the identity section of G0; we haveR2/I2 = W (k). Let ΦR2

be the Frobenius lift of R2 that is compatible with σ and takeseach wi to w

pi for i ∈ S(1, d). Let gR2

: Spec(R2) → G be the universal element. Let ∇2

be the unique connection on M ⊗W (k) R2 such that

C2 = (M ⊗W (k) R2, F1 ⊗W (k) R2, gR2

(ϕ⊗ ΦR2),∇2, λM )

is a principally quasi-polarized filtered F -crystal over R2/pR2 (one argues the exis-tence of ∇2 having the desired properties in the same way we argued its analogue forC0 in Subsection 6.2.1). From Theorem 6.2.4 applied to (R2,ΦR2

, I2), we deduce theexistence of a unique morphism e2 : Spec(R2) → Spec(R0) of W (k)-schemes which:

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(i) at the level of rings maps I0 to I2, and (ii) it is such that we have an isomor-phism e∗2(C0, (tα)α∈J )→(C2, (tα)α∈J ) that modulo I2 is the identity automorphism of(M,F 1, ϕ, λM , (tα)α∈J ).

Let k2 be the algebraic closure of the field of fractions of R2/pR2. Let g2ad ∈ Gad(k2)

be the image in Gad(k2) of the pull back of gR2to k2. As g2

ad dominates Gadk , from

Corollary 1.4.5 we get that the pull back C2,gen of C2 via the dominant morphismSpec(k2) → Spec(R2/pR2) of k-schemes is an Sh-ordinary F -crystal; it is of the form(M ⊗W (k)W (k2), g2(ϕ⊗ σk2

), GW (k2)) for some element g2 ∈ G(W (k2)). Pulling back theisomorphism e∗2(C0, (tα)α∈J )→(C2, (tα)α∈J ) via i02, we get an isomorphism

(e2 i02)∗(C0, (tα)α∈J )→i∗02(C2, (tα)α∈J ) = (C0, (tα)α∈J ).

Thus e2i02 = 1Spec(R0), cf. the uniqueness part of Theorem 6.2.4. Therefore the morphismSpec(R2/pR2) → Spec(R/pR) of k-schemes defined by e2 is dominant. This implies thatthe morphism Spec(R2/pR2) → Spec(R/pR) of k-schemes defined by e−1

0 e2 is alsodominant. From this and the fact that C2,gen is Sh-ordinary, we get that (Mgen, ϕgen, Ggen)is also Sh-ordinary.

6.3. Proof of Theorem 1.6.1

Let N 0k(v) be a connected component of Nk(v). Its field of fractions contains F (see

Section 6.1) and therefore it is geometrically connected. Let O0 be the maximal opensubscheme of N 0

k(v) such that the Newton polygons of pull backs of A through geometric

points of it are all equal, cf. Grothendieck–Katz specialization theorem [Ka2, 2.3.1 and2.3.2]. If z ∈ N (W (k)) is such that y ∈ N (k) factors through O0 (i.e., belongs to O0(k)),then (M,ϕ, G) is Sh-ordinary (cf. Proposition 6.2.9). To prove Theorem 1.6.1 we will needfew additional results.

6.3.1. Proposition. We assume that k = k and that y ∈ N (k) belongs to O0(k). Thenthere exists an isomorphism of adjoint Shimura Lie F -crystals

Iy : (Lie(Gad), ϕ)→(Lie(Gad), ϕ0 ⊗ σk)

that is defined by the composite of the Lie isomorphism ρad : Lie(GadB(k))→Lie(Gad

B(k)) (see

Subsection 6.1.6) with an inner Lie automorphism of Lie(GadB(k)) defined by an element

hy ∈ Gad(W (k))iy (see Subsection 6.1.6 for iy ∈ Gad(B(k))). Therefore I(M,ϕ, G) =I(C0).

Proof: As Lie(B0,W (k(v))) (resp. Lie(T0,W (k(v)))) is taken by ϕ0 to (resp. onto) itself, as

in Subsection 3.4.3 we argue that C0 ⊗ k is Sh-ordinary. Let gy ∈ Gad(W (k)) be as inSubsection 6.1.6. Both triples (Lie(Gad), gy(ϕ0 ⊗ σ), Gad) and (Lie(Gad), ϕ0 ⊗ σ,Gad) areSh-ordinary, cf. Proposition 6.2.9 and Corollary 3.1.5 for the first one. As 1.4.1 (a) ⇔1.4.4 (a) (see Theorem 1.4.4), we get that there exists jy ∈ Gad(W (k)) that defines aninner isomorphism of the two triples. Thus we can take hy := jyiy.

6.3.2. Proposition. We assume that k = k and that y ∈ N (k) belongs to O0(k). Thenthe Newton polygon N of (M,ϕ, G) is the same as the Newton polygon N0 of C0.

122

Proof: Let n be a positive integer such that each Newton polygon slope of either N0 orN, belongs to 1

nZ. Let M0[

1p] = ⊕γ∈QW (γ)(M0[

1p], ϕ0) be the Newton polygon slope

decomposition for (M0[1p], ϕ0). Let ν0 : Gm → GLM0[

1p ]

be the Newton cocharacter such

that β ∈ Gm(B(k(v))) acts on W (γ)(M0[1p ], ϕ0) by multiplication with βnγ . As for each

element α ∈ J we have ϕ0(vα) = vα, ν0 fixes vα and therefore it factors through GB(k(v)).

We denote also by ν0 : Gm → GB(k(v)) this factorization. Working with (M,ϕ, G), we

similarly define a Newton cocharacter ν : Gm → GB(k). Let ν1 : Gm → GB(k) be ρνρ−1,

where ρ is as in Subsection 6.1.5. Let ν1,ad and ν0,B(k)ad be the cocharacters of GadB(k)

defined by ν1 and ν0,B(k). They are Gad(B(k))-conjugate (cf. the existence of Iy; see

Proposition 6.3.1) and therefore their extensions to B(k) are G(B(k))-conjugate.

We check that the two cocharacters of Gab

B(k)defined by ν

1,B(k)and ν

0,B(k)coincide.

We first remark that they depend only on the ϕs- and the ϕs0-conjugates of the cocharacters

of GabB(k) and Gab

B(k) defined by µ and µ0 (respectively), where s runs through all positive

integers. As we have ρµB(k)ρ−1 = µ0,B(k) and ρϕρ

−1 = gy(ϕ0 ⊗ σ) (see Subsection 6.1.5),for each s ∈ N the mentioned ϕs- and the ϕs

0-conjugates coincide under the canonicalisomorphism Gab

B(k)→GabB(k) defined by ρ. Thus indeed the two cocharacters of Gab

B(k)

defined by ν1,B(k)

and ν0,B(k)

coincide.

From the last two paragraphs we get that ν1,B(k)

and ν0,B(k)

are G(B(k))-conjugate.

This implies that we have an identity N = N0 of Newton polygons.

6.3.3. Corollary. We assume that k = k and that y ∈ N (k) belongs to O0(k). We takeµ : Gm → G to be the canonical cocharacter of the Sh-ordinary F -crystal (M,ϕ, G). Letl0 be the standard generator of Lie(Gm). There exists a B(k)-linear isomorphism

hz :M ⊗W (k) B(k)→M0 ⊗W (k(v)) B(k)

that takes ϕs(dµ(l0)) to ϕs0(dµ0(l0)) for all s ∈ N ∪ 0.

Proof: The homomorphism G(B(k)) → Gad(B(k)) is surjective and therefore there existsan element h′y ∈ G(B(k)) that lifts the element hy ∈ Gad(B(k)) of Proposition 6.3.1. We

now take hz := h′y (ρ⊗ 1B(k)

) :M [ 1p]⊗B(k) B(k)→M0 ⊗W (k(v)) B(k).

The canonical cocharacters are unique, cf. Subsection 4.1.4. The canonical cocharacterof C0 factors through T0,W (k(v)) (cf. Subsection 4.3.2 applied overW (F)) and therefore (cf.Lemma 3.1.1) it is µ0 : Gm → GW (k(v)). From the last two sentences and from Proposition6.3.1, we get that hz takes ϕs(dµ(l0)) to an element l0(s) of Lie(GB(k)

) that has the same

image in Lie(Gad

B(k)) as ϕs

0(dµ0(l0)), for all s ∈ N ∪ 0. But for s ∈ N ∪ 0 the images

of l0(s) and ϕs0(dµ0(l0)) in Lie(Gab

B(k)) coincide (cf. proof of Proposition 6.3.2). Thus we

have l0(s) = ϕs0(dµ0(l0)), for all s ∈ N ∪ 0.

6.3.4. Fact. The Shimura filtered F -crystal (M,F 1, ϕ, (tα)α∈J ) attached to z ∈ N (W (k))

depends only on the GQ(A(p)f )-orbit of z. Therefore the Newton polygon of (M,ϕ) depends

also only on the GQ(A(p)f )-orbit of z.

123

Proof: It is well known that the right translations ofN by elements of GQ(A(p)f ) correspond

to passages to prime to p isogenies between abelian schemes (see [Va3, Subsect. 4.1] forthe case of complex points). From this the Fact follows.

6.3.5. End of proof of Theorem 1.6.1. We take O to be the union of O0’s, whereN 0

k(v) runs through all connected components of Nk(v). Thus O is a Zariski dense, opensubscheme of Nk(v). From Proposition 6.3.2 we get that O is the maximal open subschemeof Nk(v) such that the Newton polygons of pull backs of A through geometric points of it,

are all equal to N0. Thus O is GQ(A(p)f )-invariant, cf. Fact 6.3.4. To prove Theorem 1.6.1

we can assume k = k, cf. Fact 3.2.1 (a) in connection to the statement 1.6.1 (a). We alreadyknow that if y ∈ O(k), then (M,ϕ, G) is Sh-ordinary (see beginning of Section 6.3). Ify ∈ N 0

k(v)(k) and if (M,ϕ, G) is Sh-ordinary, then from the equivalence 1.4.1 (a) ⇔ 1.4.1 (b)

we get that for every element g1 ∈ G(W (k)) the Newton polygon of (M, g1ϕ) is above theNewton polygon of (M,ϕ); from this and the property 6.2.8 (ii) we get that the Newtonpolygon of (M,ϕ) is below all Newton polygons of pulls back of A through arbitrary k-valued points of N 0

k(v) and thus we get that y ∈ O0(k). We conclude that for a point

y ∈ N (k), its attached Shimura F -crystal (M,ϕ, G) is Sh-ordinary if and only if we havey ∈ O(k) ⊆ N (k).

If y ∈ O(k), then the statement 1.6.1 (c) holds (cf. Proposition 6.3.1). If the statement1.6.1 (c) holds, then (M,ϕ, G) is Sh-ordinary (as 1.4.1 (a) ⇔ 1.4.4 (b) and as C0 ⊗ k isSh-ordinary). If the statement 1.6.1 (c) holds, then by applying the equivalence 1.4.4 (b) ⇔1.4.4 (a) to the context of (Lie(Gad

W (k)), h(ϕ0⊗σ), GadW (k)) with h ∈ gy, 1Lie(Gad

W (k)), we get

that there exists an isomorphism Iy as in Proposition 6.3.1; thus 1.6.1 (c) ⇒ 1.6.1 (d). Ifthe statement 1.6.1 (d) holds, then by applying the equivalence 1.4.1 (a) ⇔ 1.4.1 (b) to thesame context we get that (Lie(Gad

W (k)), gy(ϕ0⊗σ), GadW (k)) is Sh-ordinary and thus (M,ϕ, G)

is Sh-ordinary (cf. Corollary 3.1.5). If I0 ≤ 2, then by applying Theorem 1.4.2 to thesame context, we get that the statement 1.6.1 (e) holds if and only if (Lie(Gad

W (k)), gy(ϕ0⊗

σ), GadW (k)) is Sh-ordinary (i.e., if and only if (M,ϕ, G) is Sh-ordinary, cf. Corollary 3.1.5).

Thus O has all properties required in order that the first two paragraphs of Theorem1.6.1 hold. The last paragraph of Theorem 1.6.1 follows from the property 6.2.8 (ii) andthe equivalence 1.4.1 (a) ⇔ 1.4.4 (a). This ends the proof of Theorem 1.6.1.

6.4. The existence of geometric lifts

We recall that z ∈ N (W (k)) is a lift of y ∈ N (k) whose attached Shimura filtered F -crystal is (M,F 1, ϕ, G). See Subsection 6.2.1 for the closed subgroup scheme N0 of G. Inthis Section we use the language of PD hulls in order to prove a converse (see Proposition6.4.6) to the below Fact.

6.4.1. Fact. Let y ∈ N (k). Let z1 ∈ N (W (k)) be another lift of y. Let (M,F 11 , ϕ, G)

be the Shimura filtered F -crystal attached to z1. Then there exists unique element n1 ∈Ker(N0(W (k)) → N0(k)) such that we have F 1

1 = n1(F1).

Proof: The tensor tα corresponds to vα via Fontaine comparison theory of z∗1(A), cf.Proposition 6.2.7 (d). Thus the inverse of the canonical split cocharacter µ1 of (M,F 1

1 , ϕ)

124

fixes tα for all α ∈ J (cf. the functorial aspects of [Wi, p. 513]); therefore µ1 factorsthrough G. Thus µ1 is a Hodge cocharacter of (M,ϕ, G) and F 1

1 is a lift of (M,ϕ, G).Therefore the Fact follows from Corollary 3.1.4.

6.4.2. On PD hulls. For l ∈ N let I [l] (resp. I[l]0 ) be the l-th ideal of the divided

power filtration of RPD (resp. of RPD0 ) defined by I (resp. by I0); see Subsection 2.8.4.

Let ΦRPD be the Frobenius lift of RPD defined by ΦR. Let the flat connection ∇RPD

triv onM⊗W (k)R

PD be as in Subsection 2.8.4. As the maximal ideal of RPD has a divided powerstructure, the pull back of (MR, ϕR,∇R, λMR

, (tRα )α∈J ) to RPD/pRPD can be identified

with (M ⊗W (k) RPD, ϕ ⊗ ΦRPD ,∇RPD

triv , λM , (tα)α∈J ) (see [BBM] and [BM]). No element

of EndW (k)(M)⊗W (k) I[1] is fixed by ϕ⊗ ΦRPD . Thus there exists a unique isomorphism

(identification)

Jy : (MR⊗RRPD, ϕR⊗ΦRPD ,∇R, λMR

, (tRα )α∈J )→(M⊗W (k)RPD, ϕ⊗ΦRPD ,∇RPD

triv , λM , (tα)α∈J )

which modulo I [1] is the identity automorphism of (M,ϕ, λM , (tα)α∈J ). Here we denotealso by ∇R the connection on MR ⊗R R

PD that is defined naturally by ∇R.

Let P big be the maximal parabolic subgroup scheme of GLM that normalizes F 1. SeeSection 6.2 for the direct summand F 1

R of MR. Let F1RPD := Jy(F

1R⊗RR

PD); it is a direct

summand of M ⊗W (k) RPD. The product morphism N1 ×W (k) (P

big ∩GSp(M,λM )) →GSp(M,λM ) is an open embedding (cf. proof of Lemma 3.1.2). Thus, as F 1 ⊗W (k) R

PD

and F 1RPD are maximal isotropic RPD-submodules of M ⊗W (k) R

PD with respect to λMwhich modulo the maximal ideal (p, I [1]) of RPD coincide, there exists a unique elementnPD1 ∈ N1(R

PD) such that we have nPD1 (F 1 ⊗W (k) R

PD) = F 1RPD .

Each W (k)-valued point z1 ∈ N (W (k)) that lifts y, factors through the naturalcomposite morphism Spec(RPD) → Spec(R) → N of W (k)-schemes; thus we also viewz1 as a W (k)-valued point of Spec(RPD). The Shimura filtered F -crystal attached toz1 is (M,n1(F

1), ϕ, G), where n1 := nPD1 z1 ∈ N1(W (k)). From Fact 6.4.1 and the

uniqueness part of Corollary 3.1.4 applied to (M,ϕ,GLM ), we get that in fact we haven1 ∈ Ker(N0(W (k)) → N0(k)) 6 N0(W (k)). As the W (k)-valued points of Spec(RPD)are Zariski dense, we conclude that in fact

nPD1 ∈ N0(R

PD).

As for all positive numbers n the Wn(k)-algebra RPD/pnRPD is the inductive limit of its

artinian localWn(k)-subalgebras, we get that nPD1 factors through the spectrum Spec(R0)

of the local ring of the completion of N0 along its identity section. This implies that nPD1

factors through Spec(RPD0 ) and therefore it gives birth to a morphism

(30) F0 : Spec(RPD) → Spec(RPD0 )

of W (k)-schemes. We can identify RPD/I [p] with R/(Ip) and RPD0 /I

[p]0 with R0/(I

p0 ). As

nPD1 z is 1M , the W (k)-homomorphism RPD

0 → RPD that defines F0, takes I[1]0 to I [1].

Thus we can speak about the dual

F00 : I/I2 → I0/I20

125

of the W (k)-linear tangent map of F0 computed at z. The composite of the W (k)-linearisomorphism I0/I2

0→I/I2 defined by e0 with F00 is naturally defined by g−1R0

−1M⊗W (k)R0

modulo I20 and thus it is a W (k)-linear automorphism. Thus F00 is a W (k)-linear isomor-

phism. It is well known that for p ≥ 3 this implies that F0 itself is an isomorphism ofW (k)-schemes.

6.4.3. Corollary. For every positive integer l the morphism

F1p

0,l : Spec(RPD/I [l][

1

p]) → Spec(RPD

0 /I[l]0 [

1

p])

of B(k)-schemes defined naturally by F0, is an isomorphism. Thus F0 is naturally therestriction of an isomorphism Spec(B(k)[[x1, . . . , xd]])→Spec(B(k)[[w1, . . . , wd]]) of B(k)-schemes.

Proof: We have natural identifications RPD/I [l][ 1p ] = B(k)[[x1, . . . , xd]]/(x1, . . . , xd)l and

RPD0 /I [l]

0 [ 1p] = B(k)[[w1, . . . , wd]]/(w1, . . . , wd)

l. The tangent map of F1p

0,l is a B(k)-linear

isomorphism as F00 : I/I2→I0/I20 is a W (k)-linear isomorphism. Thus F

1p

0,l is defined by

a B(k)-epimorphism RPD0 /I

[l]0 [ 1p ] → RPD/I [l][ 1p ] which (by reasons of dimensions of B(k)-

vector spaces) is in fact a B(k)-isomorphism. By taking the projective limit for l → ∞,the Corollary follows.

6.4.4. On the case p = 2. We refer to Section 6.1 with p = 2. Therefore iA : M ⊗W (k)

B+(W (k)) → H1 ⊗ZpB+(W (k)) is the B+(W (k))-linear monomorphism of the Fontaine

comparison theory for A. The kernel of sW (k) : B+(W (k)) ։ W (k)∧is F 1(B+(W (k)))

and therefore we can identify

gr0 := F 0(B+(W (k)))/F 1(B+(W (k))) = B+(W (k))/F 1(B+(W (k)))

with W (k)∧. Thus gr1 := F 1(B+(W (k)))/F 2(B+(W (k))) is naturally a W (k)

∧-module;

it is free of rank one. Let u0 be a generator of it. We choose β0 ∈ F 1(B+(W (k)))(see property 6.1 (d)) such that its image in gr1 is 2u0 times an invertible element iβ0

of W (k)∧

and moreover we have β0 ∈ 2F 1(B+(W (k))) (cf. [Fa2, §4, p. 125] and [Fo2,Subsect. 2.3.4]). The fact that Coker(iA) is annihilated by β0 (cf. property 6.1 (d)) implies(cf. also the strictness part of [Fa2, Thm. 5] and the invertibility of iβ0

) the following Fact:

6.4.5. Fact. We assume that p = 2. Let jA : F 0 ⊗W (k) gr1⊕F 1 ⊗W (k) gr

0 → H1 ⊗Z2gr1

be the W (k)∧-linear map induced by iA at the level of one gradings. Then jA is injective

and Coker(jA) is annihilated by 2.

6.4.6. Proposition. Let y ∈ N (k). Let n ∈ Ker(N0(W (k)) → N0(k)). We have:

(a) If p ≥ 3, then there exists a unique lift zn ∈ N (W (k)) of y whose attached Shimurafiltered F -crystal is (M, n(F 1), ϕ, G).

(b) If p = 2, then the number nn of such lifts zn is finite; if moreover k = k (resp. ifmoreover (M,ϕ) has no integral slopes), then we have nn ≥ 1 (resp. we have nn = 1).

126

Proof: The case p ≥ 3 is a direct consequence of the fact that F0 is an isomorphism ofW (k)-schemes. Thus until the end of the proof we will take p = 2. Not to introduceextra notations, to prove that nn is finite we can assume that n = 1M . Let A be anotherabelian variety over W (k) that lifts Ak and whose Hodge filtration is F 1. Let iA be the

analogue of iA but obtained working with A instead of with A. From Grothendieck–Messing deformation theory we get that the 2-divisible groups of A and A are isogenous(for instance we have such an isogeny whose crystalline realization is the multiplicationby 4 endomorphism of (M,F 1, ϕ)). Via such isogenies we can identify iA[

1β0] with iA[

1β0].

From this and Fact 6.4.5, we get that H1et(A,Z2) is a Z2-lattice of H1[ 1

2] such that we

have 2H1 ⊆ H1et(A,Z2) ⊆ 1

2H1. But the 2-divisible group of A is uniquely determined

by H1et(A,Z2), cf. [Ta1, Subsect. 4.2, Cor. 2]. Thus A itself is uniquely determined

by H1et(A,Z2), cf. Serre–Tate deformation theory. Therefore, as the number of such Z2-

lattices H1et(A,Z2) of H

1[ 12] is finite, the number n1M

is finite.

We check that nn ≥ 1 if k = k. We prove by induction on l ∈ N that there exists aW (k)-valued point zl ∈ N (W (k)) that lifts y and such that its attached Shimura filteredF -crystal is (M,F 1

l , ϕ, G), where F1l and n(F 1) coincide modulo 2l. We will also show

that we can assume that zl+1 ∈ N (W (k)) is congruent to zl modulo 2l. The case l = 1is obvious. The passage from l to l + 1 goes as follows. Let zl,l be the Wl(k)-valuedpoint of Spec(RPD) (resp. of Spec(RPD

0 )) defined by zl (resp. by F0 zl). The set ofWl+1(k)-valued points of Spec(RPD) (resp. of Spec(RPD

0 )) that lift zl,l, is parameterizedby the k-valued points of an affine space Tzl,l (resp. T0,zl,l) isomorphic to Ad

k, the origin

corresponding to zl modulo 2l+1. Let

Fzl,l : Tzl,l → T0,zl,l

be the morphism of k-schemes defined naturally by (30). We check the following twoproperties:

(a) the fibres of Fzl,l are finite;

(b) we can naturally view Fzl,l as a homomorphism of affine groups.

Based on Proposition 6.2.7 (c), to check (a) we can assume that N = M. Not tointroduce extra notations, by modifying zl accordingly we can also assume that we dealwith the fibre of Fzl,l over 0 ∈ T0,zl,l(k). We show that the assumption that this fibre isinfinite leads to a contradiction. This assumption implies the existence of an infinite num-ber of non-isomorphic principally polarized abelian schemes over W (k) that lift (A, λA)kand whose principally quasi-polarized Dieudonne filtered modules are (M, n(F 1), ϕ, λM),cf. Grothendieck–Messing and Serre–Tate deformation theories. But this contradicts therelation nn ∈ N ∪ 0. Thus (a) holds.

Not to introduce extra notations, to check (b) we can assume that zl = z andthus that n is congruent to 1M modulo 2l. We have natural identifications Tzl,l =

Spec(k[x1, . . . , xd]) = Adk and T0,zl,l = Spec(k[w1, . . . , wd]) = Ad

k. TheW (k)-homomorphism

RPD0 → RPD defined by F0 takes wi to 2l+1si,l+ui,l, where si,l, ui,l ∈ RPD and where ui,l

is a finiteW (k)-linear combination of elements that are finite product of elements in the set

127

xnjj

nj !|j ∈ S(1, d), nj ∈ N. But 2lnj

nj !is always divisible by 2l and moreover it is divisible by

2l+1 if nj /∈ 2mj |mj ∈ N ∪ 0. The W (k)-valued points of Spec(RPD) that lift zl,l arein bijection to W (k)-epimorphisms W (k)[[x1, . . . , xd]] ։W (k) that map each xi to 2lxi,l,where xi,l ∈W (k); moreover, the k-valued point of Tzl,l = Spec(k[x1, . . . , xd]) associated tosuch a W (k)-homomorphism W (k)[[x1, . . . , xd]] →W (k), is defined by the k-epimorphismk[x1, . . . , xd] ։ k that maps each xi to xi,l modulo 2. From the last three sentences weget that the k-homomorphism k[w1, . . . , wd] → k[x1, . . . , xd] that defines Fzl,l , maps each

wi to a k-linear combination of elements of the set x2mj

j |j ∈ S(1, d), mj ∈ N∪0. Thisimplies that we can naturally view Fzl,l as a homomorphism of affine groups. Thus (b)holds.

From (a) and (b) we get that Fzl,l is a finite morphism between d-dimensional varietiesover k. Thus Fzl,l(k) is surjective. Therefore there exists a W (k)-valued point zl+1 ∈

N (W (k)) which modulo 2l is congruent to zl and whose attached Shimura filtered F -crystal (M,F 1

l+1, ϕ, G) is such that F 1l+1 and n(F 1) coincide modulo 2l+1. This completes

the induction.

Let z ∈ N (W (k)) be the limit W (k)-valued point of zl’s. The Shimura filtered F -crystal attached to z is (M, n(F 1), ϕ, G) and z lifts y. Thus nn ≥ 1 if k = k.

To end the proof we are left to show that nn = 1 if p = 2 and (M,ϕ) has no integralslopes. As N (W (k)) is the subset of N (W (k)) formed by W (k)-valued points fixed underthe natural action of AutW (k)(W (k)) on N (W (k)), to check that nn = 1 we can assume

that k = k. As k = k, we have nn ≥ 1. As the natural morphism i1 : Spec(R) → Spec(R1)of W (k)-schemes is a closed embedding (see Proposition 6.2.7 (c)), each lift z1 ∈ N (W (k))of y ∈ N (k) is uniquely determined by the lift Dz1 to W (k) of the 2-divisible group of Ak

(here Dz1 is the 2-divisible group of z∗1(A)). But Dz1 is uniquely determined by its Hodgefiltration, cf. property 1.2.1 (a). From the last two sentences we get that nn ≤ 1. Thusnn = 1.

6.5. Proofs of Theorems 1.6.2 and 1.6.3

In Subsections 6.5.1, 6.5.3, 6.5.4, 6.5.7, and 6.5.8 we prove Theorem 1.6.2, 1.6.3 (a),1.6.3 (b), 1.6.3 (c), and 1.6.3 (d) (respectively). In Subsection 6.5.2 we recall few thingson Hodge cycles to be used in Subsections 6.5.3 and 6.5.5. In Subsection 6.5.5 we definethe condition W hinted at in Theorem 1.6.3 (c). See Examples 6.5.6 for two examplespertaining to the condition W. In this Section we assume that y ∈ N (k) is such thatits attached Shimura F -crystal (M,ϕ, G) is U-ordinary. Let (M,F 1, ϕ, G) be the Shimurafiltered F -crystal attached to a lift z ∈ N (W (k)) of y.

6.5.1. Proof of Theorem 1.6.2. We prove Theorem 1.6.2. Thus, if p = 2, then eitherk = k or (M,ϕ) has no integral slopes. The U-canonical lift of (M,ϕ, G) is unique, cf.statement 1.4.3 (b). Let n ∈ Ker(N0(W (k)) → N0(k)) be the unique element such thatn(F 1) is the U-canonical lift of (M,ϕ, G), cf. Corollary 3.1.4. If p = 2 let nn ∈ N be as inProposition 6.4.6 (b). If p ≥ 3, let nn := 1 (cf. Proposition 6.4.6 (a)). We have ny = nn.Thus ny ∈ N. For the rest of this Subsection we assume that either p ≥ 3 or p = 2 and(M,ϕ) has no integral slopes. Thus ny = nn = 1, cf. Proposition 6.4.6 (b). As ny = 1, to

ease notations, we will assume that n = 1M . Thus F 1 is the U-canonical lift of (M,ϕ, G).

128

Let z1 ∈ N (W (k)) be a lift of y such that its attached Shimura filtered F -crystal(M,F 1

1 , ϕ, G) has the property that (M,F 11 , ϕ) is a direct sum of filtered Diuedonne mod-

ules that have only one slope. From the uniqueness part of the statement 1.4.3 (d) weget that F 1

1 = F 1 and therefore (as ny = 1) that z = z1. From properties 1.2.1 (b) and(c) we get that Dz1 = Dz is a direct sum of p-divisible groups over W (k) that have onlyone slope. If k = k, then (M,F 1, ϕ) is cyclic (cf. end of Theorem 1.4.3) and thereforeDz1 = Dz is cyclic (cf. Fact 2.8.1 (b)). This ends the proof of Theorem 1.6.2.

6.5.2. Complements on Hodge cycles. For the rest of this Chapter we will considerthe case when k = F and (M,F 1, ϕ, G) is a U-canonical lift. We take µ : Gm → G to bethe canonical cocharacter of (M,ϕ, G); it defines the lift F 1 of (M,ϕ, G). We fix an O(v)-embedding eB(k) : B(k) → C and therefore we will speak about AC := AB(k)×B(k),eB(k)

C.

As A has level s symplectic similitude structure for all s ∈ N with (s, p) = 1, thefamily of all Hodge cycles on AC that involve no Tate twist is the pull back of the family(wα)α∈J 1 of all Hodge cycles on AB(k) that involve no Tate twist (here J 1 is a set ofindices that contains J ; thus for each element α ∈ J we will have wα := z∗E(GQ,X)(w

Aα )).

There exists a finite field k1 such that the triple (A, λA, (wα)α∈J ) is the pull back ofa triple (A, λA, (wα)α∈J ) over W (k1). This is a consequence of the facts that: (i) ny ∈ N,

and (ii) under right translations by elements of GQ(A(p)f ), the U-canonical lifts are mapped

to U-canonical lifts (cf. Fact 6.3.4). In simpler words, if H(p) is a compact, open subgroup

of GQ(A(p)f ) such that (A,Λ, (wA

α )α∈J ) is the pull back of an analogous triple over N /H(p),

then there exists a finite field k1 such that the W (k)-valued point of N /H(p) defined by zfactors through a W (k1)-valued point of N /H(p).

We can also assume that (wα)α∈J 1 is the pull back of a family (wα)α∈J 1 of Hodgecycles on AB(k1) (and therefore also that End(AC) = End(AB(k1))). We recall the standard

argument for this. Always there exists a finite field extension B(k1) of B(k1) such that(wα)α∈J 1 is the pull back of a family of Hodge cycles on AB(k1)

, cf. [De5, Prop. 2.9 and

Thm. 2.11]. But for every prime l ∈ N \ p, we can assume that B(k1) is included inthe composite field of the field extensions of B(k1) that are the fields of definition of thepoints in the set ∪s∈NAB(k1)[l

s](B(k1)). Thus we can assume that B(k1) is unramified

over B(k1). Thus by enlarging the finite field k1 we can assume that B(k1) = B(k1).

6.5.3. Proof of 1.6.3 (a). Let (M, F 1, ϕ, GW (k1)) be the Shimura filtered F -crystal over

k1 attached naturally to the pair (A, (wα)α∈J ); its extension to k is (M,F 1, ϕ, G) and thusit is a U-canonical lift (cf. Fact 3.2.1 (a)). For α ∈ J 1 let tα ∈ T (M [ 1p ]) be the de Rhamcomponent of wα. We consider the classical Hodge decomposition

(31a) M ⊗W (k1) C =M ⊗W (k) C = H1dR(AC/C) = H1(A(C),Q)⊗Q C = F 1,0 ⊕ F 0,1.

We have F 1,0 = F 1 ⊗W (k) C = F 1 ⊗W (k1) C. Let µAC: Gm → GLH1(A(C),Q)⊗QC be the

Hodge cocharacter that defines the decomposition (31a). To ease notations, we denote byGA the Mumford–Tate group of AC (i.e., the smallest subgroup of GLH1(A(C),Q) with the

property that µACfactors through GA,C) and we denote by GA the subgroup of GLM [ 1p ]

129

that fixes tα for all α ∈ J 1. The group GA is a form of GA,B(k1)(see [De5]) and thus it is

connected. As J ⊆ J 1, we have GA,B(k) 6 GB(k).

Let l ∈ N be such that k1 = Fpl . If p = 2 let ε := 4 and if p ≥ 3 let ε := 1. Let

π := (ϕ)l ∈ GLM (B(k1)).

It is well known that π is the crystalline realization of the Frobenius endomorphism of Ak1.

As in Corollary 6.1.2, for each element α ∈ J 1 we have ϕ(tα) = tα. Thus π ∈ GA(B(k1)).Let Cπ (resp. Cbig

π ) be the identity component of the centralizer of π in GB(k1) (resp.

in GLM [ 1p ]). The Lie algebra Lie(Cbig

π ) is fixed by π and thus it is B(k1)-generated by

Qp-linear combinations of endomorphisms of Ak1and therefore by elements fixed by ϕ (cf.

[Ta2]). Thus Lie(Cbigπ ) ⊆ Lie(L0

GLM(ϕ)B(k1)). As Lie(Cπ) ⊆ Lie(Cbig

π ), we conclude that

Lie(Cπ) ⊆ Lie(GB(k1)) ∩ Lie(L0GLM

(ϕ)B(k1)) = Lie(L0GW(k1)

(ϕ)B(k1)).

Thus Cπ is a subgroup of L0GW (k1)

(ϕ)B(k1) (cf. Lemma 2.3.6) and therefore it normal-

izes F 1[ 1p ]. Thus επ is the crystalline realization of an endomorphism of A, cf. Serre–

Tate and Grothendieck–Messing deformation theories. Therefore we have επ = tα forsome element α ∈ J 1. This implies that GA is a subgroup of Cπ and therefore also ofL0GW (k1)

(ϕ)B(k1). The canonical cocharacter µ : Gm → GW (k1) of (M, ϕ, GW (k1)) factors

through Z0(L0GW (k1)

(ϕ)) and µ = µW (k) factors through Z0(L0

G(ϕ)), cf. Subsection 4.1.4.

From the functorial aspects of [Wi, p. 513], we get that µB(k1) fixes each tα with α ∈ J 1

and thus that µB(k1) factors through GA. As GA 6 Cπ 6 L0GW (k1)

(ϕ)B(k1) and as µ factors

through Z0(L0GW (k1)

(ϕ)), we conclude that:

(a) the cocharacter µB(k1) : Gm → GB(k1) factors through the centers of both groups

GA and Cπ.

Thus, as µACfactors through GA,C, it commutes with µC = µC. The commuting

cocharacters µACand µC = µC of GLM⊗W (k1)C

= GLH1(A(C),Q)⊗QC act in the same way

on F 1,0 and M⊗W (k1)C/F0,1 and therefore they coincide. Thus µAC

: G → GA,C = GA,C

commutes with GA,C i.e., it factors through Z0(GA,C) = Z0(GA,C) = Z0(GA)C. Thereforeby the very definition of GA, we have GA = Z0(GA) i.e., GA is a torus. Thus the centralizerof GA in EndQ(H1(A(C),Q)) is a semisimple Q–algebra EAC

that has commutative,semisimple Q–subalgebras of rank 2r = 2dim(AC). But EAC

= End(AC) ⊗Z Q, cf. theclassical Riemann theorem of [De2, Thm. 4.7]. Therefore AC has complex multiplication.As End(AC) = End(AB(k1)) (cf. last paragraph of Subsection 6.5.2), all abelian schemes

AB(k), A, and A have complex multiplication. Thus Theorem 1.6.3 (a) holds.

6.5.4. Proof of 1.6.3 (b). We check the statement 1.6.3 (b). As cF is an endomorphismof (M,ϕ, G), it is also an endomorphism of (M,F 1, ϕ, G) (cf. Subsection 1.3.7). Thus, ifp = 2 and (M,ϕ) has no integral slopes (resp. if cF has a 0 image in EndW (k)(M/4M)),

130

then from the property 1.2.1 (c) (resp. from Grothendieck–Messing deformation theory)we get that cF is the crystalline realization of an endomorphism of Dz . Therefore eF liftsto an endomorphism of A, cf. Serre–Tate deformation theory. Thus Theorem 1.6.3 (b)holds.

6.5.5. Condition W. We use the notations of the first paragraph of Theorem 1.6.3 (c)in order to define the condition W. Let K := V [ 1

p]. Let A0 := z∗0(A). Let F 1

0 be the

Hodge filtration of H1dR(A0/V ) defined by A0. We will use the canonical and functorial

K-linear isomorphism (see [BO, Thm. 1.3]) ρ0 : H1dR(A0/V )[ 1p ]→M ⊗W (F) K. We recall

how ρ0 is constructed. Let eV ∈ N, the closed embedding cV : Spec(V ) → Spec(ReV ), andIeV ⊆ ReV be as in Subsection 2.8.7. Let v0 ∈ N (ReV ) be a lift of z0 ∈ N (V ), cf. Sub-section 2.8.7; thus z0 = v0 cV . Let (M0,ReV , F

10,ReV

,Φ0,ReV ,∇0,ReV , λM0,ReV, (t0,α)α∈J )

be the pull back of the pair (CR, (tRα )α∈J ) introduced in Section 6.2 via the morphism

Spec(ReV ) → Spec(R) ofW (k)-schemes defined by v0. Thus M0,ReV is a free ReV -module

of rank 2r, F 10,ReV

is a direct summand of it of rank r, etc. Let ∇ReVtriv be as in Subsection

2.8.4. It is known that there exist isomorphisms

J0 : (M0,ReV [1

p],Φ0,ReV ,∇0,ReV )→(M ⊗W (k) ReV [

1

p], ϕ⊗ ΦReV ,∇

ReVtriv )

of F -isocrystals over ReV /pReV (to be compared with [Fa2, beginning of §6] and [BO,Thm. 1.3]). No element of T (M)⊗W (k) IeV [

1p] is fixed by psϕ⊗ ΦReV , where s ∈ 0, 1.

Thus there exists a unique such isomorphism J0 whose pull back to an isomorphism ofF -isocrystals over k is defined by 1M [ 1p ]

; moreover J0 takes λM0,ReVto λM and t0,α to tα

for all α ∈ J . Let IV be the ideal of ReV that defines the closed embedding cV . Theisomorphism ρ0 of (31b) is obtained from J0 modulo IV [

1p ] via the canonical identification

M0,ReV /IVM0,ReV = H1dR(A0/V ); it is canonical as the triple (M0,ReV [

1p ],Φ0,ReV ,∇0,ReV )

depends only on the abelian scheme A0,V/pV .

We get that for every element α ∈ J the de Rham realization t0,α of the Hodge cyclez∗0,E(GQ,X)(w

Aα ) on A0,K , is mapped under ρ0 to tα. Thus ρ0 give birth to an isomorphism

(to be viewed in what follows as an identification)

(31b) ρ0 : (H1dR(A0/V )[

1

p], (t0,α)α∈J )→(M ⊗W (F) K, (tα)α∈J ).

In this paragraph we check that there exists an element m0 ∈ G(K) such that we haveF 10 [

1p ] = m0(F

1⊗W (k)K). We can redefine (M0,ReV , F10,ReV

,Φ0,ReV ,∇0,ReV , λM0,ReV, (t0,α)α∈J )

as the pull back of the pair (C0, (tα)α∈J ) introduced in Subsection 6.2.1 via the mor-phism Spec(ReV ) → Spec(R0) of W (k)-schemes defined by v0 and by the isomorphisme0 : Spec(R)→Spec(R0) of W (k)-schemes, cf. property 6.2.6 (b). Therefore we have anatural isomorphism

J ′0 : (M ⊗W (k) ReV , F

1 ⊗W (k) ReV , λM , (tα)α∈J )→(M0,ReV , F10,ReV , λM0,ReV

, (t0,α)α∈J ).

Thus we can take m0 to be the reduction modulo IV [1p ] of J0 (J

′0⊗1ReV [ 1p ]

) ∈ G(ReV [1p ]).

131

By enlarging the finite fields k1 and k0, we can assume that k0 = k1, that Ak1is as

in Subsection 6.5.3, and that the natural number t of Theorem 1.6.3 (c) is 1. Thus wecan use the notations of Subsection 6.5.3. As Ak1

is as in Subsection 6.5.3, the crystallinerealization of psΠ = psΠt ∈ End(Ak1

) is psπ. As psΠ lifts to an endomorphism of A0, thefunctorial aspect of ρ0 implies that the element

(31c) psπ ∈ G(K) 6 GLH1dR

(A0/V )(K) normalizes F 10 [1

p] = m0(F

1 ⊗W (k) K).

As End(Ak1)⊗ZQ is a semisimple Q–algebra whose center contains Π (see [Ta2, Thm.

1]) and which acts naturally on M [ 1p ], π is a semisimple element of GB(k1)(B(k1)). Thus

Cπ is a reductive group over B(k1). By enlarging the finite field k1 we can also assume thatπ ∈ Z0(Cπ)(B(k1)). Let Tπ be a maximal torus of Cπ ; we have π ∈ Tπ(B(k1)). Let K bea finite field extension of K such that Tπ,K is a split torus. Let P be the normalizer of F 1

in G; it is a parabolic subgroup scheme of G (cf. Subsection 2.5.3). As Cπ is centralized byµB(k1) (cf. property 6.5.3 (a)), Cπ,K normalizes N0,K . Thus Tπ,K (and therefore also π)

normalizes N0,K . From the classical Bruhat decomposition for G(K)/P (K), we get thatwe can write (cf. [Ja, Part II, 13.8])

m0 = w0n′0p0,

where (p0, n′0) ∈ P (K) × N0(K) and where w0 ∈ G(K) normalizes Tπ,K . We have (cf.

(31c))

(31d) πw0n′0(F

1 ⊗W (k) K) = (psπ)w0n′0p0(F

1 ⊗W (k) K) = w0n′0(F

1 ⊗W (k) K).

Thus n′−10 w−1

0 πw0n′0 normalizes F 1 ⊗W (k) K and therefore it belongs to P (K). Let π0 :=

w−10 πw0 ∈ Tπ(K). As n′−1

0 π0n′0π

−10 π0 ∈ P (K) and π0 ∈ P (K), we get that n′−1

0 π0n′0π

−10 ∈

P (K). But as Tπ,K normalizes N0,K , we have n′−10 π0n

′0π

−10 ∈ N0(K). Thus the element

n′−10 π0n

′0π

−10 ∈ (N0 ∩ P )(K) = 1M⊗W (k)K

is the identity. Therefore π0 commutes with

n′0. For u ∈ N there exists a unique element n′

0,u ∈ N0(K) such that n′0 = n′u

0,u. Dueto the uniqueness of n′

0,u and the fact that Tπ,K normalizes N0,K , we get that π0 and

u′0,u commute. Thus, as the centralizer of π0 in GK has a finite number of connectedcomponents, we conclude that all these elements n′

0,u and therefore also n′0 belong to

w−10 Cπ,K(K)w0. Here is the condition W for z0.

Condition W for z0. Up to a replacement of k0 by a finite field extension k1 of it(and therefore also up to a corresponding replacement of t), there exists a maximal torusTπ of the reductive group Cπ over B(k1) and there exist elements s1 = psπ, s2, . . . , sa ∈Tπ(B(k1)) (with a ∈ N) for which the following two axioms hold:

(i) each si with i ∈ 1, . . . , a is the crystalline realization of an endomorphism of A0;

(ii) the identity component of the centralizer Sπ of the semisimple elements s1, . . . , sain GB(k1), has the property that Z0(Cπ) is a subgroup of the center of w−1

0 Sπw0.

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6.5.6. Examples. (a) We check that if w0 is the identity element, then the conditionW for z0 holds. We take a = 1. The centralizer of psπ = s1 = w−1

0 s1w0 in GB(k1) has

Sπ = Cπ as its identity component and therefore Z(w−10 Sπw0) = Z(Cπ) contains Z

0(Cπ).Thus the condition W for z0 holds.

(b) If V = W (F), then we can assume that m0 ∈ Ker(N0(W (k)) → N0(k)) (cf. Fact6.4.1) and thus we can take w0 to be the identity element. Therefore the condition Wholds for z0, cf. (a).

6.5.7. Proof of 1.6.3 (c). We fix an embedding eK : K → C that extends the embeddingeB(F) = eB(k) : B(F) = B(k) → C of Subsection 6.5.2. Let A0,C := A0,K ×K,eK C. Wewill use the classical Hodge decomposition for A0,C

(31e) M ⊗W (F) C = H1dR(A0/V )⊗V C = H1(A0(C),Q)⊗Q C = F 1,0

0 ⊕ F 0,10 .

Let µA0,C: Gm → GLH1(A0(C),Q)⊗QC = GLM⊗W (F)C be the Hodge cocharacter that

defines the decomposition (31e); let GA0be the Mumford–Tate group of A0,C.

In Subsection 6.5.5 we have seen that n′0 centralizes π0 = w−1

0 πw0. A similar argumentshows that n′

0 centralizes each w−10 siw0 with i ∈ 2, . . . , a. Thus w−1

0 Sπw0 centralizesn′0. Therefore Z0(Cπ) centralizes n′

0, cf. axiom 6.5.5 (ii). As µK = µK factors throughthe center of Cπ,K (cf. property 6.5.3 (a)), we get that n′

0 is fixed by µK and thus it must

be the identity element of N0(K). From this, (31c), and (31d) we get that

(31f) w0(F1 ⊗W (k) K) = F 1

0 ⊗K K.

We have GA0,C 6 Sπ,C (to be compared with Subsection 6.5.3). Let eK : K → C be anembedding that extends eK . The cocharacter w0µCw

−10 = w0µCw

−10 factors through the

center of Sπ,C (cf. axiom 6.5.5 (ii)) and therefore commutes with µA0,C. The commuting

cocharacters w0µCw−10 = w0µCw

−10 and µA0,C

act in the same way on F 10 ⊗K C = F 1,0

0

(cf. (31f)) and therefore also on M ⊗W (F) C/F1,00 . Thus µA0,C

= w0µCw−10 = w0µCw

−10 .

Thus µA0,Cfactors through w0(Z

0(Cπ))Cw−10 and therefore (cf. axiom 6.5.5 (ii)) through

the center of Sπ,C. Thus µA0,Cfactors through Z0(GA0,C). As in the end of Subsection

6.5.3 we argue that this implies that GA0is a torus and that A0,C and A0 have complex

multiplication. From this and Example 6.5.6 (b), we get that Theorem 1.6.3 (c) holds.

6.5.8. Proof of 1.6.3 (d). We prove Theorem 1.6.3 (d); thus V = W (F). We can assumethat m0 = n′

0 ∈ Ker(N0(W (k)) → N0(k)), cf. Fact 6.4.1. Thus w0 is the identity element.From this and (31f) we get that F 1 = w0(F

1) = F 10 . Thus F 1 = F 1

0 . Therefore z0 is aU-canonical lift.

Next we assume that p = 2 and s = 0. Thus Πt is the crystalline realization ofan endomorphism of z∗0(A). By applying this with t being replaced by positive, integralmultiples of it, we get that the p-divisible group Dz0 of z∗0(A) is a direct sum of an etalep-divisible group D00 and of a p-divisible group D01 whose special fibre has only positiveNewton polygon slopes. But the filtered Dieudonne modules of D00 and D01 are cyclic(cf. end of Theorem 1.4.3 applied to the U-canonical lift (M,F 1, ϕ, G)) and therefore bothD00 and D01 are cyclic (cf. Fact 2.8.1 (b)). Thus Dz0 is cyclic.

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Let z1 ∈ N (W (k)) be another lift of y such that the p-divisible group Dz1 is cyclic.From the uniqueness parts of 1.4.3 (c) and Fact 2.8.1 (a), we get that Dz1 = Dz0 . Thisimplies that the composites of z0 and z1 with the morphism N → M, give birth to thesame W (k)-valued point of M. From this and the fact that the morphism i1 : Spec(R) →Spec(R1) of W (k)-schemes (see Section 6.2) is a closed embedding (see Proposition 6.2.7(c)), we get that z0 = z1 ∈ N (W (k)). This ends the proof of Theorem 1.6.3 (d) and thuscompletes the proof of Theorem 1.6.3.

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7 Examples and complements

We recall that p ∈ N is an arbitrary prime. In Chapters 7 to 9, the following notationsf : (GQ, X) → (GSp(W,ψ), S), Sh(GQ, X), E(GQ, X), v, k(v), O(v), L, L(p) = L⊗ZZ(p),GZ(p)

, Kp = GSp(L, ψ)(Zp), Hp = Kp ∩ GQ(Qp) = GZ(p)(Zp), r ∈ N, M, N , (A,ΛA),

(vα)α∈J , (wAα )α∈J , Lp = L ⊗Z Zp, ψ

∗, T0, B0, µ0 : Gm → T0,W (k(v)), M0 = L∗p ⊗Zp

W (k(v)) = F 10 ⊕F 0

0 , and C0 = (M0, ϕ0, GW (k(v))) will be as in Subsections 1.5.1 and 1.5.2.Until the end we will assume that (f, L, v) is a standard Hodge situation. Thus GZ(p)

is areductive group scheme over Z(p) and moreover the conditions 1.5.3 (a) and (b) hold. Let

U+0 := U+

GW (k(v))(ϕ0), I0 ∈ N ∪ 0, O ⊆ Nk(v), and N0 be as in the beginning of Section

1.6. Letq := [k(v) : Fp].

Let k be a perfect field that contains k(v). Let y ∈ N (k). Let z ∈ N (W (k)) be a lift of y.Let A, Dz, (M,F 1, ϕ, G), and (tα)α∈J be as in Subsection 1.5.3. Let λM :M ⊗W (k)M →W (k) be the perfect alternating form that is the de Rham realization of λA := z∗(ΛA).Let E(Gad

Q , Xad) be the reflex field of the adjoint Shimura pair (Gad

Q , Xad) of (GQ, X).

In this Chapter we include examples and complements to Theorem 1.6.1 to 1.6.3. InSection 7.1 we include examples that pertain to the Newton polygon N0. In Section 7.2we include a modulo p interpretation of Theorem 1.6.1 similar to the one of the statement1.4.4 (b). In Section 7.3 we present two examples on the p-ranks of pull backs of A viaSh-ordinary points of N (k). In Section 7.4 we show that for p ≥ 3 two distinct points ofO can not map to the same point of Mk(v). In Section 7.5 we take Sh(GQ, X) to be aHilbert–Blumenthal variety of dimension 3 and we give examples of U-canonical lifts of Nthat are not Sh-canonical lifts.

7.1. On the Newton polygon N0

For l ∈ N we have ϕl0 = (

∏li=1 σ

ik(v)(µ0)(

1p ))σ

lk(v), where σ

ik(v)(µ0) : Gm → GW (k(v))

is the σik(v)-conjugate of µ0 : Gm → GW (k(v)) (see (21)). But σq

k(v)(µ0) = µ0. Therefore

q times the slopes of N0 taken with multiplicities are the p-adic valuations taken withmultiplicities of the eigenvalues of the element

q∏

i=1

σik(v)(µ0)(

1

p) ∈ T0(B(k(v))) 6 GLM0

(B(k(v))).

Thus all Newton polygon slopes of N0 have as denominators divisors of q i.e., are elementsof the set 0, 1q ,

2q , . . . ,

q−1q , 1. Moreover, the Newton polygon slopes of N0 are only 0 and

135

1 if and only if for all i ∈ S(0, q − 1) we have σik(v)(µ0) = µ0 i.e., if and only if µ0 is fixed

by σk(v). This is equivalent to µ0 being defined over Zp and therefore to the equality q = 1(cf. [Mi2, top of p. 504 and Cor. 4.7]). In most cases this was obtained first in [Va1, Rm.5.4.3] (cf. also [Va3, Subsect. 1.6] and [We1]).

7.1.1. Proposition. We assume that q = 1. If p ≥ 3 (resp. if p = 2) and z is anSh-canonical lift, then the abelian variety Ak is ordinary and A is (resp. is isogenous to)the canonical lift of Ak.

Proof: As q = 1, all Newton polygon slopes of Ak are integers. Thus Ak is ordinaryand we have M = W (0)(M,ϕ) ⊕W (1)(M,ϕ). We also have F 1 = F 1 ∩W (0)(M,ϕ) ⊕F 1 ∩W (1)(M,ϕ), cf. statement 1.4.3 (d). But as ϕ(F 1) ⊆ pM and ϕ(W (0)(M,ϕ)) =W (0)(M,ϕ), we have F 1 ∩W (0)(M,ϕ) = 0. Thus F 1 is a direct summand of W (1)(M,ϕ)and thus by reasons of ranks we have F 1 =W (1)(M,ϕ). Therefore we have ϕ(F 1) = pF 1

and thus F 1 is the Hodge filtration of the canonical lift of Ak. Therefore for p ≥ 3 (resp.for p = 2) the Proposition follows from the fact that F 1 determines A uniquely (resp. fromthe first paragraph of the proof of Proposition 6.4.6).

7.1.2. Lemma. If N0 is a supersingular Newton polygon (i.e., if all its Newton polygonslopes are 1

2 ), then GQ is a torus.

Proof: The group GQ is a torus if and only if dimC(X) = 0, cf. [De3, axiom (2.1.1.3)].We have dimC(X) = 0 if and only if the cocharacter µx : Gm → GC factors throughZ0(GC) (here x ∈ X and µx is as in Subsection 1.5.1). The cocharacter µx : Gm → GC

factors through Z0(GC) if and only if the cocharacter µ0 : Gm → GW (k(v)) factors through

Z0(GW (k(v))) and therefore if and only if we have P+GW (k(v))

(ϕ0) = GW (k(v)). Thus GQ is

a torus if and only if P+GW (k(v))

(ϕ0) = GW (k(v)). If all Newton polygon slopes of N0 are 12 ,

then we have P+GLM0

(ϕ0) = GLM0and therefore we also have P+

GW (k(v))(ϕ0) = GW (k(v));

thus GQ is a torus.

7.1.3. Example. We assume that q = 2. Thus the Newton polygon slopes of N0 belongto the set 0, 12 , 1. Therefore N0 is uniquely determined by the Hasse–Witt invariantHW0

of (M0, ϕ0). From Subsection 4.1.1 (with q = 2) we get that if k = k and if z ∈ N (W (k))is an Sh-canonical lift, then (M,F 1, ϕ) is a direct sum of circular indecomposable ShimuraF -crystals whose ranks have their least common multiple equal to q = 2. This implies that(M,ϕ) is the Dieudonne module of a product of r elliptic curves over k, at least one being(cf. Proposition 7.1.1) supersingular. From Lemma 7.1.2 we get: if GQ is not a torus, thennot all Newton polygon slopes of N0 are 1

2 and thus we have HW0 ≥ 1.

7.2. Modulo p interpretation

Let n ∈ N. Let ϑ : M → M be the Verschiebung map of ϕ. By the Dieudonnetruncation modulo pn of (M,ϕ, G, λM ) we mean the quintuple

Tn(y) := (M/pnM,ϕ, ϑ, GWn(k), λM ),

where for simplicity we denote also by ϕ and ϑ their reductions modulo pn.

136

Let (M1, ϕ1, G1, λM1, (t1,α)α∈J ) be the analogue of (M,ϕ, G, λM , (tα)α∈J ) but ob-

tained working with another point y1 ∈ N (k) instead of with y ∈ N (k). Let Tn(y1) :=(M1/p

nM1, ϕ1, ϑ1, G1,Wn(k)) be the Dieudonne truncation modulo pn of (M1, ϕ1, G1, λM1).

By an inner isomorphism between Tn(y) and Tn(y1) we mean aWn(k)-linear isomor-phism h :M/pnM→M1/p

nM1 such that the following two conditions hold:

(a) it lifts to an isomorphism j0 : (M,λM , (tα)α∈J )→(M1, λM1, (t1,α)α∈J );

(b) as Z/pnZ-linear endomorphisms of M/pnM we have hϕ = ϕ1h and hϑ = ϑ1h.

7.2.1. Lemma. We assume that there exists an isomorphism

j0 : (M,λM , (tα)α∈J )→(M1, λM1, (t1,α)α∈J ).

(a) Then Tn(y) and Tn(y1) are inner isomorphic if and only if there exists an iso-morphism j1 : (M,λM , (tα)α∈J )→(M1, λM1

, (t1,α)α∈J ) such that we have j−11 ϕ1j1 = gϕ,

where the element g ∈ G(W (k)) is congruent to 1M modulo pn.

(b) If T1(y) and T1(y1) are inner isomorphic, then the FSHW maps of (M,ϕ, G)and (M1, ϕ1, G1) are naturally identified via the isomorphism Lie(Gad)→Lie(Gad

1 ) definedby j1, where j1 is as in (a) but for n = 1.

Proof: We prove (a). The “if” part is trivial. We check the “only if” part. Thus we canassume that j0 modulo pn defines an inner isomorphism h between Tn(y) and Tn(y1). Letg := j−1

0 ϕ1j0ϕ−1 ∈ G(B(k)). As h is an isomorphism, j0µj

−10 : Gm → G1 is a Hodge

cocharacter of (M1, ϕ1, G1). This implies that (gϕ)−1(M) = ϕ−1(g(M)) is ϕ−1(M) =M + 1

pF 1. Thus g normalizes M and therefore we have g ∈ G(W (k)).

Let M = F 1 ⊕ F 0 and N0 be as in Subsection 6.2.1. Let σϕ := ϕµ(p). Let g0 :=

(σϕ)−1gσϕ; it is a σ-linear isomorphism of M . We have g0 ∈ G(W (k)), cf. Lemma 2.4.6.

As h is an inner isomorphism between Tn(y) and Tn(y1), the σ-linear (resp. σ−1-linear)

endomorphisms ϕ = σϕµ(1p ) and gϕ = gσϕµ(

1p ) = σϕg0µ(

1p ) (resp. ϑ = p1Mµ(p)σ

−1ϕ and

ϑg−1 = p1Mµ(p)σ−1ϕ g−1 = p1Mµ(p)g

−10 σ−1

ϕ ) coincide modulo pn. Let g0,n ∈ G(Wn(k)) bethe reduction modulo pn of g0. We have:

(i) the element g0,n fixes F 0/pnF 0 and it is congruent to 1M/pnM modulo pn−1;

(ii) there exists an inclusion (1M/pnM − g−10,n)(F

1/pnF 1) ⊆ pn−1F 0/pnF 0.

Property (i) follows from the congruence modulo pn of the mentioned σ-linear en-domorphisms of M . Property (ii) follows from (i) and the congruence of p1Mµ(p) andp1Mµ(p)g

−10 modulo pn. From (i) and (ii) we get that g0,n ∈ Ker(N0(Wn(k)) →

N0(Wn−1(k))). Thus up to Ker(GW (k)(W (k)) → GW (k)(Wn(k)))-multiples of ϕ1 andg, we can assume that g0 ∈ N0(W (k)). Thus we can write g0 = 1M + pn−1u, whereu ∈ Lie(N0). Let g := 1M + pnu ∈ N0(W (k)). We compute

g = σϕg0σ−1ϕ = ϕµ(p)(1M + pn−1u)µ(

1

p)ϕ−1 = ϕ(1M + pnu)ϕ−1 = ϕgϕ−1.

Thus gϕ = ggϕg−1. As g and 1M are congruent modulo pn, we can take j1 := j0g−1 (with

g viewed as a W (k)-linear automorphism of M). Therefore j1 exists. Thus (a) holds.

137

To check (b), we can assume that n = 1 and that (cf. (a)) g is congruent to 1M modulop. Thus the FSHW maps of (M,ϕ, G) and (M, gϕ, G) coincide. Therefore the FSHWmaps of (M,ϕ, G) and (M1, ϕ1, G1) can be identified naturally via the Lie isomorphismLie(Gad)→Lie(Gad

1 ) defined by j1.

7.2.2. Corollary. We assume that k = k and that (M,ϕ, G) is Sh-ordinary. Let N 0k(v)

be the connected component of Nk(v) through which y factors. Then a point y1 ∈ N 0k(v)(k)

is Sh-ordinary if and only if T1(y1) is inner isomorphic to T1(y).

Proof: There exist isomorphisms (M,λM , (tα)α∈J )→(M1, λM1, (t1,α)α∈J ), cf. prop-

erty 6.2.8 (ii). Thus, as 1.6.1 (a) ⇔ 1.6.1 (c), the “if” part follows from Lemma7.2.1 (b). We now check the “only if” part. We will prove even more that in factthere exists an isomorphism j1 : (M,ϕ, G, λM , (tα)α∈J )→(M1, ϕ1, G1, λM1

, (t1,α)α∈J ).From the last part of Theorem 1.6.1 we get that there exists an isomorphism j0 :(M,ϕ, (tα)α∈J )→(M1, ϕ1, (t1,α)α∈J ). The inverse image of λM1

via j0, is a perfect al-ternating form λ′M on M .

As there exist isomorphisms between (M0 ⊗W (k(v)) B(k), ψ∗, (vα)α∈J ) and both

(M [ 1p], λM , (tα)α∈J ) and (M1[

1p], λM1

, (t1,α)α∈J ) (see Subsection 6.1.5 applied to y and

y1), we get that λ′M is aGm(B(k))-multiple of λM . As λM and λ′M are both principal quasi-

polarizations of (M,ϕ), we get that in fact λ′M is aGm(Zp)-multiple δ0 of λM . Let C0,Zpbe

the Zp form of L0G(ϕ) such that C0,Zp

(Zp) is the group of inner automorphisms of (M,ϕ, G),cf. Subsections 4.1.4 and 4.1.7. The group C0,Zp

acts on the Zp-span of λM via a surjectivecharacter χ0 : C0,Zp

։ Gm. As in Subsection 6.1.4 we argue that Ker(χ0) is a reductivesubgroup scheme of Gm. This implies that the homomorphism χ0(Zp) : C0,Zp

(Zp) →Gm(Zp) is surjective. Let c0 ∈ C0,Zp

(Zp) be such that χ0(Zp)(c0) = δ0. Let j1 := j0c0; it

defines an isomorphism (M,ϕ, G, λM , (tα)α∈J )→(M1, ϕ1, G1, λM1, (t1,α)α∈J ).

7.3. Examples on p-ranks

7.3.1. Example. We assume that there exists a simple factor G0 of GadZp

such that theimage of µ0 : Gm → GW (k(v)) in each simple factor of G0,W (k(v)) is non-trivial. Then(Lie(G0,B(k(v))), ϕ0) has Newton polygon slope −1 with positive multiplicity, cf. Fact3.4.11. Thus N0 has the Newton polygon slopes 0 and 1 with positive multiplicities.Therefore if k = k and y ∈ O(k), then the p-rank of y∗(A) is positive (cf. statement1.6.1 (b)).

7.3.2. Example. For the rest of this Section, we will assume that the GC-moduleW⊗QCis simple. From this assumption and the existence of Hodge cocharacters µx : Gm → GC

that define Hodge Q–structures on W of type (−1, 0), (0,−1) (see Subsection 1.5.1), weget that Gad

R has only one simple, non-compact factor. Let n be the number of simplefactors of Gad

R . It is known that GadQ is a simple Q–group, cf. [Pi, Prop. 5.12]. This,

the existence (of the inverses) of the cocharacters µx, and the fact that the GQ-moduleW is simple, are expressed in [Pi, Def. 4.1] by: the pair (GQ, GQ → GLW ) is a strongMumford–Tate pair over Q of weights 0, 1. Thus n is odd and 2r is an n-th power of anatural number, cf. [Pi, Props. 4.6 and 4.7]. Let G0 be the unique simple factor of Gad

Zp

such that the cocharacter µ00 : Gm → G0,W (k(v)) defined naturally by µ0 is non-trivial.

138

Let q0 ∈ N be such that G0,F has q0 simple factors. Thus G0,Fpis the ResFpq0 /Fp

of an

absolutely simple, adjoint group G00,Fpq0over Fpq0 (cf. [Ti1, Subsubsect. 3.1.2]) and G0

is the ResW (Fpq0 )/Zpof an absolutely simple, adjoint group scheme G00 over W (Fpq0 ) (cf.

[DG, Vol. III, Exp. XXIV, Prop. 1.21]). As GabQ = Gm, we have E(Gab

Q , Xab) = Q. Thus

E(GadQ , X

ad) = E(GQ, X), cf. [De2, Subsect. 3.7 and Prop. 3.8]. Therefore from [Mi2,Prop. 4.6 and Cor. 4.7] we get that B(k(v)) is the field of definition of the G0(B(k(v)))-conjugacy class of µ00. Thus q0 divides q. But G0 splits over W (Fp2q0 ), cf. Subsection3.4.4 and the fact that the existence of µ0 implies (see [Se1, Cor. 2 of p. 182]) that G00

can not be of 3D4 Lie type. Therefore we have q ∈ q0, 2q0. If q = 2q0, then there existsa positive integer m such that Gad

C is a PGLn2m+2 group (cf. [Pi, Table 4.2]).

7.3.3. Proposition. We assume that the GC-module W ⊗Q C is simple. Then theabelian varieties obtained by pulling back AO through geometric points of O, have non-zerop-ranks: if q = q0 (resp. if q = 2q0), then these p-ranks are r

2q−1 (resp. are r2q−1 (

m+12m+1)

q).

Therefore these p-ranks are 1 if and only if GadQp

is a simple Qp-group and GadC is a PGLn

2

group.

Proof: The GderB(F)-module M0 ⊗W (k(v)) B(F) is the tensor product of n simple Gder

B(F)-

modules V1, . . . , Vn such that: (i) dimB(F)(Vi) does not depend on i ∈ S(1, n), and (ii)

each representation GderB(F) → GLVi

is symplectic and at the level of Lie algebras is defined

by an irreducible representation of a simple factor fi of Lie(GderB(F)). Thus we can identify

M0⊗W (k(v))B(F) with V1⊗B(F)V2⊗B(F) · · ·⊗B(F)Vn. As Gm acts through µ0 onM0 non-trivially and via the trivial and the inverse of the identical character of Gm, there exists aunique number i0 ∈ S(1, n) such that the cocharacter µ0,B(F) acts on M0⊗W (k(v))B(F) =V1 ⊗B(F) V2 ⊗B(F) · · · ⊗B(F) Vn via the single factor Vi0 ; thus we can identify µ0,B(F) witha cocharacter of GLVi0

. Two distinct elements θ1, θ2 ∈ Gal(B(k(v))/Qp) can take underconjugation µ0,B(k(v)) to cocharacters of GB(k(v)) whose extensions to B(F) factor throughthe same factor Vi of M0 ⊗W (k(v)) B(F) = V1 ⊗B(F) V2 ⊗B(F) · · · ⊗B(F) Vn if and only if

q = 2q0 and θ1θ−12 is the non-identity element θ0 of Gal(B(k(v))/B(Fpq0 )) = Z/2Z; this

is so as G00 is an absolutely simple, adjoint group scheme over W (Fpq0 ) that splits overW (Fp2q0 ). Let I0 := i ∈ S(1, n)|fi ⊆ Lie(G0,B(F)); thus i0 ∈ I0.

We first consider the case q0 = q. For i ∈ S(1, n) let Vi = V 0i ⊕ V 1

i be the directsum decomposition normalized by all elements (σs

k(v)(µ0))B(F)(1p )’s, where V

0i (resp. V 1

i )

is the maximal B(F)-vector subspace of Vi on which each element (σsk(v)(µ0))B(F)(

1p) acts

trivially (resp. on which at least one of these elements (σsk(v)(µ0))B(F)(

1p )’s does not act

trivially). If i /∈ I0, then V1i = 0. If i ∈ I0, then dimB(F)(V

0i ) = dimB(F)(V

1i ). We have

W (0)(M0 ⊗W (k(v)) B(F), ϕ0 ⊗ σF) = V 01 ⊗B(F) V

02 ⊗B(F) · · · ⊗B(F) V

0n

and thus dimB(k(v))(W (0)(M0, ϕ0)) =12q dimB(F)(V1)

n =dimB(k(v))(M0[

1p ])

2q = r2q−1 . If this

dimension is 1, then 2r = 2q and therefore we must have q = n and dimB(F)(Vi) = 2 for

all numbers i ∈ S(1, n). Thus GadQp

is a simple Qp-group and GadC is a PGLn

2 group.

We now consider the case q = 2q0. The representations GderB(F) → GLVi

are associated

to the minuscule weight m+1 of the A2m+1 Lie type (of the simply connected semisimple

139

group cover of the simple factor of GadB(F) whose Lie algebra is fi). Thus the representations

GderB(F) → GLVi

are self-dual and have dimension(2m+2m+1

)(for instance, see [Pi, Table 4.2];

see [Bou2, plate I] for weights). We get a direct sum decomposition Vi = V 0i ⊕ V

12i ⊕ V 1

i ,

with V 0i as above and, in case i ∈ I0, with V

12

i and V 1i normalized by all above mentioned

elements and such that V 0i ⊕ V

12

i is the perpendicular of V 0i in Vi with respect to any non-

zero alternating form on Vi that is centralized by fi→Im(fi → EndB(F)(Vi)). But µ0,B(F)

(resp. (θ0(µ0,B(k(v))))B(F)) is associated to the first (resp. to the last) node of the Dynkindiagram of the A2m+1 Lie type, cf. [Pi, Table 4.2]. Thus we have a natural identification

Vi0 =∧m+1

(B(F)2m+2) such that for the standard basis e1, . . . , e2m+2 for B(F)2m+2

we have:

(a) the cocharacter µ0,B(F) acts trivially on all vectors of Vi0 that are of the formei1 ∧ ei2 ∧ · · · ∧ eim+1

with 1 < i1 < i2 < · · · < im+1 and via the inverse of the identicalcharacter of Gm on the B(k)-span of each vector of the form e1 ∧ ei2 ∧ · · · ∧ eim+1

with1 < i2 < · · · < im+1;

(b) the cocharacter (θ0(µ0,B(k(v))))B(F) acts trivially on all vectors of Vi0 that areof the form ei1 ∧ ei2 ∧ · · · ∧ eim ∧ e2m+2 with i1 < i2 < · · · < im < 2m + 2 and viathe inverse of the identical character of Gm on the B(k)-span of each vector of the formei1 ∧ ei2 ∧ · · · ∧ eim+1

with i1 < i2 < · · · < im+1 < 2m+ 2.

Thus ei1 ∧ ei2 ∧ · · · ∧ eim ∧ e2m+2|1 < i1 < · · · < im < 2m + 2 is a B(F)-basis forV 0i0. Therefore for each element i ∈ I0 we have dimB(F)(V

0i ) = dimB(F)(V

0i0) =

(2mm

). For

i /∈ I0, we have V12i = V 1

i = 0 and dimB(F)(V0i ) =

(2m+2m+1

). Thus

dimB(F)(W (0)(M0 ⊗W (k(v)) B(F), ϕ0 ⊗ σF)) =

n∏

i=1

dimB(F)(V0i ) =

(2m

m

)q0(2m+ 2

m+ 1

)n−q0

is at least 2n > 1. But 2r =(2m+2m+1

)nand therefore

(2mm

)q0(2m+2m+1

)n−q0= r

2q−1 (m+12m+1)

q.

Thus regardless of what q ∈ q0, 2q0 is, the Proposition follows from the statement1.6.1 (b).

7.3.4. Remark. If q = q0, then as we have dimB(F)(V1i ) = dimB(F)(V

0i ) for all i ∈ I0, the

multiplicity of the Newton polygon slope lq for N0 is

r(ql)2q−1 ; here l ∈ S(0, q). If q = 2q0, then

we can express the multiplicity of the same Newton polygon slope as a sum of products ofcombinatorial numbers.

7.4. On the Sh-ordinary locus O

Let NO be the open subscheme of N that contains NE(GQ,X) and whose special fibreis O. In this Section we prove the following result.

7.4.1. Theorem. If p ≥ 3, then NO is a locally closed subscheme of MO(v).

Proof: The morphism N → MO(v)is a formal immersion at all geometric points of Nk(v)

(cf. proof of Proposition 6.2.7 (a)) and defines a closed embedding morphism NE(GQ,X) →

140

ME(GQ,X). Thus we only have to show that NO → MO(v)is injective on F-valued points.

Let z1, z2 ∈ NO(W (F)) be two Sh-canonical lifts whose special fibres y1, y2 ∈ O(F) mapto the same point y ∈ MO(v)

(F). To end the proof it is enough to show that z1 and z2coincide.

For i ∈ 1, 2 let (Ai, λAi) := z∗i (A,ΛA). We identify (A1, λA1

)F with (A2, λA2)F. We

check that A1 and A2 are the same lift of A1,F = A2,F. Let l ∈ N be such that (Ai, λAi) is

the pull back to W (F) of a principally polarized abelian variety (A0,i, λA0,i) over W (Fpl),

we have (A0,1, λA0,1)F

pl= (A0,2, λA0,2

)Fpl, and moreover all Hodge cycles on Ai,B(F) are

pull backs of Hodge cycles on A0,i,B(Fpl

) (cf. Subsection 6.5.2). Let (M0, ϕ0) be the

Dieudonne module of A0,1,Fpl

= A0,2,Fpl. Let F 1

i be the Hodge filtration of M0 defined by

A0,i. The generic fibre of the inverse µi of the canonical split cocharacter of (M0, F1i , ϕ0)

factors through the center of the identity component of the centralizer of (ϕ0)l in GLM0[

1p ],

cf. property 6.5.3 (a). Therefore the generic fibres of µ1 and µ2 commute. Thus µ1 andµ2 commute. As µ1 and µ2 are commuting Hodge cocharacters of (M0, ϕ0,GLM0

), they

coincide (cf. Lemma 3.1.1). Thus we have F 11 = F 1

2 . As p ≥ 3, from Grothendieck–Messingand Serre–Tate deformation theories we get that A0,1 = A0,2. Thus A1 = A2 and thereforewe also have λA1

= λA2. As NE(GQ,X) is a closed subscheme of ME(GQ,X), the generic

fibres of z1 and z2 coincide. This implies that z1 = z2.

7.5. Example for U-ordinariness

We consider the case when f : (GQ, X) → (GSp(W,ψ), S) is a standard embeddingof a Hilbert–Blumenthal variety of dimension 3. Thus the group Gad

Q is simple over Q,

the group GderR is isomorphic to SL3

2, and we have r = 3. We also assume that our primep is such that Gder

Zpis the ResW (Fp3)/Zp

of an SL2 group scheme. Thus GderW (Fp3 )

is a

product G1 ×W (Fp3 )G2 ×W (Fp3 )

G3 of three copies of SL2 group schemes. Moreover GZp

is generated by Z(GLM0) and Gder

Zp. We have E(GQ, X) = Q and thus k(v) = Fp and

q = 1. As k(v) = Fp, we have M0 = L∗p; thus for the sake of uniformity of notations, in

this section we will use L∗p instead of M0.

Let T0 and B0 be as in Subsection 1.5.2. For i ∈ S(1, 3) let ωi ∈ Gi(W (Fp3)) bean element that normalizes T0,W (Fp3 )

∩ Gi and such that it takes B0,W (Fp3)∩ Gi to its

opposite with respect to T0,W (Fp3 )∩Gi. Let

ω ∈ (ω1, ω2, 1L∗p⊗ZpW (Fp3 )

), (ω1, 1L∗p⊗ZpW (Fp3 )

, ω3), (1L∗p⊗ZpW (Fp3 )

, ω2, ω3) ⊆ GZp(W (Fp3)).

We have L0GW (F

p3)(ω(ϕ0 ⊗ σFp3

)) = T0,W (Fp3)(to be compared with Subsection 4.3.5).

Thus, as T0 normalizes F 10 , the quadruple

Cω := (L∗p ⊗Zp

W (Fp3), F 10 ⊗Zp

W (Fp3), ω(ϕ0 ⊗ σFp3), GW (Fp3)

)

is a U-canonical lift. It is not an Sh-canonical lift as P+GW (F

p3)(ω(ϕ0⊗σFp3

)) is not contained

in the normalizer B0,W (Fp3)of F 1

0 ⊗ZpW (Fp3) in GW (Fp3 )

.

141

7.5.1. Proposition. There exist W (F)-valued points z ∈ N (W (F)) whose attachedShimura filtered F -crystals are isomorphic to Cω ⊗ F. Thus each such z is a U-canonicallift that is not an Sh-canonical lift.

Proof: Let Lωp := x ∈ L∗

p ⊗ZpW (F)|ω(1L∗

p⊗ σF)(x) = x. It is a Zp structure of

L∗p ⊗Zp

W (F) constructed as in Section 2.4; we have vα ∈ T (Lωp [

1p ]) for all α ∈ J . We

denote also by ψ∗ the perfect form on Lωp defined by ψ∗. We check that there exist

isomorphisms h : (Lωp , ψ

∗, (vα)α∈J )→(L∗p, ψ

∗, (vα)α∈J ). Such isomorphisms exist overW (k). Thus the existence of h is parameterized by a torsor of G0

Zpwhich (to be compared

with the second paragraph of Section 4.2) is trivial. Therefore such isomorphisms h exist.

As Lie(T0,W (F)) is normalized by ω(1L∗p⊗ σF), there exists a subtorus Tω

0 of GLLωp

whose extension toW (F) is T0,W (F) (cf. Lemma 2.4.1). Let Tω be a maximal torus of GZ(p)

such that theR-rank of TωR is 1 and the isomorphism h : (Lω

p , ψ∗, (vα)α∈J )→(L∗

p, ψ∗, (vα)α∈J )

takes Tω0 to Tω

Zp, cf. [Ha, Lem. 5.5.3]. We fix an embedding eB(F) :W (F) → C. Let

µω : Gm → TωC

be the cocharacter that corresponds to the cocharacter µ0,W (F) of Tω0,W (F) via h and

eB(F); we have µω := hCµ0,Ch−1C . The composite Gm → GC of µω with the natural

monomorphism TωC → GC is GC(C)-conjugate to the cocharacters µx : Gm → GC of

Subsection 1.5.1. Thus the subtorus C(ω) of TωC generated by Z(GLW )C and Im(µω) is

isomorphic to (ResC/RGm)C. From this and our assumption on the R-rank of TωR, we get

that C(ω) is the extension to C of a subtorus of TωR that is isomorphic to ResC/RGm. In

other words, µω is the Hodge cocharacter of a homomorphism hω : ResC/RGm → TωR.

Let Xω be the GQ(R)-conjugacy class of the composite of hω with the monomorphismTωR → GR. As Gad

R is a PGL32 group and due to the mentioned GQ(C)-conjugacy property,

the pair (GQ, Xω) is a Shimura pair. Even more, we can identify naturally X and Xω

with open closed subspaces of Xad. But the connected components of Xad are permutedtransitively by Gad

Z(p)(Z(p)), cf. [Va3, Cor. 3.3.3]. Thus by replacing Tω with a Gad

Z(p)(Z(p))-

conjugate of it, we can assume that X = Xω. We get an injective map of Shimura pairs

fω : (TωQ, h

ω) → (GSp(W,ψ), S)

that factors through f : (GQ, X) → (GSp(W,ψ), S). Let vω be the prime of the reflexfield E(Tω

Q, hω) of (Tω

Q, hω) such that the localization O(vω) of the ring of integers of

E(TωQ, h

ω) with respect to it is under eB(F) a subring of W (F). The smallest numbers ∈ N such that (ω(ϕ0 ⊗ σF))

s fixes under conjugation µ0,W (F) : Gm → GLL∗p⊗ZpW (F) =

GLLωp⊗ZpW (F) is s = 3. Thus we have k(vω) = Fp3 .

The torus Tω is a maximal torus of GSp(L(p), ψ). Thus Tω is the intersection ofGSp(L(p), ψ) with the double centralizer of Tω in GLL(p)

. Therefore the triple (fω, L, vω)is a standard moduli PEL situation. Let Hω

p be the unique hyperspecial subgroup ofTωQ(Qp). Let N ω be the normalization of the Zariski closure of Sh(Tω

Q, hω)/Hω

p inMO(vω)

. As (fω, L, vω) is a standard Hodge situation (cf. Remark 1.5.5 (b)), there existW (F)-valued points

z ∈ Im(N ω(W (F)) → N (W (F))).

142

Let J ω := J ∪ Lie(Tω). We can assume that for each element α ∈ J ∩ Lie(Tω) we havevα = α. Accordingly for α ∈ J ω \ J we define vα := α. Let T be the maximal torusof G such that (M,F 1, ϕ, T ) is the Shimura filtered F -crystal of z ∈ N ω(W (F)). Forα ∈ J ω let tα ∈ T (M [ 1

p]) correspond to vα via Fontaine comparison theory for A. The

extension to F of the standard Shimura F -crystal of (fω, L, vω), up to conjugation withh−1 ∈ GZp

(W (F)), is identified with (L∗p ⊗Zp

W (F)), ω(ϕ0 ⊗ σF), T0,W (F)). As T is amaximal torus of GSp(M,λM), it is well known that there exist isomorphisms (this is alsoa particular case of Theorem 8.1.5 below)

ρ : (M, (tα)α∈Jω , λM)→(L∗p ⊗Zp

W (F), (vα)α∈Jω , ψ∗).

Thus, if we take the isomorphism ρ of Subsection 6.1.5 to be defined by ρ, then we haveρϕρ−1 = gyω(ϕ0⊗σF), where gy ∈ T0(W (F)). But there exists hy ∈ T0(W (F)) that definesan inner isomorphism between (L∗

p ⊗W (k(v))W (F), ω(ϕ0 ⊗ σF), T0,W (F)) and (L∗p ⊗W (k(v))

W (F), gyω(ϕ0 ⊗ σF), T0,W (F)), cf. the equivalence 1.4.1 (a) ⇔ 1.4.4 (a). Therefore eachW (F)-valued point z ∈ Im(N ω(W (F)) → N (W (F))) has the property that its attachedShimura filtered F -crystal is isomorphic to (L∗

p ⊗W (k(v))W (F), gyω(ϕ0 ⊗ σF), GW (F)) andtherefore also to Cω ⊗ F.

7.5.2. Remark. Proposition 7.5.1 and its proof apply also to the case when the role ofSL3

2 is replaced by the one of SLm2 with m ≥ 4 (to be compared with Subsection 4.3.5).

143

8 Two special applications

In this Chapter we include two special applications of Section 1.6. In Section 8.1 weinclude complements on a motivic conjecture of Milne. In Section 8.2 we apply the Zariskidensity part of Theorem 1.6.1 to prove the existence of integral canonical models of certainShimura varieties in unramified mixed characteristic (0, p) with p ≥ 3. All notations willbe as in the first paragraph of Chapter 7. We recall that a reductive group over Qp iscalled unramified if it is quasi-split and splits over an unramified finite field extension ofQp.

8.1. Applications to a conjecture of Milne

In this Section we will assume that k = k. Let H1 := H1et(AB(k)

,Zp). Let λH1 :

H1⊗ZpH1 → Zp be the perfect form on H1 that is the Zp-etale realization of z∗(ΛA). Let

tα (resp. wα ∈ T (H1[ 1p])) be the de Rham (resp. the p-component of the etale) component

of the Hodge cycle z∗E(GQ,X)(wAα ) on AB(k). The following conjecture is only a refinement

of [Va3, Conj. 5.6.6].

8.1.1. Conjecture. There exists an isomorphism

jW (k) : (M,λM , (tα)α∈J )→(H1 ⊗ZpW (k), λH1 , (wα)α∈J ).

We refer to this Conjecture as Milne conjecture, cf. [Mi2, §4, pp. 511–512], [Va3,Conj. 5.6.6], and an unpublished manuscript of Milne available at www.arXiv.org.

We say that Conjecture 8.1.1 holds for (f, L, v) if it holds for every such field k = kand for every z ∈ N (W (k)).

8.1.2. Simple properties. (a) To prove Conjecture 8.1.1 it is enough to show that thereexists a faithfully flat, commutative W (k)-algebra C and an isomorphism jC : (M ⊗W (k)

C, λM , (tα)α∈J )→(H1 ⊗ZpC, λH1 , (wα)α∈J ). One argues this in a way similar to the last

two paragraphs of the proof of Proposition 6.2.8.

(b) The triple (H1, λH1 , (wα)α∈J ) can be identified with (L∗p, ψ

∗, (vα)α∈J ) (cf. Sub-section 6.1.5) and therefore it does not depend on z. If we have an isomorphism(M, (tα)α∈J )→(H1 ⊗Zp

W (k), (wα)α∈J ) = (L∗p ⊗Zp

W (k), (vα)α∈J ), then under it λMis mapped to a Gm(W (k))-multiple of ψ∗; but as we have a short exact sequence1 → G0

Z(p)(W (k)) → GZ(p)

(W (k)) → Gm(W (k)) → 1 (cf. Subsection 6.1.5), we can

assume this multiple is ψ∗ itself. Thus jW (k) exists if and only if there exists an isomor-phism (M, (tα)α∈J )→(H1 ⊗Zp

W (k), (wα)α∈J ) = (L∗p ⊗Zp

W (k), (vα)α∈J ).

144

(c) If Conjecture 8.1.1 holds for (f, L, v), then (as k = k) the four statements 1.6.1(a) to (d) are also equivalent to the following one:

(*) there exists an isomorphism (M, (tα)α∈J )→(M0 ⊗W (k(v)) W (k), (vα)α∈J ) that

induces an isomorphism (M,ϕ, G)→C0 ⊗ k = (M0 ⊗W (k(v)) W (k), ϕ0 ⊗ σk, GW (k)).

To check this, we first remark that an isomorphism jW (k) as in Conjecture 8.1.1 allows

us to assume that the isomorphism ρ : (M [ 1p], (tα)α∈J )→(M0 ⊗W (k(v)) B(k), (vα)α∈J ) of

Subsection 6.1.5 is such that ρ(M) = M0 ⊗W (k(v)) W (k). Similar to Subsection 6.1.6we argue that we can assume that ρµρ−1 = µ0,W (k) and thus that the element gy ofSubsection 6.1.5 belongs to GW (k)(W (k)). Thus the equivalence 1.6.1 (a) ⇔ (∗) followsfrom the equivalence 1.4.1 (a) ⇔ 1.4.4 (a) (see Theorem 1.4.4).

(d) We assume that Conjecture 8.1.1 holds for (f, L, v), that for every element α ∈ Jwe have vα ∈ T (L∗

(p)), and that GZpis the subgroup scheme of GSp(L∗

(p), ψ∗) that fixes vα

for all α ∈ J . For each α ∈ J we have wα ∈ T (H1) (cf. (b) and the relation vα ∈ T (L∗(p)))

and thus we also have tα ∈ T (M) (as Conjecture 8.1.1 holds for (f, L, v)). We use thenotations of Section 7.2; thus n ∈ N. Then Tn(y) and Tn(y1) are inner isomorphic if andonly if there exists an isomorphism (M/pnM,ϕ, ϑ, λM)→(M1/p

nM1, ϕ1, ϑ1, λM1) defined

by an isomorphism M/pnM→M1/pnM1 that takes tα modulo pn to t1,α modulo pn for all

α ∈ J .

(e) The pair (M, (tα)α∈J ) depends only on the GQ(A(p)f )-orbit of the connected com-

ponent of Nk(v) through which y factors, cf. property 6.2.8 (ii) and Fact 6.3.4. Thus fromthe Zariski density part of Theorem 1.6.1 and from (b), we get that to check Conjecture8.1.1 for (f, L, v) we can assume that k = F and that z is an Sh-canonical lift. Moreover, if

the connected components of Nk(v) are permuted transitively by GQ(A(p)f ), then to check

Conjecture 8.1.1 for (f, L, v), it suffices to show that jW (F) exists for a unique (arbitrarilychosen) W (F)-valued point z ∈ N (W (F)).

8.1.3. Fact. If p ≥ 3 and q = 1, then Conjecture 8.1.1 holds for (f, L, v).

Proof: We can assume that z ∈ N (W (k)) is an Sh-canonical lift of y ∈ N (k), cf. property8.1.2 (e). Thus Ak is an ordinary abelian variety and A is the canonical lift of Ak, cf.Proposition 7.1.1. Let F 0 be the direct supplement of F 1 inM such that we have ϕ(F 0) =F 0. Let iA and β0 be as in Section 6.1. As µ : Gm → G is the inverse of the canonicalsplit cocharacter of (M,F 1, ϕ), it fixes F 0. We have (see either the constructions in [Fa2,§6] or [BM, Cor. 2.2.4] where the modulo pn version of this is proved)

iA(F0 ⊗W (k) B

+(W (k))⊕1

β0F 1 ⊗W (k) B

+(W (k))) = H1 ⊗ZpB+(W (k)).

As F 0 ⊗W (k) B+(W (k)) ⊕ 1

β0F 1 ⊗W (k) B

+(W (k)) = µ(β0)(M ⊗W (k) B+(W (k))), we get

the existence of jB+(W (k)). Thus the Fact follows from the property 8.1.2 (a).

8.1.4. A relative context. We consider now the following situation. We assume thatf : (GQ, X) → (GSp(W,ψ), S) factors through an injective map of Shimura pairs

f1 : (G1,Q, X1) → (GSp(W,ψ), S)

145

and that the Zariski closure G1,Z(p)of G1,Q in GLL(p)

is a reductive group scheme overZ(p). As we have an injective map of Shimura pairs (GQ, X) → (G1,Q, X1), the reflex fieldE(G1,Q, X1) of (G1,Q, X1) is a subfield of E(GQ, X). Let v1 be the prime of E(G1,Q, X1)that divides v. We list two statements:

(i) The triple (f1, L, v1) is a standard Hodge situation and Conjecture 8.1.1 holds forit;

(ii) One of the following two conditions holds:

(ii.a) we have Z0(GZ(p)) = Z0(G1,Z(p)

) and there exists a product decomposition Gder1,W (F) =∏n

i=1G(i)1 such that for all numbers i ∈ S(1, n) there exists a closed embedding

monomorphism G(i)1 →

∏j∈Ii

GLNi,j, where Ii is a finite set and where each Ni,j

is a free W (F)-module of finite rank, with the property that the GderF -module (and

thus also the G(i)1,F-module) Ni,j/pNi,j is simple for all j ∈ Ii;

(ii.b) the group scheme GZ(p)is the centralizer in G1,Z(p)

of a torus T1,Z(p)of G1,Z(p)

andeither p ≥ 3 or p = 2 and the Newton polygon N0 (of C0) has no integral Newtonpolygon slopes.

8.1.5. Theorem. If conditions 8.1.4 (i) and (ii) hold, then Conjecture 8.1.1 holds for(f, L, v).

Proof: We choose the family of tensors (vα)α∈J of T (W ∗) such that there exists a subsetJ1 of J with the property that G1,Q is the subgroup of GLW that fixes vα for all α ∈ J1.Let O(v1) be the localization of the ring of integers of E(G1,Q, X1) with respect to v1. LetN1 be the normalization of the Zariski closure of Sh(G1,Q, X1)/G1,Z(p)

(Zp) in MO(v1). Let

z1 ∈ N1(W (k)) be the composite of z with the natural morphism N → N1. Let V be afinite, discrete valuation ring extension of W (k) such that there exists an isomorphism(32a)jV : (M ⊗W (k) V, λM , (tα)α∈J1

)→(H1 ⊗ZpV, λH1 , (wα)α∈J1

) = (L∗p ⊗Zp

V, ψ∗, (vα)α∈J1),

cf. condition 8.1.4 (i) and property 8.1.2 (b) (it is significantly more convenient here towork with a flexible jV and not with a rigid jW (k)). Let K := V [ 1p ]. For each element

α ∈ J let t′α ∈ T (L∗p ⊗Zp

K) be the tensor such that jV takes tα to t′α. Thus foreach α ∈ J1 we have t′α = vα ∈ T (L∗

p ⊗ZpK). From the existence of an isomorphism ρ :

(M [ 1p], (tα)α∈J )→(M0⊗W (k(v))B(k), (vα)α∈J ) (see Subsection 6.1.5), we get the existence

of an element g ∈ G1,V (K) such that we have g(t′α) = vα for all α ∈ J (equivalently forall α ∈ J \ J1); more precisely, under the canonical identification M0 ⊗W (k(v)) W (k) =

L∗p ⊗Zp

W (k), we can take g = (ρ⊗ 1K) j−1V ∈ G1,V (K). If we have

(32b) g ∈ GV (K)G1,V (V ),

then by composing jV with a V -linear automorphism of L∗p ⊗Zp

V defined by an el-

ement of G1,V (V ) that corresponds to (32b), we get an isomorphism jV : (M ⊗W (k)

V, (tα)α∈J )→(H1 ⊗ZpV, (wα)α∈J ) = (L∗

p ⊗ZpV, (vα)α∈J ). As in the property 8.1.2 (b)

we argue that we can assume that jV takes λM to λH1 = ψ∗. Thus based on the property

146

8.1.2 (a), to end the proof of the Theorem we only have to show that g ∈ GV (K)G1,V (V )and therefore from now on we will ignore the alternating forms.

Case A.We first consider the case when the condition 8.1.4 (ii.a) holds. Let H1 := GV (V ).As gjV Gj

−1V g−1 is a reductive group scheme that extends GK , the subgroup

H2 := h ∈ GV (K)|h(g(L∗p ⊗Zp

V )) = g(L∗p ⊗Zp

V ) = gH1g−1 6 GV (K)

is hyperspecial. The hyperspecial subgroups H1 and H2 of GV (K) are GadV (K)-conjugate,

cf. [Ti2, end of Subsect. 2.5]. Thus by replacing V with a finite, discrete valuation ringextension of it, we can assume that there exists an element g1 ∈ GV (K) such that we haveg1(H2)g

−11 = H1. By replacing g with g1g we can assume that H1 = H2 = gH1g

−1.

In this and the next paragraph, we show that g normalizes H(i)1 := G

(i)1,V (V ) for all

i ∈ S(1, n). Let H(i)2 := gH

(i)1 g−1. Let G

0,(i),scW (F) be the maximal direct factor of the simply

connected semisimple group scheme cover G,scW (F) of G

derW (F) such that the natural homomor-

phism G0,(i),scB(F) → G

(i)1,B(F) has a finite kernel. Let G(i) be the semisimple group scheme over

W (F) that is the quotient of GderW (F) through the Zariski closure of Ker(Gder

B(F) → G(i)1,B(F))

in GderW (F). We can redefine G(i) as the quotient of G

0,(i),scW (F) through a finite, flat subgroup

scheme of Z(G0,(i),scW (F) ) and thus the existence of G(i) is implied by [DG, Vol. III, Exp.

XXII, Cor. 4.3.2]. Let H(i) := G(i)(V ). It is the only hyperspecial subgroup of G(i)(K) to

which H1 maps, cf. Lemma 2.3.10. As H1 = gH1g−1, we get H(i) = gH(i)g−1. Thus H

(i)1

and H(i)2 are hyperspecial subgroups of G

(i)1 (K) that contain H(i).

For j ∈ Ii and s ∈ 1, 2 let Li,j,s be a V -lattice of Ni,j ⊗W (k) K such that we have

H(i)s 6

∏j∈Ii

GLLi,j,s(V ), cf. [Ja, Part I, 10.4]. As H(i) normalizes ⊕j∈I1Li,j,s, from [Va3,

Prop. 3.1.2.1 a)] we get that ⊕j∈I1Li,j,s is a G(i)V -module. The G

(i)k -module Ni,j ⊗W (F) k

is irreducible (cf. condition 8.1.4 (ii.a)) and thus the G(i)k -module Li,j,s⊗V k is isomorphic

to Ni,j ⊗W (F) k and therefore it is irreducible (cf. [Ja, Part I, 10.9]). Thus we haveLi,j,1 = π

mi j

V Li j 2, where mi j ∈ Z and where πV is a fixed uniformizer of V . Therefore

the product group∏

j∈IiGLLi,j,s

(V ) does not depend on s. Thus we have H(i)1 = H

(i)2

and therefore g normalizes H(i)1 .

Thus the image gad of g in Gad1,V (K) normalizes the unique hyperspecial subgroup

Gad1,V (V ) of Gad

1,V (K) that contains Im(H(i)1 → Gad

1,V (K)) for all i ∈ S(1, n) (for the unique-

ness part, cf. Lemma 2.3.10 applied to the epimorphism Gder1,V ։ Gad

1,V ). From [Va3, Prop.

3.1.2.1 a)] we get that gad defines via inner conjugation an automorphism of Gad1,V . Thus

we have gad ∈ Gad1,V (V ). By replacing V with a finite, discrete valuation ring extension

of it, we can assume that there exists an element g−12 ∈ G1,V (V ) such that its image in

Gad1 (V ) is gad. By replacing g with gg2 we can assume that gad is the identity element i.e.,

g ∈ Z(G1,V )(K). By replacing V with a finite, discrete valuation ring extension of it wecan write g = g3g4, where g3 ∈ Z0(G1,V )(K) = Z0(GV )(K) and where g4 ∈ Z(Gder

1,V )(V ).Thus g ∈ GV (K)G1,V (V ).

147

Case B. We consider the case when the condition 8.1.4 (ii.b) holds. To show that Con-jecture 8.1.1 holds for (f, L, v) we can assume that z is an Sh-canonical lift, cf. property8.1.2 (e). We choose the set J such that we have J ⊇ J1 ∪ Lie(T1,Z(p)

) and vα = α

for all α ∈ Lie(T1,Z(p)). There exists a maximal torus T of G such that (M,ϕ, T ) is a

Shimura F -crystal, cf. Subsection 4.3.2. Let J be the union of J and of the set of allelements of Lie(T ) fixed by ϕ. For α ∈ Lie(T ), we define tα = α. We can assume thatfor α ∈ J ∩ Lie(T ), the two definitions of tα define the same tensor of T (M [ 1p ]). For

α ∈ J \ J , let vα ∈ End(H1) be the endomorphism that corresponds to tα via Fontainecomparison theory (see Subsection 6.1.7) and let t′α ∈ T (L∗

p ⊗ZpK) be the tensor such

that jV takes tα to t′α. Thus we have g(t′α) = vα, for all α ∈ J . Moreover, the Zariskiclosure in GLH1⊗ZpV

= GLL∗p⊗ZpV

of the subgroup of GLH1⊗ZpK= GLL∗

p⊗ZpKthat

fixes vα (resp. t′α) for all α ∈ J , is a maximal torus T et1 (resp. T crys

1 ) of GV and thereforealso of G1,V . More precisely, T et

1 is the extension to V of the torus Tet of Lemma 6.1.8

and T crys1 := jV T j

−1V . Thus Lie(T et

1 ) (resp. Lie(T crys1 )) is V -generated by the vα’s (resp.

the t′α’s) with α ∈ (J \ J1) ∪ Lie(T1,Z(p)). But T et

1 and T crys1 are GV (V )-conjugate, cf.

Subsection 2.3.8 (d). Thus by replacing g with gg1, where g1 ∈ GV (V ) 6 G1,V (V ), we canassume that T et

1 = T crys1 . Due to the V -generation property of Lie(T et

1 ) and Lie(T crys1 ),

we get that g normalizes T et1 = T crys

1 . Thus g = g2g3, where g2 ∈ T et1 (K) 6 GV (K)

and where g3 is a V -valued point of the normalizer of T et1 in G1,V . Therefore we have

g ∈ GV (K)G1,V (V ).

8.1.6. Remarks. (a) There exists a variant of the Case B of the proof of Theorem 8.1.5which uses the fact that every two tori T (1) and T (2) of G1,V with the properties that wehave T (i) = Z0(C(i)) (i ∈ 1, 2), where C(i) is the centralizer of T (i) in G1,V , and thattheir generic fibres are G1,V (K)-conjugate, are G1,V (V )-conjugate. This can be checkedby reduction to the case of classical semisimple group schemes over V . Both this newapproach and the proof of Theorem 8.1.5 under the condition 8.1.4 (ii.b), work even for

p = 2 provided the connected components of Nk(v) are permuted transitively by GQ(A(p)f ).

This is so as (cf. property 8.1.2 (e)) we can assume that z ∈ N (W (k)) is such that thevariant 6.1.9 (b) applies to it.

(b) Let Y be an abelian variety over a number field E. We fix an embedding E → Cand we identify E with Q. Let (sα)α∈JY

be the family of all Hodge cycles on Y thatinvolve no Tate twist. To ease notations, we will assume that (sα)α∈JY

is equal to thefamily of all Hodge cycles on AE that involve no Tate twist. Let LY := H1(Y (C),Z). LetWY := LY ⊗Z Q = H1(Y (C),Q). For each element α ∈ JY let sBα be the Betti realizationof sα; thus s

Bα is a tensor of the tensor algebra ofW ∗

Y ⊕WY . Let OE be the ring of integersof E. Let mY ∈ N be such that Y extends to an abelian scheme Y over Spec(OE [

1mY

]).

Let DY := H1dR(Y/OE[

1mY

]). There exists a finite field extension E1 of E such that there

exists an E1-linear isomorphism iE1: W ∗

Y ⊗Q E1→DY ⊗OE [ 1mY

] E1 that takes sBα to the

de Rham component sdRα of sα for all α ∈ JY . This is a standard consequence of the factthat under the de Rham C-linear isomorphism W ∗

Y ⊗Q C→DY ⊗OE [ 1mY

] C, sBα is mapped

to sdRα for all α ∈ JY (cf. [De5, §1 and §2]). Let nY ∈ mY N be such that we have

148

iE1(L∗

Y ⊗Z OE1[ 1nY

]) = DY ⊗OE [ 1mY

] OE1[ 1nY

]; here OE1is the ring of integers of E1.

We now fix f and L but we allow p to vary. The connected components of NC are

permuted transitively by GQ(A(p)f ), cf. [Va3, Lem. 3.3.2]. It is well known that this

implies that for p >> 0 the connected components of Nk(v) are also permuted transitively

by GQ(A(p)f ) (cf. [Va3, Rm. 3.4.6 and Cor. 3.4.7] and [Gr1, Thm. (9.7.7)]). Thus the

Conjecture 8.1.1 holds for (f, L, v) if p >> 0 and if Conjecture 8.1.1 holds for one W (F)-valued point z ∈ N (W (F)), cf. property 8.1.2 (e). Thus from the previous paragraph weget that Conjecture 8.1.1 holds for (f, L, v) provided p >> 0.

8.1.7. On applying 8.1.5. The usefulness of Conjecture 8.1.1 stems from its motivicaspect that will be used in future work by Milne and us in connection to the Langlands–Rapoport conjecture and by us to achieve the project 1.9 (b). We include a practical formof Subsection 8.1.4 that suffices for applications to this last conjecture. We assume thatthe following three conditions hold:

(i) the centralizer CZ(p)of GZ(p)

in GLL(p)is a reductive group scheme;1

(ii) we have Z0(GQ) = Z0(G1,Q), where G1,Q is the identity component of the cen-tralizer D(ψ)Q of CQ in GSp(W,ψ);

(iii) if p = 2, then we have G1,Q = D(ψ)Q.

Let X1 be the G1,Q(R)-conjugacy class of monomorphisms ResC/RGm → G1,R thatcontains X (i.e., contains the composite of any monomorphism ResC/RGm → GR thatis an element of X with the monomorphism GR → G1,R). We get an injective mapf1 : (G1,Q, X1) → (GSp(W,ψ), S) of Shimura pairs through which f : (GQ, X) →(GSp(W,ψ), S) factors, cf. [Va3, Rm. 4.3.12]. Let the prime v1 of E(G1,Q, X1) be asin Subsection 8.1.4. The group scheme CZ(p)

is the group scheme of invertible elementsof the semisimple Z(p)-subalgebra B1 of EndZ(p)

(L(p)) whose elements are the elements ofLie(CZ(p)

). As B1 is normalized by the involution ′ of Subsection 1.5.1, the Zariski closure

G1,Z(p)of G1,Q in GLL(p)

is a reductive group scheme. To check this can assume that B1[1p ]

is simple and for this well known case we refer to [Ko2]. Thus the quadruple (f1, L, v1,B1)is a standard PEL situation which is as well a standard Hodge situation, cf. Remark 1.5.5(b). Thus the condition 8.1.4 (i) holds if and only if Conjecture 8.1.1 holds for (f1, L, v1).

The fact that Conjecture 8.1.1 holds for (f1, L, v1) is well known. For instance, dueto existence of the isomorphism ρ of Subsection 6.1.5, the arguments of the proof of [Ko2,Lem. 7.2] apply but with Qp being replaced by B(k). We now include a new self-containedand short argument for this. Let V , K, and πV be as in the proof of Theorem 8.1.5.Let L0,V := L∗

(p) ⊗W (k) V = M0 ⊗W (k(v)) V . We will work with an isomorphism ρ :

(M [ 1p ], (tα)α∈J )→(M0⊗W (k(v))B(k), (vα)α∈J ) that takes λM to ψ∗ (see Subsection 6.1.5).

Let LV := ρ(M)⊗W (k) V . Thus LV is a V -lattice of M0 ⊗W (k(v))K that is normalized byB1 ⊗Z(p)

V , that has the property that the Zariski closure of GK in GLLVis a reductive

group scheme (ρGρ−1)V , and that is such that we have a perfect alternating form ψ∗ :

1 Due to Lemma 2.3.11, this always holds but we will not stop to detail this here.

149

LV ⊗V LV → V . We will show these three properties of LV imply the existence of anelement g ∈ G(K) such that we have g−1(LV ) = L0,V .

We can replace V by any finite, discrete valuation ring extension of it (to be comparedwith the property 8.1.2 (a)). Thus, as in the first paragraph of the Case A of the proofof Theorem 8.1.5, we argue that we can assume that (LV is such that) GV is the Zariskiclosure of GK in GLLV

. Let LV = ⊕j∈χ(T0,V )LjV and L0,V = ⊕j∈χ(T0,V )L

j0,V be the direct

sum decompositions such that the maximal torus T0,V of GV acts on LjV and Lj

0,V via the

character j of T0,V . The B1 ⊗Z(p)V -modules Lj

V and Lj0,V are simple. Thus there exists

mj ∈ Z such that we have LjV = π

mj

V Lj0,V . Let g ∈ GLLV

(K) be the semisimple element

such that it acts on LjV as the multiplication with π

−mj

V ; we have g−1(LV ) = L0,V . Theelement g centralizes B1 ⊗Z(p)

V . Moreover, as ψ∗ induces perfect forms on LV and L0,V ,

g fixes ψ∗. As πmj

V is never a root of unity different from 1, the Zariski closure of the setgm|m ∈ Z in GLLV [ 1p ]

is a torus of GK (and not only a group of multiplicative type).

Thus we have g ∈ G(K). Therefore jV := g−1 (ρ ⊗ 1K) is an isomorphism as in (32a).Thus Conjecture 8.1.1 holds for (f1, L, v1), cf. property 8.1.2 (a).

We conclude that the condition 8.1.4 (i) is implied by conditions (i) to (iii). We nowcheck that the condition 8.1.4 (ii.a) also holds. The group scheme Gder

1,W (F) is a product

of semisimple group schemes that are SLn (with n ≥ 2), Sp2n (with n ≥ 1), or SO2n

(with n ≥ 2) group schemes. For each such factor G(i)1 of Gder

1,W (F), the G(i)1 -module

M0 ⊗W (k(v)) W (F) is a direct sum of trivial G(i)1 -modules and of standard simple G

(i)1 -

modules whose ranks are n for the case of SLn and are 2n for the case of either Sp2n or

SO2n. Let Ni be a direct summand of M0 ⊗W (k(v)) W (F) that is a simple G(i)1 -module.

The resulting representation ρi : GderW (F) → GLNi

is irreducible (over B(F) this is implied

by the condition (ii)). Thus ρi,B(F) is a tensor product of irreducible representationsassociated to minuscule weights, cf. property 1.5.6 (ii); the number of the simple factorsof the adjoint group of Im(Gder

B(F) → GLNi[1p ]) is equal to the number of factors of the

mentioned tensor product. From this and [Ja, Part I, 10.9], we get that the representationρi,F has a composite series whose factors are isomorphic with the factors of a compositionseries of the representation over F which is a tensor product of representations of theform ρ,F, where ρ is as in Lemma 2.3.11, and therefore which is a tensor product ofirreducible representations (cf. Lemma 2.3.11). Thus the representation ρi,F is irreducible.Therefore the condition 8.1.4 (ii.a) is also implied by conditions (i) to (iii). From Theorem8.1.5 we get:

8.1.8. Theorem. If conditions 8.1.7 (i) to (iii) hold, then Conjecture 8.1.1 holds for(f, L, v).

8.1.9. Example. Let (GadQ , X

ad) be a simple, adjoint Shimura pair of An, Bn, Cn, DHn ,

or DRn Shimura type; see Subsection 1.5.6. Let vad be a prime of E(Gad

Q , Xad) that divides

a prime p ≥ 5. We assume that the group GadQp

is unramified. The constructions of [Va3,

Subsects. 6.5 and 6.6.5] show that there exists a standard Hodge situation (f, L, v) suchthat the notations match (i.e., (Gad

Q , Xad) is the adjoint Shimura pair of (GQ, X), etc.) and

150

the representation GW (F) → GLL(p)⊗Z(p)W (F) is a direct sum of irreducible representations

whose restrictions to GderW (F) are associated to minuscule weights of some direct factor of

the simply connected semisimple group scheme cover of GderW (F) that has an absolutely

simple, adjoint. Thus L(p) ⊗Z(p)F is a semisimple GF-module, cf. Lemma 2.3.8. Thus CF

is reductive. Moreover CF and CB(F) have equal dimensions. Thus CW (F) is a reductivegroup scheme and therefore the condition 8.1.7 (i) holds. By replacing GQ with the largergroup that is the subgroup of G1,Q generated by Z0(G1,Q) and by Gder

Q we can also assumethat the condition 8.1.7 (ii) also holds: under this replacement, the Zariski closure GZ(p)

of GQ in GLL(p)is isomorphic to the quotient of Z0(G1,Z(p)

) × GderZ(p)

by a central finite,

flat subgroup scheme isomorphic to Z0(G1,Z(p)) ∩Gder

Z(p), and thus GZ(p)

continues to be a

reductive group scheme over Z(p) (cf. [DG, Vol. III, Exp. XXII, Cor. 4.3.2]). As p ≥ 5,the condition 8.1.7 (iii) is vacuous. Thus Conjecture 8.1.1 holds for (f, L, v), cf. Theorem8.1.8.

8.1.10. Remark. Subsections 8.1.2 to 8.1.9 are far from a complete proof of Conjecture8.1.1. But they form the simplest approach to Conjecture 8.1.1, suffice for most applica-tions (even for p = 2), and are close in spirit to [Ko2, Lem. 7.2]. We refer to [Va7] (resp.to [Va8]) for a complete proof of Conjecture 8.1.1 for p ≥ 3 (resp. for all primes p ≥ 2),that uses much heavier crystalline cohomology theories.

8.2. Applications to integral canonical models

We consider a standard Hodge situation (f, L, v) together with a map (G1,Q, X1) →(GQ, X) of Shimura pairs such that the following three conditions hold:

(i) we have E(G1,Q, X1) = E(GQ, X);

(ii) the group G1,Qpis unramified;

(iii) the homomorphism G1,Q → GQ is an epimorphism whose kernel is a torus T1with the property that the group H1(E, T1,E) is trivial for every field E of characteristic0.

Due to the condition (i) we identify v with a prime of E(G1,Q, X1). Let G1,Zpbe

a reductive group scheme over Zp that extends G1,Qp, cf. [Ti2, Subsubsects. 1.10.2

and 3.8.1]. The normalization of GZp×Zp

Gab1,Zp

in G1,Qpis a reductive group scheme

over Zp that extends G1,Qp. Not to introduce extra notations, we will assume that this

normalization is G1,Zpitself; thus we have an epimorphism G1,Zp

։ GZp. Let H1,p :=

G1,Zp(Zp). The Zariski closure of T1,Qp

in Z0(G1,Zp) (equivalently in G1,Zp

) is a torus overZp (cf. Fact 2.3.12). The quotient group scheme G1,Zp

/T1,Zpis reductive (cf. [DG, Vol.

III, Exp. XXII, Cor. 4.3.2]) and has GQpas its generic fibre. The natural homomorphism

G1,Zp/T1,Zp

→ GZpis an isomorphism, cf. Lemma 2.3.9. Thus we have a short exact

sequence 1 → T1,Zp(Zp) → G1,Zp

(Zp) → GZp(Zp) → 1 and therefore H1,p maps onto

Hp. The holomorphic map X1 → X induced by the epimorphism G1,Q ։ GQ is abijection (cf. [Mi1, Lem. 4.12]) and thus an isomorphism. We have a functorial morphismSh(G1,Q, X1)/H1,p → Sh(GQ, X)/H1 = NE(GQ,X) of E(GQ, X)-schemes (cf. [De2, Cor.5.4]) that is pro-finite (as X→X1). Therefore we can speak about the normalization N1

of N in (the ring of fractions of) Sh(G1,Q, X1)/H1,p.

151

8.2.1. Lemma. Let s2 : (T2, h2) → (T3, h3) be a map between two 0 dimensionalShimura pairs. We assume that T2 surjects onto T3 and that Ker(s2) is T1. Let T2,pbe a compact, open subgroup of T2(Qp) that contains the unique hyperspecial subgroupT1,p of T1(Qp). Let T3,p be the image of T2,p in T3(Qp). The functorial morphismSh(T2, h2) → Sh(T3, h3) (see [De2, Cor. 5.4]) gives birth naturally to a morphisms2,p : Sh(T2, h2)/T2,p → Sh(T3, h3)E(T2,h2)/T3,p of E(T2, h2)-schemes. Then themap s2,p(B(F)) : Sh(T2, h2)/T2,p(B(F)) → Sh(T3, h3)E(T2,h2)/T3,p(B(F)) is onto.

Proof: The groups T2(Q), T2(Qp), and T2,p surject onto T3(Q), T3(Qp), and T3,p (re-spectively). As we have T1(Qp) = T1(Q)T1,p (cf. [Mi2, Lem. 4.10]), the homomorphisms2(Qp) gives birth to an identification

(33) T2(Qp)/T2,pT2(Q) = T3(Qp)/T3,pT3(Q).

Each B(F)-valued point of Sh(T3, h3)E(T2,h2)/T3,p factors through an F -valued point of

Sh(T3, h3)E(T2,h2)/T3,p with F as a subfield of Q. Moreover, from the reciprocity map(see [Mi1, §1, pp. 163–164]) and the class field theory (see [Lan, Ch. XI, §4, Thm. 4]) we getthat each F -valued point of Sh(T3, h3)E(T2,h2)/T3,p lifts to a point of Sh(T2, h2)/T2,pwith values in an abelian extension of F which (due to (33)) is unramified above all primesof F that divide p. As B(F) has no unramified finite field extensions, we conclude thatthe map s2,p(B(F)) is onto.

8.2.2. Proposition. The image Im(N1(W (F)) → N (F)) is Zariski dense in Nk(v).

Proof: It suffices to show that every Sh-canonical lift z ∈ N (W (F)) belongs to the im-age Im(N1(W (F)) → N (W (F))), cf. the Zariski density part of Theorem 1.6.1. Wefix an O(v)-embedding eB(F) : B(F) → C. Let T3 be the Mumford–Tate group ofz∗(A)C. As z∗(A)C has complex multiplication (cf. Theorem 1.6.3 (a)), the group T3is a torus. Let h3 : ResC/RGm → T3,R be the homomorphism that defines the HodgeQ–structure on the Betti homology and cohomology of z∗(A)C. We identify naturallythe Shimura pair (T3, h3) with a special pair of the Shimura pair (GQ, X). Let T2be the inverse image of T3 in G1,Q. As X→X1, there exists a unique element h2 ∈ X1

that maps to h3 ∈ X ; thus the homomorphism h2 : ResC/RGm → G1,R factors through

T2,R. We have Sh(GQ, X)/Hp(C) = GZ(p)(Z(p))\(X × GQ(A

(p)f )) (see Subsection 1.5.1).

As G1,Q(A(p)f ) surjects onto GQ(A

(p)f ), up to GQ(A

(p)f )-translations we can assume that

zE(GQ,X) is a B(F)-valued point of Sh(T3, h3)/T3,p. As T1,Qpis unramified, T1,Q ex-

tends to a torus T1,Z(p)over Z(p) (cf. [Va3, Lem. 3.1.3]). But T1,Z(p)

splits over a Ga-lois cover of Spec(Z(p)). Thus the reflex field E(T1, h1) i.e., the field of definition ofh1, is unramified above p. But E(T2, h2) is the composite field of E(T1, h1) andE(T3, h3) and therefore it is unramified above all primes of E(T3, h3) that divide p.Thus we can view zE(GQ,X) as a B(F)-valued point of Sh(T3, h3)E(T2,h2)/T3,p. As wehave a natural composite morphism Sh(T2, h2)/T2,p → Sh(T3, h3)E(T2,h2)/T3,p →Sh(G1,Q, X1)E(T2,h2)/H2,p of E(T2, h2)-schemes, the point zE(GQ,X) lifts to a B(F)-valued point of Sh(T3, h3)E(T2,h2)/T3,p (cf. Lemma 8.2.1) and therefore also of N1.This implies that z ∈ Im(N1(W (F)) → N (W (F))).

152

8.2.3. Corollary. We recall that (f, L, v) is a standard Hodge situation. Let o be theorder of the center Z(Gsc

Q) of the simply connected semisimple group cover GscQ of Gder

Q . Weassume that p ≥ 3 and that o is a power of 2. Then the O(v)-scheme N1 together with the

natural right action of G1,Q(A(p)f ) on it is an integral canonical model of (G1,Q, X1, H1,p, v)

in the sense of [Va3, Defs. 3.2.3 6) and 3.2.6].

Proof: From [Va3, Rm. 6.2.3.1] and the assumption on o, we get that each connectedcomponent of N1,C = Sh(G1,Q, X1)C/H1,p is a pro-finite Galois cover of a connectedcomponent of NC = Sh(GQ, X)C/Hp, the Galois group being a 2m-torsion pro-finite group(for some fixed m ∈ N). Moreover, N1,C surjects onto NC (see [Mi1, Lem. 4.13]). The

connected components of N1,C are permuted transitively by G1,Q(A(p)f ) (see [Va3, Lem.

3.3.2]) and N1,B(F) has B(F)-valued points (cf. Proposition 8.2.2). We conclude thateach connected component of N1,B(F) is geometrically connected and therefore a pro-finiteGalois cover of a connected component of NB(F), the Galois group being as above. In theproof of [Va3, Lem. 6.8.1] we checked that there exists no finite W (F)-monomorphismW (F)[[x1, . . . , xd]] → E1 which by inverting p becomes an etale cover of degree 2, whereE1 is a commutative, integral W (F)-algebra equipped with a section E1 ։ W (F). Fromthe last two sentences we get that the morphism N1 → N is a pro-etale cover above everyW (F)-valued point in Im(N1(W (F)) → N (W (F))). Thus, based on the classical puritytheorem of [Gr2, Exp. X, Thm. 3.4 (i)] and on Proposition 8.2.2, we conclude that N1 isa pro-etale cover of N . But N is a regular, formally smooth O(v)-scheme (cf. Definition1.5.3) and therefore N1 is as well a regular, formally smooth O(v)-scheme.

Let H(p) be a compact, open subgroup of GQ(A(p)f ) such that N is a pro-etale cover

of N /H(p) and the O(v)-scheme N /H(p) is of finite type, cf. [Va3, proof of Prop. 3.4.1].

Let H(p)1 be a compact, open subgroup of G1,Q(A

(p)f ) that maps to H(p). It is easy to

see that the quotient N1/H(p)1 of N1 by H

(p)1 exists: it is the normalization of N /H(p)

in the ring of fractions of N1,E(GQ,X). From the last two sentences and the fact that

N1 is a pro-etale cover of N , we get that N1 is a pro-etale cover of N1/H(p)1 and that

N1/H(p)1 is an etale cover of N /H(p). Therefore N1/H

(p)1 is a smooth O(v)-scheme of

finite type. Moreover, if H′(p)1 is a compact open subgroup of H

(p)1 , then the natural

epimorphism N1/H′(p)1 → N1/H

(p)1 is an etale cover. Thus N1 is a smooth integral model

of Sh(G1,Q, X1)/H1,p over O(v) in the sense of either [Mi1, Def. 2.2] or [Va3, Def. 3.2.32)]. The O(v)-scheme N1 also has the extension property of [Va3, Def. 3.2.3 3)], cf. [Va3,Rm. 3.2.3.1 5)]. Thus N1 is an integral canonical model of (G1,Q, X1, H1,p, v).

8.2.4. Applications to [Va3]. We continue to assume that p ≥ 3 and that the order o ofCorollary 8.2.3 is a power of 2. This assumption on o holds if all simple factors of (Gad

Q , Xad)

are of A2n−1, Bn, Cn, DRn , or DH

n type. We consider now a map (G2,Q, X2) → (GQ, X) ofShimura pairs such that the homomorphism Gder

2,Q → GderQ is an isogeny, the group G2,Qp

is unramified over Qp, and there exists a hyperspecial subgroup H2p of G2,Qp(Qp) that

maps to Hp. Let v2 be a prime of the reflex field E(G2,Q, X2) of (G2,Q, X2) that divides v.Let O(v2) be the localization of the ring of integers of E(G2,Q, X2) with respect to v2. Wenow choose the map of Shimura pairs (G1,Q, X1) → (GQ, X) such that (besides conditions

153

8.2 (i) to (iii)) we also have Gder1,Q = Gder

2,Q, cf. [Va3, Rm. 3.2.7 10)].

We check the following claim that the normalization of NO(v2)in Sh(G2,Q, X2)/H2p is:

(i) a pro-etale cover of an open closed subscheme of NO(v2)and (ii) an integral canonical

model of (G2,Q, X2, X2, v2). As N1 is a pro-etale cover of N and as we have E(G2,Q, X2) ⊇E(G1,Q, X1) = E(GQ, X) andGder

1,Q = Gder2,Q, the claim follows from the proof of either [Va3,

Lem. 6.2.3] or [Va6, Prop. 2.4.3], applied to quadruples of the form (G2,Q, X2, H2p, v2)instead of to triples of the form (G2,Q, X2, H2p).

This claim proves [Va3, Thm. 6.1.2*] in the cases when all simple factors of (GadQ , X

ad)

are of A2n−1, Bn, Cn, DRn , or DH

n type. In particular, this completes the (last step of the)proof (see [Va3]) of the existence of integral canonical models of those Shimura varietieswhose adjoints have all simple factors of some DH

n type (n ≥ 4) with respect to primes ofcharacteristic at least 5, cf. [Va3, Subsects. 1.4, 6.1.2.1, 6.4.2, and 6.8.0].

8.2.5. Remark. Any attempt of handling [Va3, Thm. 6.1.2*] only at the level of discretevaluation rings is pointless (see [Va3, Subsect. 6.2.7]); thus the proof of [Mo, Prop. 3.22]is wrong. This explains why the proof of [Va3, Thm. 6.1.2*] was carried out in [Va3,Subsect. 6.8] only up to [Va3, Subsubsect. 6.8.6]. Also the descent arguments of [Mo,Prop. 3.10] are incorrect: the counterexample of [BLR, Ch. 6, Sect. 6.7] makes sense inmixed characteristic. These two errors of [Mo] can not be “patched”. Moreover, they alsoinvalidate [Mo, Prop. 3.18 and Cor. 3.23] i.e., essentially all of [Mo, §3] that did not follow[Va3] and its preliminary versions. Thus the claim of [Mo, Rm. 3.24] of a “very different”presentation from the earlier versions of [Va3], is meaningless.

154

9 Generalization of the properties 1.1 (f) to (h)

to the context of standard Hodge situations

In this Chapter we generalize properties 1.1 (f) to (h) to the geometric context of astandard Hodge situation (f, L, v). In Sections 9.1 and 9.2 we generalize properties 1.1 (f)and (g) to the context of (f, L, v). In Sections 9.4 to 9.8 we generalize the property 1.1(h) to the context of (f, L, v). Theorem 9.4 considers the case when the nilpotent class I0of Section 1.6 is at most 1. In Section 9.8 we consider the case when I0 ≥ 2. Differentnotations and definitions needed to state and prove the results of Sections 9.4 to 9.8 aregathered in Section 9.3. In Sections 9.5 to 9.7 we include several complements to Section9.2 and Theorem 9.4 that pertain to the case p = 2, to ramified valued points, to functorialaspects, etc.

We often identify a formal Lie group F over Spf(W (k)) with a quadruple of theform (Spf(R), eR, IR,MR), where eR : Spf(W (k)) → Spf(R), IR : Spf(R)→Spf(R), and

MR : Spf(R)×Spf(W (k))Spf(R) → Spf(R) are morphisms of formal schemes over Spf(W (k))that satisfy the standard axioms of a group.

We will use all notations of the first paragraph of Chapter 7. Until the end we willassume that k = k, that y ∈ N (k) is an Sh-ordinary point, and that z ∈ N (W (k)) isan Sh-canonical lift of y. Until the end let µ : Gm → G be the canonical cocharacterof (M,ϕ, G) (see Subsection 4.1.4), let M = F 1 ⊕ F 0 be the direct sum decompositionnormalized by µ, let the group schemes P+

G(ϕ), U+

G(ϕ), L0

G(ϕ), P−

G(ϕ), and U−

G(ϕ) be as

in Section 2.6, and let

u+ := Lie(U+

G(ϕ)) and u− := Lie(U−

G(ϕ)).

Let γ1 < γ2 < . . . < γc be the Newton polygon slopes of N0 listed increasingly. Letγ0 ∈ Q ∩ (−∞, γ1). We have a direct sum decomposition

(M,ϕ) = ⊕cs=1(W (γs)(M,ϕ), ϕ)

into F -crystals over k that have only one Newton polygon slope, cf. statements 1.4.3 (d)and 1.6.1 (b). Let

(34) 0 =Wγ0(M,ϕ) ⊆Wγ1

(M,ϕ) ⊆Wγ2(M,ϕ) ⊆ . . . ⊆Wγc

(M,ϕ) =M

be the ascending Newton polygon slope filtration of (M,ϕ) (see Section 2.1 for notations).For s ∈ S(1, c) we identify naturally W (γs)(M,ϕ) with Wγs

(M,ϕ)/Wγs−1(M,ϕ).

155

We denote also by ϕ the σ-linear endomorphism of M/Wγs(M,ϕ) induced by ϕ.

Let ϕ(−) be the σ-linear endomorphism of EndW (k)(M) that takes x ∈ EndW (k)(M) top(ϕ(x)) = ϕ(px).

Until the end, the following notations d ∈ N ∪ 0, R = W (k)[[x1, . . . , xd]],R0 = W (k)[[w1, . . . , wd]], I = (x1, . . . , xd) ⊆ R, ΦR, mR : Spec(R) → N , N1, N0, Q,I0 = (w1, . . . , wd) ⊆ R0, ΦR0

, C0 = (M ⊗W (k) R0, F1 ⊗W (k) R0, gR0

(ϕ ⊗ ΦR0),∇0, λM ),

gR0∈ N0(R0), and e0 : Spec(R)→Spec(R0) will be as in Subsections 6.2, 6.2.1, and

6.2.6. We get a morphism mR e−10 : Spec(R0) → N of W (k)-schemes. Until the end let

Spf(RPD), Spf(RPD0 ), and F0 : Spec(RPD) → Spec(RPD

0 ) be as in Subsection 6.4.2. Weidentify naturally 1M with a Spf(W (k))-valued point of either Spf(R0) or Spf(R

PD0 ). The

morphism F0 defines naturally a morphism of formal schemes

Ff0 : Spf(RPD) → Spf(RPD

0 ).

If p ≥ 3, then F0 is an isomorphism of W (k)-schemes (cf. end of Subsection 6.4.2) and

therefore also Ff0 is a formal isomorphism over Spf(W (k)). We consider the ΦR0

-linearendomorphism

Φ := ϕ(−)⊗ ΦR0: EndW (k)(M)⊗W (k) R0 → EndW (k)(M)⊗W (k) R0.

Let (D,ΛD) be the principally quasi-polarized p-divisible group over N of (A,ΛA).

9.1. Generalization of the property 1.1 (f)

We have P−

G(ϕ) 6 Q, cf. Proposition 4.3.1 (a); thus P−

G(ϕ) normalizes F 0. At the level

of unipotent radicals we have N0 ⊳ U−

G(ϕ), cf. Proposition 2.3.3 applied over B(k). But

U−

G(ϕ) is also a closed subgroup scheme of the unipotent radical U−

GLM(ϕ) of P−

GLM(ϕ).

Thus N0 6 U−GLM

(ϕ). We have (cf. Example 2.6.4)

(35) Lie(U−GLM

(ϕ)) = ⊕s1, s2∈S(1,c), s1<s2HomW (k)(W (γs2)(M,ϕ),W (γs1)(M,ϕ)).

We conclude that N0 normalizes the ascending filtration (34) and moreover it acts triviallyon its factors. In particular, we have a filtration

Wγ1(M,ϕ)⊗W (k) R0 ⊆ Wγ2

(M,ϕ)⊗W (k) R0 ⊆ . . . ⊆Wγc(M,ϕ)⊗W (k) R0 =M ⊗W (k) R0

left invariant by gR0(ϕ⊗ ΦR0

). We consider the p-divisible group over Spec(R0)

D(z) := (mR e−10 )∗(D).

Let D(z)W (k) be the pull back of D(z) to W (k) via the W (k)-epimorphism R0 ։

R0/I0 = W (k); it is a p-divisible group over W (k) that is canonically identified with Dz

and therefore whose filtered Dieudonne module is (M,F 1, ϕ). For s ∈ S(1, c+1) let D(z)(s)k

be the p-divisible subgroup of the special fibre D(z)k ofD(z)W (k) whose Dieudonne module

156

is (M/Wγs−1(M,ϕ), ϕ) (cf. classical Dieudonne theory; see [Fo1, Ch. III, Prop. 6.1 iii)]).

If s ∈ S(2, c+ 1), then D(z)(s)k is a p-divisible subgroup of D(z)

(s−1)k (cf. loc. cit.).

9.1.1. Lemma. For s ∈ S(1, c+1) there exists a unique p-divisible subgroup D(z)(s)W (k) of

D(z)W (k) whose special fibre is D(z)(s)k .

Proof: The uniqueness part is implied by Grothendieck–Messing deformation theory (werecall that the maximal ideal of W2(k) has a canonical nilpotent divided power structure).As the cocharacter µ : Gm → G factors through L0

G(ϕ) (see Subsection 4.1.4) and therefore

also through P−GLM

(ϕ), from Example 2.6.4 we get that the direct summand Wγs−1(M,ϕ)

ofM is normalized by µ and therefore the triple (M/Wγs−1(M,ϕ), F 1/F 1∩Wγs−1

(M,ϕ), ϕ)

is a filtered Dieudonne module. Therefore, if D(z)(s)W (k) exists, then its Hodge filtration

is F 1/F 1 ∩Wγs−1(M,ϕ). Thus, if p ≥ 3 or if p = 2 and γ1 > 0, then the existence of

D(z)(s)W (k) is well known (cf. properties 1.2.1 (b) and (c)). We now consider the case when

p = 2 and γ1 = 0. Let D(1)W (k) := D(z)W (k). As D

(2)W (k) we take the kernel of a natural

epimorphism D(z)W (k) ։ (Qp/Zp)m1 , where m1 is the rank of Wγ1

(M,ϕ). By induction

on s ∈ S(3, c + 1), we construct D(z)(s)W (k) as a p-divisible subgroup of D(z)

(s−1)W (k) : as

D(z)(s−1)k is connected, the property 1.2.1 (c) implies the existence of a unique p-divisible

subgroup D(z)(s)W (k) of D(z)

(s−1)W (k) that lifts D(z)

(s)k .

9.1.2. Proposition. There exists a decreasing Newton polygon slope filtration

(36) 0 = D(z)(c+1) → D(z)(c) → . . . → D(z)(2) → D(z)(1) = D(z)

of p-divisible groups over Spec(R0) such that for every number s ∈ S(1, c) the factor

F (z)(s) := D(z)(s)/D(z)(s+1)

is the pull back of a p-divisible group F (z)(s)W (k) over W (k) which is cyclic and whose fibre

over k has all Newton polygon slopes equal to γs.

Proof: From [Fa2, Thm. 10] and [dJ, Lem. 2.4.4] we get that there exists a p-divisible

group D(z)(s) over Spec(R0) that lifts D(z)(s)W (k) and whose filtered F -crystal over R0/pR0

is

((M/Wγs−1(M,ϕ))⊗W (k) R0, (F

1/F 1 ∩Wγs−1(M,ϕ))⊗W (k) R0, gR0

(ϕ⊗ ΦR0),∇

(s)0 ).

Here ∇(s)0 is the unique connection on (M/Wγs−1

(M,ϕ)) ⊗W (k) R0 with respect towhich gR0

(ϕ ⊗ ΦR0) is horizontal, cf. [Fa2, Thm. 10]. But the Dieudonne F -crystal

D(D(z)(s)R0/pR0

) is uniquely determined by this filtered F -crystal, cf. [BM, Prop. 1.3.3].

Thus D(z)(s)R0/pR0

is uniquely determined, cf. [BM, Thm. 4.1.1] and the fact that

w1, . . . , wd is a finite p-basis of R0/pR0. As the ideal pI0 has a natural nilpotent dividedpower structure δv modulo each ideal pIv

0 (with v ∈ N) and as R0 = proj.lim.v∈NR0/pIv,

157

from Grothendieck–Messing deformation theory we get that D(z)(s) itself is uniquely de-termined.

We consider the W (k)-linear automorphism

a(s)M :M ⊕M/Wγs−1

(M,ϕ)→M ⊕M/Wγs−1(M,ϕ)

that takes (x, y) ∈ M ⊕M/Wγs−1(M,ϕ) to (x, x(s) + y), where x(s) is the image of x in

M/Wγs−1(M,ϕ). As∇0⊕∇(s)

0 is the unique connection on (M⊕M/Wγs−1(M,ϕ))⊗W (k)R0

with respect to which gR0(ϕ ⊗ ΦR0

) ⊕ gR0(ϕ ⊗ ΦR0

) is horizontal (cf. Fact 2.8.5) and

as a(s)M ⊗ 1R0

is fixed by gR0(ϕ ⊗ ΦR0

) ⊕ gR0(ϕ ⊗ ΦR0

), the connection ∇0 ⊕ ∇(s)0 on

(M ⊕ M/Wγs−1(M,ϕ)) ⊗W (k) R0 is fixed by a

(s)M ⊗ 1R0

. This means that ∇0 induces

naturally a connection on Wγs−1(M,ϕ)⊗W (k) R0 and that ∇

(s)0 is obtained from ∇0 via a

natural passage to quotients. Thus we have an epimorphism of F -crystals over R0/pR0

e(s)M : (M⊗W (k)R0, gR0

(ϕ⊗ΦR0),∇0) ։ ((M/Wγs−1

(M,ϕ))⊗W (k)R0, gR0(ϕ⊗ΦR0

),∇(s)0 ).

From the fully faithfulness part of [BM, Thm. 4.1.1], we get the existence of a

unique homomorphism e(s)R0/pR0

: D(z)(s)R0/pR0

→ D(z)R0/pR0of p-divisible groups over

R0/pR0 such that D(e(s)R0/pR0

) is defined naturally (cf. [BM, Prop. 1.3.3]) by e(s)M .

Due to the existence of the nilpotent divided power structures δv’s, from Grothendieck–

Messing deformation theory and [dJ, Lem. 2.4.4] we get that e(s)R0/pR0

and the embedding

D(z)(s)W (k) → D(z)W (k) lift uniquely to a homomorphism e(s) : D(z)(s) → D(z). From

Nakayama lemma we get that e(s) is a closed embedding. If s ∈ S(2, c+ 1), then e(s)M fac-

tors through e(s−1)M and this implies that D(e

(s)R0/pR0

) factors through D(e(s−1)R0/pR0

). From

this and the fully faithfulness part of [BM, Thm. 4.1.1], we get that e(s)R0/pR0

factors through

e(s−1)R0/pR0

. From this and Grothendieck–Messing deformation theory applied as above, we

get that e(s) itself factors through e(s−1). Thus the Newton polygon slope filtration (36)exists.

All Newton polygon slopes of F (z)(s)k = D(z)

(s)k /D(z)

(s+1)k are equal to γs, cf.

constructions. Let F (z)(s)W (k) be the unique p-divisible group over W (k) whose fil-

tered Dieudonne module is (W (γs)(M,ϕ), F 1 ∩ W (γs)(M,ϕ), ϕ), cf. properties 1.2.1(b) and (c). From the cyclic part of Theorem 1.4.3 and Fact 2.8.1 (b) we get that

F (z)(s)W (k) is cyclic. The filtered F -crystal of F (z)(s) is naturally defined by the triple

(W (γs)(M,ϕ) ⊗W (k) R0, (F1 ∩ W (γs)(M,ϕ)) ⊗W (k) R0, ϕ ⊗ ΦR0

) and therefore, in the

same way we argued the existence and the uniqueness of e(s), we get that F (z)(s) is the

pull back to R0 of F (z)(s)W (k).

The pull back of the descending Newton polygon slope filtration (36) via the isomor-phism e0 : Spec(R)→Spec(R0) of W (k)-schemes, is a descending Newton polygon slope

filtration of m∗R(D) whose factors are the pull backs to R of F (z)

(s)W (k)’s (with s ∈ S(1, c));

158

this represents the geometric generalization of the property 1.1 (f) to the context of thestandard Hodge situation (f, L, v).

9.1.3. Corollary. The connection ∇0 on M ⊗W (k) R0 is of the form ∇R0triv + Θ0, where

Θ0 ∈ u− ⊗W (k) (⊕di=1R0dwi) and where ∇R0

triv is as in Subsection 2.8.4.

Proof: Let Θ0 := ∇0 −∇R0

triv ∈ EndW (k)(M)⊗W (k) (⊕di=1R0dwi). As we have ∇0(tα) = 0

for all α ∈ J (cf. Proposition 6.2.2 (a)), we have Θ0 ∈ Lie(G) ⊗W (k) (⊕di=1R0dwi). But

for s ∈ S(1, c), the connection ∇0 induces a connection on Wγs−1(M,ϕ) ⊗W (k) R0 (cf.

proof of Proposition 9.1.2). As F (z)(s) is the pull back to R0 of F (z)(s)W (k) (cf. Proposition

9.1.2), the connections on (Wγs(M,ϕ)/Wγs−1

(M,ϕ))⊗W (k) R0 induced by ∇0 and ∇R0triv,

are equal. The last two sentences imply that Θ0 ∈ Lie(U−GLM

(ϕ))⊗W (k) (⊕di=1R0dwi), cf.

(35). But u− = Lie(G) ∩ Lie(U−GLM

(ϕ)) and thus we have Θ0 ∈ u− ⊗W (k) (⊕di=1R0dwi).

9.1.4. Proposition. The direct sum F (z)(0)W (k) := ⊕c

s=1F (z)(s)W (k) is isomorphic to the

unique cyclic p-divisible group F(0)0,W (k) over W (k) whose filtered Dieudonne module is

(M0 ⊗W (k(v)) W (k), F 10 ⊗W (k(v)) W (k), ϕ0 ⊗ σk).

Proof: Let l0 be the standard generator of Lie(Gm). The filtered Dieudonne modules of

F (z)(0)W (k) and F

(0)0,W (k) are (M,F 1, ϕ) and (M0 ⊗W (k(v))W (k), F 1

0 ⊗W (k(v))W (k), ϕ0 ⊗ σk)

(respectively) and therefore (to be compared with Subsection 4.1.1) they depend only onthe iterates of dµ(l0) and dµ0(l0) (respectively) under the actions of ϕ and ϕ0 (respectively).Thus from Corollary 6.3.3 we get that these two filtered Dieudonne modules are isomorphic.Therefore the Proposition follows from Fact 2.8.1 (b) applied to the direct summands of

F (z)(0)W (k) and F

(0)0,W (k) whose special fibres have only one Newton polygon slope.

9.1.5. Two Witt rings. Let RP be the p-adic completion of the commutative W (k)-

algebra obtained from R by adding the variables x1

pn

i , i ∈ S(1, d) and n ∈ N, that obey the

following rules: they commute and we have (x1

pn

i )p = x1

pn−1

i . Thus RP/pRP = (R/pR)perf

and RP is (W (k)-isomorphic to) the ringW (RP/pRP ) of Witt vectors with coefficients inthe perfect ring RP/pRP . Let ΦRP be the canonical Frobenius lift of RP ; it is compatible

with σ and takes each x1

pn

i to x1

pn−1

i . Similarly we define R0P and its canonical Frobeniuslift ΦR0P starting from R0 and wi’s (instead of R and xi’s).

9.1.6. Proposition. The principally quasi-polarized p-divisible group

(m∗R(D,ΛD))RP/pRP

is constant i.e., it is isomorphic to the pull back to RP/pRP of the principally quasi-polarized p-divisible group (m∗

R(D,ΛD))k of (A, λA)k.

Proof: We first show that the pull back (C0P ,ΛC0P) to R0P/pR0P of the principally quasi-

polarized F -crystal over R0/pR0 defined by C0 is constant i.e., it is the pull back of aprincipally quasi-polarized F -crystal over k. As R0P is the ring of Witt vectors with

159

coefficients in the perfect ring R0P/pR0P , we can speak about the action of (ϕ⊗ΦR0P )−1

on EndW (k)(M) ⊗W (k) R0P [1p] and therefore also on the abstract group GLM (R0P [

1p]).

For each positive integer i let

gi,R0P := (ϕ⊗ ΦR0P )−i(gR0

) ∈ U−

G(ϕ)(R0P ).

The belongness part is due to the facts that ϕ−1(u−) ⊆ u− and that gR0∈ N0(R0) ⊳

U−

G(ϕ)(R0) modulo I0 is 1M . As all Newton polygon slopes of (u−, ϕ−1) are positive, the

elementg∞,R0P := (

∏

i∈N

g−1i,R0P

)−1 ∈ U−

G(ϕ)(R0P )

is well defined and it fixes λM . We have an equality

g∞,R0P gR0(ϕ⊗ ΦR0P )(g∞,R0P )

−1 = ϕ⊗ ΦR0P

(to be compared with (19)). Thus the fact that (C0P ,ΛC0P) is constant follows once we

remark that the p-adic completion Ω∧R0P/W (k) of ΩR0P/W (k) is the trivial R0P -module.

By pull back via the isomorphism Spec(RP/pRP )→Spec(R0P/pR0P ) defined by e0 :Spec(R)→Spec(R0), we get that the principally quasi-polarized F -crystal over RP/pRPof (m∗

R(D,ΛD))RP/pRP is constant. Thus D((m∗R(D,ΛD))RP/pRP ) is as well constant

i.e., it is isomorphic to the pull back to RP/pRP of D((m∗R(D,ΛD))k). The k-algebra

RP/pRP is normal, integral, and perfect. Thus from [BM, Thm. 4.1.1] we get that(m∗

R(D,ΛD))RP/pRP is isomorphic to the pull back to RP/pRP of (m∗R(D,ΛD))k.

9.1.7. On F0. In this Subsection we make explicit the construction of the morphismF0 : Spec(RPD) → Spec(RPD

0 ) of W (k)-schemes introduced in Subsection 6.4.2. LetePD0 : Spec(RPD)→Spec(RPD

0 ) be the isomorphism of W (k)-schemes defined naturallyby the isomorphism e0 : Spec(R)→Spec(R0) of W (k)-schemes (see Subsection 6.2.6 andProposition 6.2.7 (a)). We now compute the endomorphism

E0 := F0 (ePD0 )−1 : Spec(RPD

0 ) → Spec(RPD0 )

of W (k)-schemes.

Due to the property 6.2.6 (b), E0 is defined naturally by the unique isomorphism(37)

(M⊗W (k)RPD0 , gR0

(ϕ⊗ΦRPD0

),∇0, λM , (tα)α∈J )→(M⊗W (k)RPD0 , ϕ⊗ΦRPD

0,∇

RPD0

triv , λM , (tα)α∈J )

that lifts 1M , where gR0∈ N0(R0) 6 N0(R

PD0 ) (see Subsection 6.2.1) and where we denote

also by ∇0 its natural extension to a connection on M ⊗W (k) RPD0 . As (u−, ϕ(−)) is a

Dieudonne module and as gR0∈ N0(R0) ⊳ U−

G(ϕ)(R0), the following element

uRPD0

:= (∏

i∈N∪0

(ϕ⊗ ΦRPD0

)i(gR0))−1 ∈ U−

G(ϕ)(RPD

0 )

160

is well defined and we have an equality

uRPD0

(gR0(ϕ⊗ ΦRPD

0))u−1

RPD0

= ϕ⊗ ΦRPD0.

Thus the element uRPD0

defines the isomorphism (37). It also defines naturally a morphism

vRPD0

: Spec(RPD0 ) → U−

G(ϕ)/P big ∩ U−

G(ϕ)

of W (k)-schemes, where P big is the normalizer of F 1 in GLM . By identifying Spec(R0)with the spectrum of the local ring of the completion of (the open subscheme N0 of)U−

G(ϕ)/P big ∩ U−

G(ϕ) along the W (k)-section defined by 1M ∈ U−

G(ϕ)(W (k)) and by re-

marking (as in Subsection 6.4.2) that vRPD0

factors through this spectrum and therefore

also through Spec(RPD0 ), we get that vRPD

0factors naturally through an endomorphism

Spec(RPD0 ) → Spec(RPD

0 ) of W (k)-schemes which is precisely E0.

9.2. Generalization of the property 1.1 (g)

Let V be a finite, discrete valuation ring extension of W (k). Let K := V [ 1p ]. Let

z ∈ N (V )

be a lift of y ∈ O(k). To generalize the property 1.1 (g) to the context of z, we canassume that k has a countable transcendental degree. Thus we can speak about a fixedO(v)-monomorphism eK : K → C. Let K = B(k) be the algebraic closure of K = eK(K)

in C. Let Gal(K) := Gal(K/K). As in the property 1.1 (g), the T ∗p ’s will denote duals of

Tate modules.

Based on Subsection 6.1.5 we will identify: (i) H1et(z

∗(A)K ,Zp) = T ∗p (z

∗(D)K) withL∗p, (ii) GLT∗

p (z∗(D)K) with GLL∗

p, and (iii) the maximal integral, closed subgroup scheme

of GLT∗p (z∗(D)

K) that fixes the p-component of the etale component of z∗E(GQ,X)(w

Aα ) for

all α ∈ J with GZp. We consider the p-adic Galois representation

ρz : Gal(K) → GLT∗p (z∗(D)

K)(Zp) = GLL∗

p(Zp).

Due to the identification of (iii), we have Im(ρz) 6 GZp(Zp).

The filtered Dieudonne module of F (z)(s)W (k) = D(z)

(s)W (k)/D(z)

(s+1)W (k) is cyclic (cf.

Proposition 9.1.4) and it is (W (γs)(M,ϕ), F 1 ∩ W (γs)(M,ϕ), ϕ). Thus there exists amaximal torus of GLW (γs)(M,ϕ) that normalizes F 1 ∩ W (γs)(M,ϕ) and whose Lie al-gebra is normalized by ϕ, cf. Lemma 2.8.2. Thus the p-adic Galois representation

ρs : Gal(B(k)) → GLT∗p (F (z)

(s)

K)of F (z)

(s)B(k) factors through the group of Zp-valued points

of a maximal torus Tmax,(s)Zp

of GLT∗p (F (z)

(s)

K). This holds even for p = 2 due to the fact

that F (z)(s)k has only one Newton polygon slope γs (to be compared with Lemma 6.1.7.1).

161

The Zariski closure of Im(ρs) in Tmax,(s)Zp

has a connected fibre over Qp (cf. [Wi, Prop.

4.2.3]) and therefore it is a subtorus T(0,s)Zp

of Tmax,(s)Zp

(cf. Fact 2.3.12).

The Zariski closure Sz of Im(ρz) in GZpnormalizes T ∗

p (z∗(e∗0(D(z)(1)/D(z)(s)))K) for

all numbers s ∈ S(1, c). The image of Sz(Zp) in

GLT∗p (F (z)

(s)

K)= GLT∗

p (z∗(e∗0(D(z)(1)/D(z)(s+1)))K)/T∗

p (z∗(e∗0(D(z)(1)/D(z)(s)))K)

is the torus T(0,s)Zp

. Thus the group scheme Sz is solvable.

Let P−z be the maximal, flat, closed subgroup scheme of GZp

that normalizes the

filtration (T ∗p (z

∗(e∗0(D(z)(1)/D(z)(s)))K))s∈S(1,c) of T∗p (z

∗(D)K) = L∗p. We have Sz 6 P−

z .

The Lie subalgebra Lie(P−

G(ϕ)) of EndW (k)(M) is normalized by ϕ(−) and µ. Thus

(Lie(P−

G(ϕ))⊗W (k) R0, (Lie(P

−

G(ϕ)) ∩ F i(EndW (k)(M)))i∈−1,0,1 ⊗W (k) R0, gR0

Φ,∇0)

is a filtered F -subcrystal of

End(C0)(1) := (EndW (k)(M)⊗W (k) R0, (Fi(EndW (k)(M))⊗W (k) R0)i∈−1,0,1, gR0

Φ,∇0).

Here (F i(EndW (k)(M)))i∈−1,0,1 is the intersection of the filtration of T (M) defined byF 1 with EndW (k)(M) = M ⊗W (k) M

∗. Thus, as e0 is an isomorphism of W (k)-schemes,

P−z,Qp

corresponds via Fontaine comparison theory for the abelian variety AB(k) to the

parabolic subgroup P−

G(ϕ)B(k) of GB(k) and therefore it is a parabolic subgroup of GQp

.

As P−z is the Zariski closure of P−

zQpin GZp

, it is a parabolic subgroup scheme of GZp(cf.

property 2.3.8 (a)).

The next Proposition is the geometric generalization of the property 1.1 (g) to thecontext of the standard Hodge situation (f, L, v).

9.2.1. Proposition. There exists a connected, solvable, smooth, closed subgroup schemeWz of GZp

such that the image of ρz : Gal(K) → GLL∗pis a subgroup of Wz(Zp).

Proof: Let U−z be the unipotent radical of P−

z . Let Wz be the flat, closed subgroup schemeof P−

z generated by Sz and U−z . We have a short exact sequence

(38) 1 → U−z → Wz → T

(0)Zp

→ 1,

where T(0)Zp

is the closed, flat subgroup scheme of∏

s∈S(1,c) T(0,s)Zp

that is the Zariski closure

of∏

s∈S(1,c) ρs. But T(0)Qp

is connected (cf. [Wi, Prop. 4.2.3]) and thus T(0)Zp

is a torus (cf.

Fact 2.3.12). Thus Wz is a connected, solvable, smooth, closed subgroup scheme of Pz andthus also of GZp

. But Im(ρz) 6 Sz(Zp) and therefore we have Im(ρz) 6Wz(Zp).

9.2.2. Lemma. We recall that z is a Sh-canonical lift. We have:

(a) The group Sz,Qpis a torus.

162

(b) If p ≥ 3 or if p = 2 and (M,ϕ) has no integral Newton polygon slopes, then Sz

itself is a torus.

Proof: From [Wi, Prop. 4.2.3] we get that Sz,Qpis connected. Let T be a maximal torus

of G such that (M,F 1, ϕ, T ) is an Sh-canonical lift, cf. Subsection 4.3.2. Let Tet,Qpbe

the torus of GLH1et(A

B(k),Qp) that corresponds to TB(k) via Fontaine comparison theory for

AB(k), cf. variant 6.1.9 (a). As Sz,Qp6 Tet, we get that Sz,Qp

is a torus. Thus (a) holds.

We check (b). The Zariski closure Tet of Tet,Qpin GZp

is a torus, cf. Lemma 6.1.7.1.

As Sz is the Zariski closure of the torus Sz,Qpin GZp

and thus in Tet, it is also a torus (cf.Fact 2.3.12).

9.3. Basic notations and definitions

We now include few notations, definitions, and simple properties to be used in thenext three Sections. The upper right index ∧ will refer to the p-adic completion. As inSection 5.3, let CG be the category of abstract groups. Let δp be the natural divided powerstructure of the ideal pW (k) of W (k). We consider the small crystalline site

Cris(Spec(k)/Spec(W (k)))

of Spec(k) with respect to (Spec(W (k)), pW (k), δp); we recall that it is the full subcategoryof CRIS(Spec(k)/Spec(W (k))) whose objects are pairs (Spec(k) → Z, δZ), where Z =Spec(RZ) is a local W (k)-scheme whose residue field is k and which is annihilated by somepositive, integral power of p and where δZ is a divided power structure on the maximalideal of RZ which is compatible with δp. We emphasize that Z has only one point (as themaximal ideal of RZ is the nilradical of RZ).

9.3.1. Definition. By a formal group on Cris(Spec(k)/Spec(W (k))) we mean a con-travariant functor

Cris(Spec(k)/Spec(W (k))) → CG

that is representable by a quadruple (Spf(RPD), eRPD , IRPD ,MRPD ), where RPD is as

in Subsection 2.8.6 and the three morphisms eRPD : Spf(W (k)) → Spf(RPD), IRPD :

Spf(RPD)→Spf(RPD), and MRPD : Spf((RPD ⊗W (k) RPD)∧) → Spf(RPD) of W (k)-

schemes define the identity section, the inverse, and the multiplication of the group law(respectively).

We will often identify such a functor with the quadruple (Spf(RPD), eRPD , IRPD ,MRPD ).

9.3.2. Example. We recall that we identify Spec(R0) with the spectrum of the local ringof the completion of N0 along its identity section, cf. Subsection 6.2.1. Thus Spf(RPD

0 )extends uniquely to a quadruple

FPD0 := (Spf(RPD

0 ), 1M , IRPD0,MRPD

0)

that defines a commutative formal group on Cris(Spec(k)/Spec(W (k))) whose evaluationat the pair (Spec(k) → Z, δZ) is the subgroup

Spec(RPD0 )(Z) = Spec(R0)(Z) = Ker(N0(Z) → N0(k))

163

of N0(Z).

9.3.3. Extra notations. LetDy := Spf(R)

be the formal scheme defined by the completion R of the local ring of y in NW (k). For a liftzi ∈ N (W (k)) of y ∈ N (k) = NW (k)(k), with i as an index, we denote also by zi the formalclosed embeddings Spf(W (k)) → Dy and Spf(W (k)) → Spf(RPD) defined naturally by zi.

Let Gad =∏

i∈I Gi be the product decomposition into absolutely simple, adjoint

group schemes overW (k). Let S− be the solvable, smooth, closed subgroup scheme of Gad

such that (Lie(S−), ϕ(−)) is the Dieudonne module of the standard non-positive p-divisiblegroup of (Lie(Gad), ϕ, Gad) (see Definitions 5.1.1). Let s− := Lie(S−). Let

T (1)

be the unique maximal torus of S− whose Lie algebra is W (1)(s−, ϕ(−)) = W (0)(s−, ϕ)

(i.e., whose Lie algebra is normalized by ϕ). If the torus T(0)Zp

of GZpis as in the proof

of Proposition 9.2.1, then the torus T (1) corresponds naturally to Im(T(0)Zp

→ GadZp

) viaFontaine comparison theory with Zp coefficients for the abelian scheme A.

Let r(1) ∈ N ∪ 0 be the rank of t(1) := Lie(T (1)). We have U−

G(ϕ) ⊳ S− and S−

is the semidirect product of T (1) and U−

G(ϕ). Thus we can identify T (1) with S−/U−

G(ϕ).

Therefore we have a direct sum decomposition

s− = u− ⊕ t(1)

of t(1)-modules. Let F 0 := F 0(Lie(Gad)) be the Lie algebra of the centralizer of µ in Gad.

LetD+W (k) andD

−W (k) be the positive and the negative (respectively) p-divisible groups

over W (k) of (M,ϕ, G) or equivalently of (Lie(Gad), ϕ, Gad) (see Definitions 5.1.1). FromProposition 6.3.1 we get that D+

W (k) can be identified with the positive p-divisible group

D+0,W (k) over W (k) of C0⊗k. Thus the first formal Lie group F1 of C0⊗k (see Sections 5.3

and 5.4) is isomorphic to the formal Lie group of D+W (k). Therefore our notations match

with the ones of Section 1.7.

9.3.4. On Ff0 . Let zi ∈ N (W (k)). Let ni ∈ Ker(N0(W (k)) → N0(k)) be the unique

element such that the Shimura filtered F -crystal attached to zi is (M,ni(F1), ϕ, G), cf.

Fact 6.4.1. We refer to ni as the canonical unipotent element of zi.

From the very definition of the morphism F0 : Spec(RPD) → Spec(RPD0 ) of Subsection

6.4.2 of W (k)-schemes, we get that the map

Ff0(Spf(W (k))) : Spf(RPD)(Spf(W (k))) → Spf(RPD

0 )(Spf(W (k)))

takes zi ∈ Spf(RPD)(Spf(W (k))) to the element of Spf(RPD0 )(Spf(W (k))) defined by ni.

164

9.4. Theorem (the existence of canonical formal Lie group structures for I0 ≤ 1)

We recall that k = k and that z ∈ N (W (k)) is an Sh-canonical lift. If I0 ≤ 1, then:

(a) There exists a unique formal Lie group Fz = (Dy, z, IR,MR) whose pull back FPDz

via the morphism Spf(RPD) → Dy is such that the morphism Ff0 : Spf(RPD) → Spf(RPD

0 )defines a homomorphism

FPDz → FPD

0

of formal groups on Cris(Spec(k)/Spec(W (k))).

(b) The formal Lie group Fz is isomorphic to the first formal Lie group F1 of C0 ⊗ k.

Proof: The case I0 = 0 is trivial as we have Dy = Spf(W (k)). Thus we can assume thatI0 = 1. We prove (a). The uniqueness of MR and IR follows from Corollary 6.4.3. Forinstance, the uniqueness of IR is implied by the following two facts:

(i) for each positive integer l, the isomorphism I1p

R,l : Spec(RPD/I [l][ 1

p])→Spec(RPD/I [l][ 1

p])

defined by IR is equal to F1p

0,l I1p

R0,l (F

1p

0,l)−1, where F

1p

0,l is as in Corollary 6.4.3 and where

I1p

R0,l: Spec(RPD

0 /I[l]0 [ 1p ])→Spec(RPD

0 /I[l]0 [ 1p ]) is the isomorphism defined by IRPD

0of Ex-

ample 9.3.2;

(ii) IR is uniquely determined by (I1p

R,l)l∈N (being a restriction of the automorphism

of Spec(B(k)[[x1, . . . , xd]]) defined naturally by the inductive limit of the I1p

R,l’s).

We start the checking of the existence of MR and IR. We consider an isomorphismIy : (Lie(Gad), ϕ)→(Lie(Gad

W (k)), ϕ0⊗σ) of adjoint Shimura Lie F -crystals as in Proposition

6.3.1. We have Iy(u+) = Lie(U+

0,W (k)). The nilpotent class of U+0,W (k) is I0 = 1, cf. the

very definition of I0 in Section 1.6. This implies that U+0,W (k) is commutative. Thus u+

is commutative. Due to the oppositions of Section 2.7, u− is also commutative (cf. theproperty 3.6.1 (c)).

We have a short exact sequence of filtered F -crystals over R0/pR0

(39) 0 → (u− ⊗R0, (u− ∩ F 0)⊗R0,Φ,∇0) → (s− ⊗R0, (s

− ∩ F 0)⊗R0, gR0Φ,∇0) →

→ (t(1) ⊗R0, t(1) ⊗R0,Φ,∇0) → 0.

In (39) the tensorizations are over W (k) and the connections ∇0 are the ones inducednaturally (cf. Corollary 9.1.3) by the connection ∇0 on M ⊗W (k) R0 that defines thefiltered F -crystal C0 over R0/pR0. As the Lie algebra u− is commutative, for the first andthird factor we do not have to insert gR0

∈ U−

G(ϕ)(R0). As in Lemma 9.1.1 and Proposition

9.1.2 we argue that to (39) corresponds a short exact sequence

(40) 0 → µµµr(1)

p∞ → Dbigext → D−

R0→ 0

of p-divisible groups over Spec(R0) that is uniquely determined by the property that mod-ulo I0 it splits (this last condition holds automatically if either p ≥ 3 or p = 2 and

165

(u−, ϕ(−)) does not have Newton polygon slope 0). The push forward of (40) via an

epimorphism e(1) : µµµr(1)

p∞ ։ µµµp∞ is of the form

(41) 0 → µµµp∞ → Dext → D−R0

→ 0.

The Cartier dual of (41) is

(42) 0 → (D−R0

)t → Dtext → Qp/Zp → 0.

Such an epimorphism e(1) : µµµr(1)

p∞ ։ µµµp∞ is defined by an element

t(1) ∈ t(1)

that is fixed by ϕ (equivalently we have ϕ(−)(t(1)) = pt(1)) and such that the W (k)-module

t(1)(1) :=W (k)t(1)

is a direct summand of t(1). We say e(1) is a good epimorphism if for all elements i ∈ Ithe component of t(1) in Lie(Gi) generates a direct summand of Lie(Gi) (i.e., the imageIm(t(1)(1) → Lie(Gi)) is a direct summand of Lie(Gi)). As t(1)(1) is a direct summandof t(1), Dextk is a minimal non-positive p-divisible group of (M,ϕ, G) in the sense ofDefinitions 5.1.1. From now on we will take e(1) to be a good epimorphism. This impliesthat the action of t(1) on u− is a W (k)-linear automorphism, cf. property 5.2.2 (b).

We recall that N0 ⊳ U−

G(ϕ) (see Section 9.1) and that µ : Gm → G normalizes both

Lie(N0) and u−. Thus we have a direct sum decomposition u− = Lie(N0)⊕ (u−∩ F 0). LetU−−

G(ϕ) (resp. N−

0 ) be the unipotent radical of the maximal parabolic subgroup scheme

of GLu−⊕t(1)(1) (resp. of GLLie(N0)⊕t(1)(1)) that normalizes Lie(N0). Thus U−−

G(ϕ) (resp.

N−0 ) acts trivially on Lie(N0) and on u− ⊕ t(1)(1)/Lie(N0) (resp. and on (Lie(N0) ⊕

t(1)(1))/Lie(N0)). We have a natural closed embedding monomorphism N−0 → U−−

G(ϕ).

As N0 acts on both u− ⊕ t(1)(1) and Lie(N0) ⊕ t(1)(1) via inner conjugation, we have anatural homomorphism

h0 : N0 → N−0 .

It is well known that dh0 : Lie(N0) → Lie(N−0 ) is such that for x ∈ Lie(N0) and y ∈

Lie(N0) ⊕ t(1)(1) we have dh0(x)(y) = [x, y]. Taking y = t(1) and using the fact thatthe action of t(1) on u− (and therefore also on Lie(N0)) is a W (k)-linear automorphism,we get that the image of dh0 is a direct summand of the Lie algebra of GLLie(N0)⊕t(1)(1)

and therefore also of Lie(N−0 ). But both N0 and N−

0 have relative dimensions d. Thusdh0 is an isomorphism. This implies that the fibres of h0 are finite and therefore h0 is aquasi-finite morphism. But N0,B(k) has no non-trivial finite subgroup. Thus h0,B(k) is aclosed embedding and therefore (by reasons of dimensions) an isomorphism. Thus h0 isalso birational. From Zariski main theorem (see [Gr1, Thm. (8.12.6)]) we get that h0 isan open embedding. As N−

0 has connected fibres we get that h0 is an isomorphism.

As gR0∈ N0(R0) is a universal element (see Section 6.2) and as h0 : N0 → N−

0 is anisomorphism, the Kodaira–Spencer map of (41) is injective modulo I0 (the argument for

166

this is the same as for K0 in the proof of Proposition 6.2.2 (c)). Thus the image of theKodaira–Spencer map of (41) is a direct summand (of the corresponding codomain) thathas rank equal to d = rkW (k)(Lie(N0)) = rkW (k)(u

−/(u−∩ F 0)) and therefore equal to the

dimension of (D−k )

t. From this and the dimension part of Lemma 5.3.3, we get that (42)is the universal extension of Qp/Zp by (D−

W (k))t. Thus (41) (i.e., the Cartier dual of (42))

is the universal extension of D−W (k) by µµµp∞ and therefore Spf(R0) has a natural structure

of a formal Lie group (Spf(R0), 1M , I(1)R0,M

(1)R0

) over Spf(W (k)).

We check that this formal Lie group structure on Spf(R0) does not depend on the

choice of a good epimorphism e(1) : µµµr(1)

p∞ ։ µµµp∞ . We consider another good epimorphism

e(2) : µµµr(1)

p∞ ։ µµµp∞ that is defined by an element t(2) ∈ t(1) which satisfies the same

conditions as t(1) does. Let t(1)(2) :=W (k)t(2). We get a W (k)-linear isomorphism

i−1,2 : u− ⊕ t(1)(1)→u− ⊕ t(1)(2)

via the rules: t(1) is mapped to t(2) and [t(1), x] is mapped to [t(2), x] for all x ∈ u−.Thus, as i−1,2 ⊗ 1R0

commutes with gR0Φ, it induces an isomorphism between the short

exact sequence of filtered F -crystals over R0/pR0 that defines (40) and the analogous shortexact sequence of filtered F -crystals over R0/pR0 we get by working with t(2) instead oft(1). This implies that the isomorphism class of the short exact sequence 0 → D(D−

R0) →

D(Dext) → D(µµµp∞) → 0 defined naturally by (40) does not depend on the choice of e(1)(cf. also [BM, Prop. 1.3.3]). From this, [BM, Thm. 4.1.1], and Grothendieck–Messingdeformation theory, we get that the isomorphism class of (40) itself does not depend on

e(1). Thus the formal Lie group structure (Spf(R0), 1M , I(1)R0,M

(1)R0

) on Spf(R0) does notdepend on the choice of a good epimorphism e(1).

We know that (D−W (k))

t is naturally identified with D+W (k), cf. Subsection 5.2.4 and

Section 5.4. Let (Dy, z, IR,MR) be the formal Lie group structure on Dy that is the trans-

port of structure of (Spf(R0), 1M , I(1)R0,M

(1)R0

) via the isomorphism ef0 : Spf(R)→Spf(R0)associated naturally to e0 : Spec(R)→Spec(R0). As (42) is the universal extension of

Qp/Zp by (D−W (k))

t, the formal Lie group (Spf(R0), 1M , I(1)R0,M

(1)R0

) over Spf(W (k)) is nat-

urally identified (cf. Lemma 5.3.3) with the formal Lie group of (D−W (k))

t and therefore

also of D+W (k). Thus the formal Lie group (Dy, z, IR,MR) is isomorphic to F1 i.e., (b)

holds.

To end the proof of (a) we only have to check that the pull back FPDz of (Dy, z, IR,MR)

via Spf(RPD) → Dy is such that the morphism Ff0 defines a morphism FPD

z → FPD0

that is a homomorphism of formal groups on Cris(Spec(k)/Spec(W (k))). Let z1, z2 ∈Dy(Spf(W (k))). Let z3 := z1 + z2 ∈ Dy(Spf(W (k))) be the sum with respect to MR.Let z4 := −z1 ∈ Dy(Spf(W (k))) be the inverse of z1 with respect to IR. Let s−(1) :=u− ⊕ t(1)(1). We consider the short exact sequence

(43) 0 → (u−, u− ∩ F 0, ϕ(−)) → (s−, F 1i , ϕ(−)) → (t(1), t(1), ϕ(−)) → 0

of filtered Dieudonne modules that is obtained by pulling back (39) via e0zi : Spec(W (k)) →Spec(R0) (the resulting connections being trivial, are ignored). Let

(44) 0 → (u−, u− ∩ F 0, ϕ(−)) → (s−(1), F 1i (1), ϕ(−)) → (t(1)(1), t(1)(1), ϕ(−)) → 0

167

be the short exact sequence obtained from (43) via a natural pull back through themonomorphism (t(1)(1), t(1)(1), ϕ(−)) → (t(1), t(1), ϕ(−)) of filtered Dieudonne modules.Here F 1

i is a direct summand of s− and F 1i (1) := F 1

i ∩ s−(1).

For each number i ∈ S(1, 4) let ni ∈ Ker(N0(W (k)) → N0(k)) be the canonicalunipotent element of zi, cf. Subsection 9.3.4. As both (39) and (43) are obtained naturallyfrom C0 via passages to Lie subalgebras of EndW (k)(M) and to quotients, we get that F 1

i

is the inner conjugate of s− ∩ F 0 through ni (equivalently, it is h0(W (k))(ni)(F1i ), where

h0(W (k)) : N0(W (k))→N0(W (k)) is the isomorphism defined by h0). Let si ∈ pLie(N0)be such that we have an equality ni = 1M + si. Let l0 ∈ t(1) be the image of the standardgenerator of Lie(Gm) through dµ. We denote by ∗(l0) ∈ s− the inner conjugate of l0 viaan element ∗ ∈ N0(W (k)); we have ∗(l0) = h0(W (k))(∗)(l0) = l0 + [∗ − 1M , l0]. Thus F 1

i

is the 0 eigenspace of the action of ni(l0) on s− (here we have 0 and not 1 as ϕ(−) is ptimes ϕ). Let li := ni(l0)− l0 = [si, l0] ∈ u− ⊆ s−.

But (44) for i = 3 is obtained by performing the following three operations in theorder listed (cf. Yoneda addition law for short exact sequences):

(O) take the direct sum of (44) for i ∈ 1, 2, take a pull back via the diagonal mapt(1)(1) → t(1)(1)⊕ t(1)(1), and take a push forward via the addition map u− ⊕ u− → u−.

These three operations preserve the natural action of N0 on s− and therefore they aredefined byW (k)-linear maps that preserve the Lie brackets. Thus F 1

3 (1) is the 0 eigenspaceof the action on s−(1) of

l0 + l1 + l2 = l0 + [s1, l0] + [s2, l0] = (1M + s1 + s2)(l0) = n1n2(l0).

Thus F 13 (1) = n3(s

−(1) ∩ F 0) is n1n2(s−(1) ∩ F 0). As h0 is an isomorphism, we get that

(45) n3 = n1n2.

By replacing the triple (n3, n1, n2) with the triple (1M , n1, n4), we get that 1M = n1n4;

thus we have n4 = n−11 . Therefore Ff

0 defines a morphism FPDz → FPD

0 that inducesa homomorphism of groups FPD

z (Spf(W (k))) → FPD0 (Spf(W (k))). As the W (k)-valued

points of Spec(RPD) are Zariski dense, we conclude that Ff0 defines a morphism FPD

z →FPD

0 that is a homomorphism of formal groups on Cris(Spec(k)/Spec(W (k))). Therefore(a) also holds.

9.5. Complements

In this Section we include different complements to Section 9.2 and to Theorem 9.4.We first consider a product decomposition D+

W (k) =∏

j∈J D+j,W (k) into p-divisible groups

over W (k) such that for each element j ∈ J the special fibre D+j,k of D+

j,W (k) has only

one Newton polygon slope γ+j ∈ (0, 1] (see the property 5.2.2 (c) and Section 5.4). LetF1 =

∏j∈J F1,j be the corresponding product decomposition into formal Lie groups over

Spf(W (k)) (see Subsection 5.3.1 and 5.4). If I0 ≤ 1, we consider the product decompositionFz =

∏j∈J Fz,j that corresponds to the isomorphism of Theorem 9.4 (b) and the product

decomposition F1 =∏

j∈J F1,j.

168

9.5.1. Proposition. Let m0(1) be the multiplicity of the Newton polygon slope 1 of(Lie(Gad

W (k(v))), ϕ0) (equivalently of (Lie(U+0 ), ϕ0)). We assume that p = 2 and I0 ≤ 1.

Then the Sh-canonical lifts of y (when viewed as Spf(W (k))-valued points of Dy) are theSpf(W (k))-valued torsion points of Fz and they are all of order 1 or 2. Moreover, thenumber ny of Theorem 1.6.2 is 2m0(1).

Proof: The subgroup

T := Ker(FPDz (Spf(W (k))) → FPD

0 (Spf(W (k)))) 6 FPDz (Spf(W (k))) = Fz(Spf(W (k)))

is finite (cf. Proposition 6.4.6) and the torsion subgroup of FPD0 (Spf(W (k))) is trivial.

Thus T is the torsion subgroup of FPDz (Spf(W (k))). The Spf(W (k))-valued points of Fz

that correspond to Sh-canonical lifts of y, are precisely the elements of T (cf. Subsection9.3.4). Thus ny = |T|. We check that T is isomorphic to (Z/2Z)m0(1).

Let Tj be the torsion subgroup of Fz,j(Spf(W (k))). We have T =∏

j∈J Tj. Letzj ∈ Tj be a non-identity element. To zj corresponds a non-split short exact sequence 0 →

D+j,W (k) → D

+(0)j,W (k) → Qp/Zp → 0 of p-divisible groups over W (k) (cf. Lemma 5.3.3) with

the property that its corresponding short exact sequence of filtered Dieudonne modulessplits. Thus the Newton polygon slope γ+j is 1, cf. property 1.2.1 (c). As γ+j = 1, D+

j,W (k)

is of multiplicative type and therefore F1,j is a formal torus. A one dimensional formaltorus over Spf(W (k)) has precisely two Spf(W (k))-valued torsion points. As Fz,j→F1,j,we get that zj is of order 2.

But m0(1) is the multiplicity of the Newton polygon slope 1 of (Lie(Gad), ϕ) (cf.Proposition 6.3.1) and thus it is the number of µµµp∞ copies of the decomposition (cf. Lemma

5.1.3 ) of D+W (k) into circular indecomposable p-divisible groups. Thus

∏j∈J, γ+

j=1 Fz,j is

a m0(1) dimensional formal torus over Spf(W (k)). Therefore T→(Z/2Z)m0(1) and thusny = |T| = 2m0(1).

9.5.2. Simple facts and remarks. (a) The following five statements are equivalent:

(i) we have N0 = U−

G(ϕ);

(ii) all Newton polygon slopes of (Lie(Gad), ϕ) (equivalently of (Lie(G), ϕ)) are inte-gral;

(iii) all Newton polygon slopes of (Lie(GadW (k(v))), ϕ0) (equivalently of (Lie(GW (k(v))), ϕ0))

are integral;

(iv) the residue field k(vad) of the prime vad of E(GadQ , X

ad) divided by v, is Fp;

(v) the p-divisible group D+W (k) (equivalently the p-divisible group D+

0,W (k) overW (k)

of C0 ⊗ k, cf. Proposition 6.3.1) is of multiplicative type.

The equivalence (ii)⇔(iii) follows from Proposition 6.3.1. The equivalence (iii)⇔(iv)is obtained as in the beginning of Section 7.1. The equivalences (iv)⇔(v) and (v)⇔(i) aretrivial. Thus, if k(vad) = Fp, then (Dy, z, IR,MR) is a formal torus over Spf(W (k)).

(b) We include an example. We consider the case when

GadR →SO(2ℓ+ 1)2R ×R SO(2, 2ℓ− 1)2R

169

and GadZp

is the Weil restriction of a split, absolutely simple, adjoint group scheme over

W (Fp4) (of Bℓ Lie type). Thus d = dimC(X) = 2(2ℓ − 1) (see [He, Ch. X, §6, TableV]) and I0 = 1 (see Subsections 3.4.7 and Corollary 3.6.3). The group scheme Gad

W (k) has

four simple factors permuted by the automorphism of GadW (k) defined naturally by σ and

denoted also by σ. If the two simple factors of GadW (k) in which µ0,W (k) has a non-trivial

image, are not (resp. are) such that σ takes one directly to the other one, then fromExample 5.2.7 applied with n = 4 and d = 2 (resp. and d = 1), we get that D+

W (k) is the

direct sum of 2(2ℓ − 1) (resp. of 2ℓ − 1) copies of the circular indecomposable p-divisiblegroup D 1

2(resp. D2,4) over W (k) introduced at the end of Example 5.2.7. Thus D+

W (k) is

self-dual and therefore isomorphic to D−W (k).

(c) We can define directly the short exact sequence (42) as follows. Let E(1) be adirect supplement of t(1) in Lie(L0

Gad(ϕ)) normalized by ϕ and by a maximal torus of Gad

that contains T (1). Let E−(1) := u− ⊕ E(1). It is easy to check that N0 normalizes E−(1)

(the argument for this is similar to the one used in Subsection 1.2.5 to get an isomorphismq1 : U−

1 →U−+1 and to the one used to check properties (15) and (16)). We have a short

exact sequence compatible with ϕ of N0-modules

(46) 0 → Lie(P−

Gad(ϕ))/E−(1) → Lie(Gad)/E−(1) → u+ → 0.

By extending (46) to R0, by allowing gR0(ϕ⊗ΦR0

) to act on this extension, and by insertingthe connections ∇0 and the natural filtrations, we get the analogue of (39) that defines theCartier dual of (40) modulo the identification of D+

W (k) with (D−W (k))

t.

(d) We assume that I0 = 1. If p = 2 and z1 ∈ Dy(Spf(W (k))) defines an Sh-canonicallift of y different from z, then z1 is of order 2 (cf. Proposition 9.5.1) and therefore fromthe uniqueness part of Theorem 9.4 (a) we get that Fz1 is obtained naturally from Fz viaa translation through z1.

(e) Let p = 2. As (M,F 1, ϕ) is cyclic (cf. end of Theorem 1.4.3), there exists a unique

cyclic p-divisible group DcyclicW (k) over W (k) whose filtered Dieudonne module is (M,F 1, ϕ)

(cf. Fact 2.8.1 (a)). We decompose DcyclicW (k) = D0,cyclic ⊕ D

(0,1),cyclicW (k) ⊕ D1,cyclic

W (k) , where

D0,cyclicW (k) is etale, D

(0,1),cyclick has no integral Newton polygon slopes, and D1,cyclic

W (k) is of

multiplicative type. Due to this decomposition and the property 1.2.1 (c), λM is the

crystalline realization of a unique principal quasi-polarization of DcyclicW (k) . Each automor-

phism aM of (M,ϕ, G) normalizes F 1 (see Subsection 4.1.5); thus the property 1.2.1 (c)

also implies that aM is the crystalline realization of an automorphism of DcyclicW (k) . If the

p-divisible group Dz of z∗(A) is isomorphic to DcyclicW (k) , then z is the unique Sh-canonical

lift of y with this property (cf. Theorem 1.6.3 (d)). If we have a standard moduli PELsituation (f, L, v,B), then N is a moduli space of principally-polarized abelian varietiesendowed with endomorphisms (see [Zi1], [RZ], and [Ko2]) and therefore from Serre–Tatedeformation theory we get that z exists. One can use [Va7, Thm. 1.2] to show that infact such a z always exists. If I0 = 1 and if such a z is assumed to exist, then as in the

170

classical context of Section 1.1 one works only with Fz (i.e., if ny > 1 one disregards theFz1 ’s, where z1 is as in (d)).

(f) For i ∈ 0, 1 let A+i be the i-th group of automorphisms of the positive p-divisible

group D+0,W (k) over W (k) of C0⊗ k with respect to C0 ⊗ k, cf. Subsection 5.1.4. The group

A+2 := Aut(D+

0,W (k)) acts on F1. If I0 = 1, then the isomorphism Fz→F1 of Theorem

9.4 (b) is canonical up to a composite with an automorphism of F1 defined by an elementof A+

0 , cf. Remark 5.2.3 and the proofs of Proposition 6.3.1 and Theorem 9.4; thus fori ∈ 0, 1, 2 we can speak about the canonical subgroup Az,i of automorphisms of Fz thatcorresponds to A+

i .

(g) We have natural homomorphisms

(47) End(D+0,W (k)) → End(F1) and End(D+

0,k) → End(F1 ×Spf(W (k) Spf(k))

that are unique up to conjugation through A+0 . There exist endomorphisms eFr0 ∈

End(D+0,k) that are defined by endomorphisms cFr0 of (Lie(U+

0,W (k)), ϕ0 ⊗ σ) whose im-

ages are equal to (ϕ0 ⊗ σ)(Lie(U+0,W (k))). This is so as the fact that D+

0,W (k) is cyclic

(see Lemma 5.1.3 and 5.4) implies that there exists a W (k)-basis B0 for Lie(U+0,W (k)) with

respect to which the matrix representation of ϕ0 ⊗ σ has entries in Zp (in fact we canchoose B0 such that all these entries are either 0 or p); thus we can choose cFr0 such that itacts on elements of B0 in the same way as ϕ0 ⊗ σ does. Each endomorphism eFr0 is uniqueup to a (left) composite with an element of Aut(D+

0,k).

If I0 = 1 and k(vad) 6= Fp, then we do not know if we can use the images of (47) todefine good Frobenius lifts of Dy.

(h) Sections 9.1 and 9.2 and Theorem 9.4 do not extend to the case when z is a U-canonical lift of a U-ordinary point y ∈ Nk(v)(k) that is not a Sh-ordinary point (the mostone can do is to identify suitable formal subschemes of Dy that have canonical formal Liegroup structures; to be compared with Subsection 9.8.2 below). However, in future workwe will also generalize the property 1.1 (i) to the context of U-canonical lifts. Moreover,one can define canonical unipotent elements of W (k)-valued lifts of y as in Subsection9.3.4.

(i) We have natural variants of Theorem 9.4 for k 6= k but the resulting formal Liegroups Fz will depend on y ∈ O(k) (to be compared with Subsection 1.2.5).

(j) The descending Newton polygon slope filtration (36) depends on the injective mapf : (GQ, X) → (GSp(W,ψ), S) and therefore it is not intrinsically associated to the Sh-ordinary point y ∈ N (k). Thus from many points of view it is more appropriate to worknot with (36) but with the analogous ascending Newton polygon slope filtration of thefiltered F -crystal over R0/pR0 defined naturally by

(48) (s− ⊗W (k) R0, (s− ∩ F 0)⊗W (k) R0, gR0

Φ).

In future work we will show that this new ascending Newton polygon slope filtration ofF -crystals attached to (48) is intrinsically associated to y and this will allow us to achievethe project 1.9 (c).

171

(k) One proves formula (1) by following the last part of the proof of Theorem 9.4.

(l) If the group GadQ is non-trivial and if all simple factors of (Gad

Q , Xad) are of A2, Bℓ,

Cℓ, orDRℓ type, then we always have I0 = 1 (cf. the structure of the number of elements |E|

of the set E of Subsection 3.4.8). We add here that each simple factor (GQ, X) of (GadQ , X

ad)

that is of DRℓ type, gives birth to simple factors of (Lie(Gad

W (F)), ϕ0 ⊗ σF, GadW (k)) that are

either trivial (i.e., are as in Subsection 3.2.2) or are such that their type is of the form(Dℓ, 1, η1, . . . , ηn) with η1, . . . , ηn ∈ 0, 1. For ℓ ≥ 5 this is so, as for any element x ∈ Xthe image of dµx in a simple factor f of Lie(GC) has a centralizer in f whose derived Liealgebra is of Dℓ−1 Lie type (cf. the very definition of the DR

ℓ type in Subsection 1.5.6).For ℓ = 4 we will only add here that one defines the DR

4 type (see [De3, Table 2.3.8]) sothat the mentioned property holds (for all potential standard Hodge situations).

9.6. Ramified valued points

Let V and K be as in Section 9.2. Let eV , ReV , IeV , Spf(ReV ), ΦReV , and dV :Spf(V ) → Spf(ReV ) be as in Subsection 2.8.7. Let z1, z2 : Spf(V ) → Dy. If I0 ≤ 1,let z3 := z1 + z2 be the addition with respect to Fz (of Theorem 9.4 (a)). There existformal morphisms vi : Spf(ReV ) → Dy such that we have vi dV = zi, cf. Subsection2.8.7. If I0 ≤ 1, we can assume that v3 = v1 + v2 is the addition with respect to Fz (thismakes sense as for each positive integer s, ReV /p

sReV is an inductive limit of artinian,local k-algebras). We denote also by zi (resp. by vi) the V -valued (resp. the ReV -valued)point of N defined by zi (resp. by vi), where i ∈ 1, 2, 3 if I0 ≤ 1 and i ∈ 1, 2if I0 ≥ 2. Let (Mi,ReV , F

1i,ReV

,Φi,ReV ,∇i,ReV , λMi,ReV, (ti,α)α∈J ) be the pull back of

the pair (CR, (tRα )α∈J ) introduced in Section 6.2 via vi (to be compared with Subsection

6.5.5). We emphasize that the direct summand F 1i,ReV

of Mi,ReV depends on vi while

(Mi,ReV ,Φi,ReV ,∇i,ReV , λMi,ReV, (ti,α)α∈J ) and F 1

i,ReV⊗ReV V depend only on zi. There

exists a unique isomorphism (to be compared with Subsection 6.5.5)

Ji : (Mi,ReV [1

p],Φi,ReV ,∇i,ReV )→(M ⊗W (k) ReV [

1

p], ϕ⊗ ΦReV ,∇

ReVtriv )

of F -isocrystals over ReV /pReV whose pull back to an isomorphism of F -isocrystals overk is defined by 1M [ 1p ]

. There exists no element of T (M)⊗W (k) IeV [1p] fixed by psϕ⊗ΦReV

(s ∈ 0, 1). Thus the isomorphism Ji takes λMi,ReVto λM and ti,α to tα for all α ∈ J .

9.6.1. Proposition. There exists a unique element ni ∈ N0(ReV [1p ]) such that we have

Ji(F1i,ReV

[ 1p ]) = ni(F1 ⊗W (k) ReV [

1p ]). If I0 ≤ 1, then we moreover have

(49) n3 = n1n2.

Proof: We can redefine (Mi,ReV , F1i,ReV

,Φi,ReV ,∇i,ReV , λMi,ReV, (ti,α)α∈J ) as the pull

back of the pair (C0, (tα)α∈J ) introduced in Subsection 6.2.1 via the morphism vi,0 :Spec(ReV ) → Spec(R0) of W (k)-schemes defined by vi and by the isomorphism e0 :Spec(R)→Spec(R0), cf. property 6.2.6 (b). Thus we have a natural isomorphism

J ′i : (M⊗W (k)ReV , F

1⊗W (k)ReV , gi,ReV (ϕ⊗ΦReV ),∇′i,ReV

)→(Mi,ReV , F1i,ReV

,Φi,ReV ,∇i,ReV )

172

that takes ti,α to tα for all α ∈ J and λMi,ReVto λM , where ∇′

i,ReVis the connection on

M ⊗W (k)ReV that is the natural extension of ∇0 and where the element gi,ReV ∈ G(ReV )is computed as follows. We write gi,ReV = gcorri,ReV

g0,ReV , where g0,ReV ∈ N0(ReV ) isthe composite of vi,0 with the element gR0

∈ N0(R0) of Subsection 6.2.1 and where the

correction factor gcorri,ReV∈ G(ReV ) is computed as follows. There exists an ReV -linear

automorphism hcorri,ReVof (M + 1

pF1) ⊗W (k) σReV such that for each element x ∈ M we

have

hcorri,ReV(x⊗ 1) :=

∑

i1,i2,...,id∈N∪0

(

d∏

l=1

∇0(d

dwl)il)(x⊗ 1)

d∏

l=1

sillil!,

where sl := ΦReV (wi,l) − wpi,l ∈ pReV (cf. [De4, Formula (1.1.3.4)] or [Fa2, §7]); here the

element wi,l ∈ ReV is the image of wl ∈ R0 via the morphism vi,0 of W (k)-schemes. Letga0,ReV

: (M + 1pF 1) ⊗W (k) σReV →M ⊗W (k) ReV be the ReV -linear isomorphism defined

by g0,ReV (ϕ ⊗ ΦReV ). Then (cf. the standard way of pulling back filtered F -crystals; seethe last two references, etc.) we have:

gcorri,ReVga0,ReV

= ga0,ReVhcorri,ReV

: (M +1

pF 1)⊗W (k) σReV →M ⊗W (k) ReV .

From the shape of the connection ∇0 (see Corollary 9.1.3) and from the fact that N0 6

U−

G(ϕ), we easily get that gcorri,ReV

∈ U−

G(ϕ)(ReV ). Thus gi,ReV ∈ U−

G(ϕ)(ReV ).

Let P := G ∩ P big; it is the normalizer of F 1 in G. As in Subsection 9.1.7, we define

ui,ReV := (∏

s∈N∪0

(ϕ⊗ ΦReV )s(gi,ReV ))

−1 ∈ U−

G(ϕ)(ReV [

1

p]);

the ReV [1p]-linear automorphism Ji (J ′

i ⊗ 1ReV [ 1p ]) of M ⊗W (k) ReV [

1p] is precisely

the element ui,ReV . Thus Ji(F1i,ReV

[ 1p]) = ui,ReV (F

1 ⊗W (k) ReV [1p]). As U−

G(ϕ) is

the semidirect product of U−

G(ϕ) ∩ P and N0, we can write ui,ReV = niu

0i,ReV

, where

ni ∈ N0(ReV [1p ]) and where u0i,ReV

∈ U−

G(ϕ)(ReV [

1p ]) normalizes F 1 ⊗W (k) ReV [

1p ]. Thus

Ji(F1i,ReV

[ 1p ]) = ni(F1 ⊗W (k) ReV [

1p ]) i.e., the element ni exists. The uniqueness of ni

follows from the fact that the product morphism N0 ×W (k) P → G is an open embedding(see proof of Lemma 3.1.2).

We now assume that I0 ≤ 1. To check formula (49), we consider a commutativediagram (see Subsection 2.8.7)

Spec(ReV )[peV ]−→ Spec(ReV )yvi(peV )

yvi

Spec(RPD)can−→ Spec(R).

Due to this diagram and the uniqueness properties of Ji and of the isomorphism Jy ofSubsection 6.4.2, we get that Ji⊗ReV [ 1p ] [peV ]ReV [

1p ] is the pull back of Jy via vi(peV )B(k).

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Thus (49) holds after pull backs through B(k)-valued points of Spec(ReV [1p ]) that are

defined by W (k)-valued points of Spec(ReV ) which factor through [peV ] : Spec(ReV ) →Spec(ReV ), cf. formula (45). As these B(k)-valued points are Zariski dense in Spec(ReV ),formula (49) holds.

9.6.2. Definition. We refer to [Fo3] for filtered modules over K. Let cV : Spec(V ) →Spec(ReV ) be as in Subsection 2.8.7. As (Mi,ReV ,Φi,ReV ,∇i,ReV , λMi,ReV

, (ti,α)α∈J ) and

F 1i,ReV

⊗ReV V depend only on zi, the element ni,K := c∗K(ni) ∈ N0(K) also depends only

on zi. We call the quintuple (M [ 1p ], ϕ, ni,K(F 1 ⊗W (k) K), λM , (tα)α∈J ) as the polarized

filtered module with tensors of z∗i,K(A,ΛA, (wAα )α∈J ) or of zi ∈ N (V ). Moreover, we call

ni,K ∈ N0(K) as the canonical unipotent element of zi ∈ N (V ).

9.6.3. A formula. We assume that I0 ≤ 1. The following formula implied by (49)

(50) n3,K = n1,Kn2,K

is the rational form of (45) for Spf(V )-valued points of Fz. It is independent on Fz (i.e.,for p = 2 it does not depend on the choice of an Sh-canonical lift z of y).

If z2 = −z1 (i.e., if z3 is the identity element), then from formula (50) we get thatn3,K is the identity element and thus that n2,K = n−1

1,K .

We have the following Corollary to Theorem 1.6.3 and Proposition 9.6.1.

9.6.4. Corollary. We assume that k = F. We use the notations and the assumptionsof the first paragraph of Theorem 1.6.3 (c) for our present Sh-ordinary point y ∈ O(k) =O(F) ⊆ N (F) and for a lift z0 ∈ N (V ) of it. Then A0 = z∗0(A) has complex multiplication.

Proof: Let n0 ∈ N0(K) be the canonical unipotent element of z0 ∈ N (V ). Its existenceimplies that the element w0 of Subsection 6.5.5 is the identity element. Thus from Example6.5.6 (a) we get that the condition W holds for z0. Therefore the Corollary follows fromTheorem 1.6.3 (c).

We also have the following Corollary to Section 9.2 and Proposition 9.6.1.

9.6.5. Corollary. We use the notations of Section 9.2, with z ∈ N (V ) replaced by eitherz1 ∈ N (V ) or z ∈ N (W (k)). Let n1,K be the canonical unipotent element of z1. If I0 ≤ 1(resp. if I0 ≥ 2), then the following three statements are equivalent:

(a) the solvable group Sz1,Qpis a torus;

(b) the Spf(V )-valued point z1 ∈ Fz(Spf(V )) is torsion (resp. there exists a positiveinteger m such that we have nm

1,K = 1M⊗W (k)K);

(c) we have n1,K = 1M⊗W (k)K .

Proof: As N0(K) has only one torsion element, the implication (b)⇒(c) follows from for-mula (50) if I0 ≤ 1 (resp. is obvious if I0 ≥ 2). The quintuple (M [ 1p ], ϕ, F

1[ 1p ], λM , (tα)α∈J )

is the polarized filtered module with tensors of z∗(A,ΛA, (wAα )α∈J ). If (c) holds,

then (M [ 1p ], ϕ, n1,K(F 1 ⊗W (k) K), λM , (tα)α∈J ) is the extension to K of the quintuple

174

(M [ 1p ], ϕ, F1[ 1p ], λM , (tα)α∈J ) and therefore the p-adic Galois representations over Qp de-

fined by ρz1 and by the restriction of ρz to Gal(K), can be identified. Thus Sz1,Qpis

isomorphic to Sz,Qpand therefore (cf. Lemma 9.2.2 (a)) it is a torus. Thus (c)⇒(a).

We are left to check that (a) implies (b). It suffices to show that (a) implies (c) andthat (c) implies (b). We check that (a) implies (c). Both z and z1 factor through the sameconnected component of NW (k). Thus we have a canonical identification

H1et(z

∗(A)B(k)

,Zp) = H1et(z

∗1(A)K,Zp).

Due to the descending Newton polygon slope filtration of Proposition 9.1.2, under thisidentification the parabolic subgroup schemes P−

z1and P−

z of GZpand the quotient torus

T(0)Zp

of Wz1 and Wz (see (38)) are also identified. Therefore also Wz1,Qpand Wz,Qp

arenaturally identified. Under the identificationWz1,Qp

=Wz,Qp, the maximal tori Sz1,Qp

andSz,Qp

of Wz,Qpare Wz,Qp

(Qp)-conjugate (cf. [Bo, Ch. V, Thm. 19.2]). Thus the p-adicGalois representations over Qp defined by ρz1 and by the restriction of ρz to Gal(K), areGQp

(Qp)-conjugate. Therefore we have an isomorphism τi from (M [ 1p ], ϕ, ni,K(F 1 ⊗W (k)

K), λM , (tα)α∈J ) to the extension to K of (M [ 1p ], ϕ, F1[ 1p ], λM , (tα)α∈J ), cf. Fontaine

comparison theory. The isomorphism τi is defined by an element of G(B(k)) fixed by ϕ andthus (cf. Subsection 4.1.5 and Lemma 4.1.6) by an element of G(B(k)) that also normalizesF 1[ 1p ]. Therefore we have n1,K(F 1 ⊗W (k) K) = F 1 ⊗W (k) K and thus n1,K = 1M⊗W (k)K .

Therefore (c) holds (i.e., (a) implies (c)).

We check that (c) implies (b). Obviously this implication holds if I0 ≥ 2. Thus wecan assume I0 ≤ 1 and therefore we can appeal to Theorem 9.4 and to the decompositionFz =

∏j∈J Fz,j. The subset of Fz(Spf(V )) formed by those Spf(V )-valued points z2

for which we have n2,K = 1M⊗W (k)K , is a subgroup TV of Fz(Spf(V )) (cf. Subsection9.6.3). We check that TV is a finite group and therefore a torsion group. For j ∈ Jlet Tj,V be the image of TV in Fz,j(Spf(V )) via the natural projection Fz ։ Fz,j . Werecall Fz,j is the formal Lie group of D+

j,W (k) and therefore of extensions of Qp/Zp by

D+j,W (k), cf. Lemma 5.3.3. Thus each element x ∈ Tj,V defines uniquely a Zp-lattice Lx

of Tp(D+j,B(k))[

1p ]⊕ Tp(Qp/Zp)[

1p ] that is normalized by Gal(K), that has Tp(D

+j,B(k)) as a

direct summand, and that maps onto Tp(Qp/Zp). Thus, if x0 is a generator of Tp(Qp/Zp),then Lx is determined uniquely by each element x1 ∈ Lx that maps onto x0. We write

x1 = x0 + pmxx′0, where x′0 ∈ Tp(D+j,B(k)) \ pTp(D

+j,B(k)) and where mx ∈ Z.

As Lx is Gal(K)-invariant and has Tp(D+j,B(k)) as a direct summand, if mx < 0 then x′0

modulo p−mx is fixed by Gal(K). As the Newton polygon slope γ+j belongs to the interval

(0, 1], the p-divisible group D+j,k over k is connected and thus the number of such pairs

(mx, x′0) with mx < 0 is finite. If mx ≥ 0, then Lx = Tp(D

+j,B(k))⊕ Tp(Qp/Zp). Thus the

set of such Zp-lattices Lx is finite and therefore Tj,V is a finite group. This implies thatTV is a finite group. As (c) holds, the element z1 ∈ TV has finite order i.e., (b) holds.

9.6.6. Remarks. (a) Both Corollary 9.6.5 (c) and the part of the proof of Corollary9.6.5 that involves p-adic Galois representations over Qp, are equivalent to the existence

175

of a Q–isogeny between (z∗(D,ΛD))V and z∗1(D,ΛD) under which (the p-component ofthe etale component of) the Hodge cycles z∗E(GQ,X)(w

Aα ) and z∗1,E(GQ,X)(w

Aα ) get identified

for all α ∈ J . Thus, as the special fibres of (A, λA)V and z∗1(A,Λ) coincide, Corollary9.6.5 (c) is equivalent to the existence of a Z[ 1

p]-isogeny between (A, λA)V and z∗1(A,Λ)

under which z∗E(GQ,X)(wAα ) corresponds to z∗1,E(GQ,X)(w

Aα ) for all α ∈ J . Therefore, if

k = F, then the three statements of Corollary 9.6.5 are also equivalent to the statementthat z∗1(A) has complex multiplication (cf. Theorem 1.6.3 (a) applied to A).

(b) If p = 2 and I0 ≤ 1, then the torsion subgroup of Fz(Spf(V )) is intrinsicallyassociated to Dy (i.e., it does not depend on choice of the Sh-canonical lift z of y, cf.Corollary 9.6.5 (c)).

9.7. A functorial property

One can list many functorial properties of the formal Lie groups of Theorem 9.4. Theyare all rooted in Subsection 5.4.1, Theorem 9.4 (a), and properties (49) and (50). To beshort, we include here only one simple but useful property.

We assume that f : (GQ, X) → (GSp(W,ψ), S) factors through an injective mapf ′ : (G′

Q, X′) → (GSp(W,ψ), S) and that the triple (f ′, L, v′) is also a standard Hodge

situation, where v′ is the prime of E(G′Q, X

′) divided by v. We use an upper right primeindex to denote the analogues of different notations we used for (f, L, v) but obtainedworking with (f ′, L, v′). Let C′

0 be the abstract Shimura F -crystal of (f ′, L, v′), let N ′

be the normalization of the Zariski closure of Sh(G′Q, X

′)/H ′p in MO(v′)

, where H ′p :=

G′Q(Qp) ∩ Kp, etc. We also assume that N0 is equal to the Newton polygon N′

0 of C′0.

Let z′ ∈ N ′(W (k)) and y′ ∈ N ′(k) be the composites of z ∈ N (W (k)) and y ∈ N (k)(respectively) with the natural morphism N → N ′. As N0 = N′

0, y′ is an Sh-ordinary

point. Let G′ be the reductive closed subgroup scheme of GLM such that (M,ϕ, G′) isthe Shimura F -crystal attached to y′. We have G 6 G′ and thus F 1 is a lift of (M,ϕ, G′).From the uniqueness part of the statement 1.4.3 (d) we get that (M,F 1, ϕ, G′) is an Sh-canonical lift. Thus z′ is an Sh-canonical lift. The composite of µ : Gm → G with theclosed embedding monomorphism G → G′, is the canonical cocharacter of (M,ϕ, G′) (cf.Subsection 4.3.2). This implies that we have a natural closed embedding monomorphismh0,y : N0 → N ′

0.

The natural morphism N → N ′ gives birth to a morphism ey : Dy → Dy′ betweenthe formal schemes over Spf(W (k)) associated to the completions of the local rings of yand y′ in NW (k) and N ′

W (k) (respectively). We have:

9.7.1. Proposition. We assume that the Newton polygons N0 and N′0 are equal. Then the

formal morphism hy : Dy → Dy′ is a formally closed embedding. If moreover I0 = I′0 = 1,then hy defines a monomorphism Fz → Fz′ of formal Lie groups over Spf(W (k)).

Proof: The first part follows from the fact that the morphism i1 : Spec(R) → Spec(R1) ofW (k)-schemes (see Section 6.2) is a closed embedding (see Proposition 6.2.7 (c)) and fromthe analogue of this but applied in the context of y′ ∈ N ′(k). As N0 is a closed subgroup

176

scheme of N ′0, from Subsection 9.3.4 we easily get that we have a commutative diagram

Spf(RPD)hPDy−→ Spf(R′PD)

yFf0

yF′f0

Spf(RPD0 )

hPD0,y−→ Spf(R′PD

0 );

here hPDy and hPD

0,y are defined by hy and h0,y (respectively).

We now assume that I0 = I′0 = 1. We know that Ff0 and F′f

0 define homo-morphisms of formal groups on Cris(Spec(k)/Spec(W (k))) (cf. Theorem 9.4 (a)) andthat hPD

0,y is a monomorphism of formal groups on Cris(Spec(k)/Spec(W (k))). From

this and Corollary 6.4.3 we easily get that hPDy defines a homomorphism of formal

groups on Cris(Spec(k)/Spec(W (k))) (even if p ≥ 2); for p ≥ 3 we actually have

hPDy = (F′f

0 )−1 hPD0,y Ff

0 . Thus hy defines a monomorphism Fz → Fz′ .

We do not stop to check here that the monomorphism Fz → Fz′ is defined (up tonatural identifications) by the monomorphism f+

t : D+W (k) → D+′

W (k) constructed as in

Subsection 5.4.1.

9.8. Four operations for the case I0 ≥ 2

In this Section we describe the four main operations in the study of Dy in the casewhen I0 ≥ 2. We refer to future work for the formalism that uses these operations to definenatural formal Lie group structures on Dy even if I0 ≥ 2. Let ΦRPD

0be the Frobenius lift

of RPD0 defined by ΦR0

(see the conventions of Subsection 2.8.6). Until the end we willassume that I0 ≥ 2.

9.8.1. The set J . Let T be a maximal torus of G such that (M,F 1, ϕ, T ) is an Sh-canonical lift, cf. Subsection 4.3.2. Thus µ : Gm → G factors through T (cf. Subsection4.3.2) and ϕ normalizes Lie(T ). Let T (1) := Im(T → Gad). Let J be the finite set whoseelements j parameterize all smooth, unipotent, closed subgroup schemes U−

j of U−

G(ϕ) that

have the following three properties:

(a) the group scheme U−j is normalized by T ;

(b) the Lie algebra u−j := Lie(U−j ) is left invariant by ϕ(−);

(c) N0,j := N0 ∩ U−j is a product of Ga group schemes.

Let n0,j := Lie(N0,j). One can check that (a) implies (c) but this is irrelevant for

what follows. Due to Proposition 6.3.1, the number |J | is independent of the Sh-ordinarypoint y ∈ N (k).

9.8.2. Operation I: canonical formal subschemes of Dy. We can assume that for

each element j ∈ J the Frobenius lift ΦR0of R0 leaves invariant the ideal defining the

closed subscheme Spec(R0,j) of Spec(R0) that is the spectrum of the local ring of thecompletion of N0,j along its identity section. Thus we can speak about the Frobenius liftΦR0,j

of R0,j that is defined naturally by ΦR0via a passage to quotients. Let I0,j be the

177

ideal of R0,j that is the image of I0 in R0,j; we have R0,j/I0,j = W (k). The pull back ofC0 via the closed embedding Spec(R0,j) → Spec(R0) is of the form

C0,j := (M ⊗W (k) R0,j, F1 ⊗W (k) R0,j, gR0,j

(ϕ⊗ ΦR0,j),∇0,j, λM ),

where gR0,j: Spec(R0,j) → N0,j 6 N0 is the universal morphism of N0,j . Let FPD

0,j bethe formal group on Cris(Spec(k)/Spec(W (k))) obtained as in Example 9.3.2 but workingwith N0,j instead of N0. Let

Dy,j = Spf(Rj)

be the formal closed subscheme of Dy that is the pull back of Spf(R0,j) via the isomorphism

ef0 : Spf(R)→Spf(R0) over Spf(W (k)) defined naturally by e0 : Spec(R)→Spec(R0). Theformal morphism z : Spf(W (k)) → Dy factors through each Dy,j = Spf(Rj) and theresulting morphism z : Spf(W (k)) → Dy,j will be denoted as well by z.

Let u+j be the opposite of u−j with respect to T . We identify naturally u+j and

u−j with Lie subalgebras of Lie(Gad). Let F 1(Lie(Gad)) be the maximal Lie subalge-

bra of Lie(Gad) on which µ acts via the inverse of the identical character of Gm. LetD+

j,W (k) be the connected p-divisible group over W (k) whose filtered Dieudonne module is

(u+j , F1(Lie(Gad))∩ u+j , ϕ). Let D

−j,W (k) be the p-divisible group over W (k) whose filtered

Dieudonne module is (u−j , F0 ∩ u−j , ϕ(−)) (see Subsection 9.3.3 for F 0).

9.8.2.1. Lemma. There exist natural isomorphisms (D+j,W (k))

t→D−j,W (k).

Proof: We consider a direct sum decomposition (u+j , ϕ) = ⊕j′∈D(j)(u+j′ , ϕ) such that

for each element j′ ∈ D(j), the T(1)B(k)-submodules of u+j [

1p ] that are of rank 1, are

permuted transitively by ϕ (cf. property 5.2.1 (v) and Section 5.4). We emphasizethat the set D(j) is not always a subset of J , cf. the exceptional cases of the prop-erty 5.2.1 (i). Let (u−j , ϕ(−)) = ⊕j′∈D(j)(u

−j′ , ϕ(−)) be the decomposition that corre-

sponds to (u+j , ϕ) = ⊕j′∈D(j)(u+j′ , ϕ) via the opposition with respect to the torus T (1).

To these decompositions of (u+j , ϕ) and (u−j , ϕ(−)) correspond product decompositions

D+j,W (k) =

∏j′∈D(j)D

+j′,W (k) and D−

j,W (k) =∏

j′∈D(j)D−j′,W (k) (respectively). But there

exist isomorphisms (D+j′,W (k))

t→D−j′,W (k), cf. Subsection 5.2.4. From this the Lemma

follows.

9.8.2.2. Standard formal closed embeddings. Let

J2 := (j1, j2) ∈ J2|U−j1

6 U−j2.

Let (j1, j2) ∈ J2. As U−j1

6 U−j2, we also have N0,j1 6 N0,j2 . Due to the property 9.8.1

(a), the W (k)-submodule u−j1 of u−j2 is a direct summand and therefore also n0,j1 is adirect summand of n0,j2 . Thus to the closed embedding monomorphism N0,j1 → N0,j2

corresponds naturally a formal closed embedding ij1,j2 : Dy,j1 → Dy,j2 .

9.8.2.3. Theorem. Let j ∈ J be an element for which the following condition holds:

178

(i) the group scheme U−j is commutative and there exists an element t(1)j ∈ Lie(T (1))

that is fixed by ϕ and such that the action of t(1)j on u−j is a W (k)-linear automorphism.

We recall that k = k and that z ∈ N (W (k)) is an Sh-canonical lift. We have:

(a) There exists a unique formal Lie group Fz,j = (Dy,j, z, IRj,MRj

) whose pull

back FPDz,j via the morphism Spf(RPD

j ) → Dy,j has the property that the morphism Ff0 :

Spf(RPD) → Spf(RPD0 ) induces naturally a homomorphism FPD

z,j → FPD0,j of formal groups

on Cris(Spec(k)/Spec(W (k))).

(b) The formal Lie group Fz,j is isomorphic to the formal Lie group F1,j of D+j,W (k).

Proof: The proof of this Theorem is the same as the proof of Theorem 9.4, the roles of u−

and t(1) being replaced by the ones of u−j and t(1)j.

9.8.3. Operation II: quotient formal subschemes. Let (j1, j2) ∈ J2. As n0,j1 isa direct summand of n0,j2 , the quotient vector group scheme N0,j2,j1 := N0,j2/N0,j1 isalso a product of Ga group schemes. Let Spec(R0,j2,j1) be the spectrum of the local ringof the completion of N0,j2,j1 along its identity section. Pulling back the natural W (k)-epimorphism Spec(R0,j2) ։ Spec(R0,j2,j1) via e0 : Spec(R)→Spec(R0), we get a naturalW (k)-epimorphism Spec(Rj2) ։ Spec(Rj2,j1). To it corresponds a formal epimorphismqj2,j1 : Dy,j2 ։ Dy,j2,j1 . Its composite with z : Spf(W (k)) → Dy,j2 will be denoted as wellby z : Spf(W (k)) → Dy,j2,j1 .

9.8.3.1. Partial sections. Let j3 ∈ J be such that u−j1 ⊕ u−j3 is a direct summand of u−j2 .Thus also n0,j1 ⊕ n0,j3 is a direct summand of n0,j2 . Therefore we have a natural closedembedding monomorphism N0,j1 ×W (k) N0,j3 6 N0,j2 of vector group schemes such thatthe quotient group scheme N0,j2/[N0,j1×W (k)N0,j3 ] is also a product of Ga group schemes.Thus we can view N0,j3 as a closed subgroup scheme of N0,j2,j1 . This implies that

ej1,j2,j3 := qj2,j1 ij3,j2 : Dy,j3 → Dy,j2,j1

is a formal closed embedding (and thus a partial section of qj2,j1).

9.8.3.2. Theorem. Let (j1, j2) ∈ J2. We assume that the quotient group schemeU−j2/U−

j1is commutative and that there exists an element t(1)j2,j1 ∈ Lie(T (1)) that is

fixed by ϕ and such that the action of t(1)j2,j1 on u−j2/u−j1

is a W (k)-linear automor-

phism. Let Ff0,j2,j1

: Spf(RPDj2,j1

) → Spf(RPD0,j2,j1

) be the morphism induced naturally by

F0 : Spf(RPD) → Spf(RPD0 ). We have:

(a) There exists a unique formal Lie group Fz,j2,j1 = (Dy,j2,j1 , z, IRj2,j1,MRj2,j1

)

whose pull back FPDz,j2,j1

via the morphism Spf(RPDj2,j1

) → Dy,j2,j1 is such that the morphism

Ff0,j2,j1

: Spf(RPDj2,j1

) → Spf(RPD0,j2,j1

) defines naturally a homomorphism FPDz,j2,j1

→ FPD0,j2,j1

of formal groups on Cris(Spec(k)/Spec(W (k))).

(b) As (u+j1 , F1(Lie(Gad))∩u+j1 , ϕ) is a direct summand of (u+j2 , F

1(Lie(Gad))∩u+j2 , ϕ),

there exists a unique p-divisible group D+j2,j1,W (k) over W (k) such that D+

j2,W (k) is isomor-

phic to D+j1,W (k) ⊕D+

j2,j1,W (k). The formal Lie group Fz,j2,j1 is isomorphic to the formal

Lie group of D+j2,j1,W (k) = D+

j2,W (k)/D+j1,W (k).

179

Proof: The proof of this Theorem is the same as the proof of Theorem 9.4, the roles of u−

and t(1) being replaced by the ones of u−j2/u−j1

and t(1)j2,j1 .

9.8.4. Operation III: actions by formal Lie groups. For j ∈ J , let Z(Uj) be themaximal flat, closed subscheme of Uj that commutes with Uj . Let

Z(J2) := (j1, j2) ∈ J2|U−j1

6 Z(U−j2).

Let (j1, j2) ∈ Z(J2) be such that the condition 9.8.2.3 (i) holds for j1. Let Spf(R0,j1×j2) :=Spf(R0,j1) ×Spf(W (k)) Spf(R0,j2). Let ΦR0,j1×j2

be the Frobenius lift of R0,j1×j2 that isnaturally induced by ΦR0,j1

and ΦR0,j2. Let I0,j1×j2 be the ideal of R0,j1×j2 generated

naturally by I0,j1 and I0,j2 ; we have R0,j1×j2/I0,j1×j2 = W (k). We have two naturalprojections

s1,j1×j2 : Spec(R0,j1×j2) ։ Spec(R0,j1) and s2,j1×j2 : Spec(R0,j1×j2) ։ Spec(R0,j2).

Let gR0,j1×j2:= (gR0,j1

s1,j1×j2)(gR0,j2 s2,j1×j2) ∈ N0(R0,j1×j2) and

C0,j1×j2 := (M ⊗W (k) R0,j1×j2 , F1 ⊗W (k) R0,j1×j2 , gR0,j1×j2

(ϕ⊗ ΦR0,j1×j2),∇0,j1×j2 , λM ).

The connection ∇0,j1×j2 on M ⊗W (k) R0,j1×j2 is uniquely determined by the requirementthat gR0,j1×j2

(ϕ⊗ΦR0,j1×j2) is horizontal with respect to ∇0,j1×j2 , cf. [Fa2, Thm. 10]. As

in Proposition 6.2.2 (a) we argue that ∇0,j1×j2 annihilates tα for all α ∈ J . Thus fromTheorem 6.2.4 applied to (R0,j1×j2 ,ΦR0,j1×j2

, I0,j1×j2) we get the existence of a uniquemorphism v0,j1×j2 : Spec(R0,j1×j2) → Spec(R0) such that: (i) at the level of rings mapsI0 to I0,j1×j2 , and (ii) we have an isomorphism v∗0,j1×j2

(C0, (tα)α∈J )→(C0,j1×j2 , (tα)α∈J )

that lifts the identity automorphism of (M,F 1, ϕ, λM , (tα)α∈J ).

Pulling back v0,j1×j2 via e0 : Spec(R)→Spec(R0), we get a morphism vj1×j2 :Spec(Rj1×j2) = Spec(Rj1)×W (k) Spec(Rj2) → Spec(R). Let Ij1×j2 be the ideal of Rj1×j2

that corresponds naturally to I0,j1×j2 via e0; we have Rj1×j2/Ij1×j2 = W (k). To vj1×j2

corresponds a formal morphism

tj1×j2 : Spf(Rj1×j2) = Dy,j1 ×Spf(W (k)) Dy,j2 → Dy = Spf(R)

over Spf(W (k)) and a morphism vPDj1×j2

: Spec(RPDj1×j2

) → Spec(RPD) of W (k)-schemes.

9.8.4.1. Theorem. We assume that (j1, j2) ∈ Z(J2) and that the condition 9.8.2.3 (i)holds for j1. Then tj1×j2 : Dy,j1 ×Spf(W (k)) Dy,j2 → Dy factors through Dy,j2 and theresulting morphism

tj1×j2 : Fz,j1 ×Spf(W (k)) Dy,j2 → Dy,j2

defines an action of the formal Lie group Fz,j1 on the formal scheme Dy,j2 .

Proof: The unique isomorphism

(M ⊗W (k) RPD0,j1×j2

, gR0,j1×j2(ϕ⊗ ΦRPD

0,j1×j2

),∇0,j1×j2 , λM , (tα)α∈J )→

180

(M ⊗W (k) RPD0,j1×j2

, ϕ⊗ ΦRPD0,j1×j2

,∇RPD

0,j1×j2

triv , λM , (tα)α∈J )

that lifts 1M , is defined by the element (to be compared with Subsection 9.1.7)

uRPD0,j1×j2

:= (∏

i∈N∪0

(ϕ⊗ ΦRPD0,j1×j2

)i(gR0,j1×j2))−1 ∈ U−

j2(RPD

0,j1×j2).

As U−j1

6 Z(U−j2), the element uRPD

0,j1×j2

is the product of (∏

i∈N∪0(ϕ⊗ΦRPD0,j1

)i(gR0,j1))−1

sPD1,j1×j2

with (∏

i∈N∪0(ϕ⊗ ΦRPD0,j2

)i(gR0,j2))−1 sPD

2,j1×j2, where the morphisms

sPD1,j1×j2 : Spec(RPD

0,j1×j2) → Spec(RPD0,j1) and sPD

2,j1×j2 : Spec(RPD0,j1×j2) → Spec(RPD

0,j2)

of W (k)-schemes are defined naturally by s1,j1×j2 and s2,j1×j2 (respectively). This im-plies that the morphism F0 vPD

j1×j2factors through Spec(RPD

0,j2). From this and Corollary

6.4.3 we get that for every positive integer l, the morphism Spec(RPDj1×j2

/I[l]j1×j2

[ 1p]) →

Spec(RPD/I [l][ 1p ]) defined naturally by tj1×j2 factors through the closed subscheme

Spec(RPDj2

/I [l][ 1p ]). Thus tj1×j2 factors through Dy,j2 .

To check that tj1×j2 defines an action of the formal Lie group Fz,j1 on the formalscheme Dy,j2 , we will use Spf(W (k))-valued points. Let z1, z2 ∈ Dy,j1(Spf(W (k)))and z3 ∈ Dy,j2(Spf(W (k))). Let n1, n2 ∈ Ker(N0,j1(W (k)) → N0,j1(k)) and n3 ∈

Ker(N0,j2(W (k)) → N0,j2(k)) be such that for i ∈ S(1, 3) the lift of (M,ϕ, G) de-fined by zi is ni(F

1) (cf. Subsection 9.3.4). The lift of (M,ϕ,G) defined by z2z3 :=tj1×j2(Spf(W (k)))(z2, z3) ∈ Dy,j2(Spf(W (k))) is n2n3(F

1), cf. the mentioned productproperty of uRPD

0,j1×j2

.

Thus the lifts of (M,ϕ,G) defined by (z1z2)z3 and z1(z2z3) are both equal ton1n2n3(F

1). If p ≥ 3 or if p = 2 and (M,ϕ) has no integral Newton polygon slopes,then from Proposition 6.4.5 we get

(51) (z1z2)z3 = z1(z2z3) ∈ Dy,j2(Spf(W (k))).

Using Corollary 6.4.3 one checks (as usual) that formula (51) holds even if p = 2 and(M,ϕ) has integral Newton polygon slopes. Thus tj1×j2 is an action of the formal Liegroup Fz,j1 on the formal scheme Dy,j2 .

9.8.5. Remark. One can get a slightly more general form of Theorem 9.8.4.1 by combin-ing Theorems 9.8.3.2 and 9.8.4.1.

9.8.6. Operation IV: bilinear commutating formal morphisms. Let

J3 := (j1, j2, j3) ∈ J3|0 6= [u−j1 , u−j2] ⊆ u−j3.

Let (j1, j2, j3) ∈ J3 be such that: (i) the condition 9.8.2.3 (i) holds for j1, j2, and j3, and(ii) the group schemes U−

j1, U−

j2, and U−

j3are commutative. Similar to (24) and (25), based

on Theorem 9.8.2.3 (b) we get that we have a natural bilinear commutating morphism

cj1,j2,j3 : Fz,j1 ×Spf(W (k)) Fz,j2 → Fz,j3 .

181

9.8.7. Example. Let n ∈ N. We assume that (GadQ , X

ad) is a simple, adjoint Shimura

pair of DHℓ type (ℓ ≥ 4) and that Gad

Zpis the ResW (Fpn )/Zp

of an absolutely simple, split,

adjoint group scheme G′W (Fpn ) over W (Fpn). Let G := GW (k). If Gad =

∏i∈I Gi is the

product decomposition into absolutely simple, adjoint groups, then I has n elements, theLie type X of each Gi is Dℓ, and there exists a permutation Γ of I such that for eachelement i ∈ I we have (ϕ0 ⊗σk)(Lie(Gi)[

1p]) = Lie(GΓ(i))[

1p]. In order to use the notations

of Section 3.4, we will take I = S(1, n), Γ = (1 · · ·n), and g := Lie(Gad). As G′W (Fpn ) is

split, the order e of Subsection 3.4.4 is 1. Let Iy : (Lie(Gad), ϕ)→(Lie(Gad), ϕ0 ⊗ σk) bethe isomorphism of Proposition 6.3.1.

We emphasize that we choose the torus T of G such that Iy(Lie(T(1))) = Lie(T ),

where T := Im(T0,W (k) → Gad); thus we will also use the notations of Subsection 3.4.3(which are with respect to T ). The set E of Subsection 3.4.8 has at least two elements (cf.Corollary 3.6.3 and the fact that I0 ≥ 2) and is a subset of 1, ℓ−1, ℓ (as X = Dℓ). From[Se1, p. 186 and Cor. 2 of p. 182] applied to GQp

, we get that for ℓ ≥ 5 the set E can notcontain either 1, ℓ−1 or 1, ℓ and that for ℓ = 4 the set E can not be 1, 3, 4. Moreover,for ℓ = 4 we can choose the numberings in Subsection 3.4.5 such that we have E = 3, 4.Thus as E has at least two elements and as X = Dℓ, we conclude that E = ℓ−1, ℓ. ThusI0 = 2, cf. Corollary 3.6.3. Next we apply Subsections 9.8.1 to 9.8.6.

(a) Let S−i (ℓ−1, ℓ) be the set of roots in Φ−

i that have in their expressions −αℓ−i(i)−αℓ(i). Let S

−i (ℓ−1) (resp. S−

i (ℓ−1)) be the set of roots in Φ−i that have in their expressions

−αℓ−i(i) (resp. −αℓ(i)). Let j1, j2, j3 ∈ J be such that

Iy(u−j1) = ⊕n

i=1 ⊕α∈S−i(ℓ−1,ℓ) gi(α),

Iy(u−j2) = ⊕n

i=1 ⊕α∈S−i(ℓ−1)\S−

i(ℓ−1,ℓ) gi(α), and Iy(u

−j3) = ⊕n

i=1 ⊕α∈S−i(ℓ)\S−

i(ℓ−1,ℓ) gi(α).

Let j4, j5, j6 ∈ J be such that we have u−j4 = u−j1 ⊕ u−j2 , u−j5

= u−j1 ⊕ u−j3 , and u−j6 = u−.

(b) We take t(1)j1 = t(1)j2 = t(1)j4 (resp. t(1)j3 = t(1)j5) such that Iy(t(1)j1)(resp. Iy(t(1)j3)) commutes with each gi(α) with α ∈ ∆i \ ℓ − 1 (resp. with α ∈∆i \ ℓ) and for every element x ∈ gi(αℓ−1(i)) (resp. for every element x ∈ gi(αℓ(i)))we have [Iy(t(1)j1), x] = x (resp. [Iy(t(1)j3), x] = x). For each i ∈ S(1, 5), using thementioned expression for t(1)ji = φ(t(1)ji), we get that the condition 9.8.2.3 (i) holds forji (to be compared with the property 5.2.1 (iv)). For each i ∈ S(1, 5) the Lie algebrau−ji is commutative (cf. the property 3.6.1 (a) applied over B(k)) and therefore U−

jiis

commutative. Thus for each i ∈ S(1, 5) there exists naturally a formal closed subschemeDy,ji of Dy that has a canonical structure Fz,ji of a formal Lie group whose identity sectionis z and which is isomorphic to the formal Lie group of D+

ji,W (k), cf. Theorem 9.8.2.3. As

u−j4 = u−j1 ⊕ u−j2 and u−j5 = u−j1 ⊕ u−j3 , we have natural decompositions

Fz,j4 = Fz,j1 × Fz,j2 and Fz,j5 = Fz,j1 ×Fz,j3 .

(c) From the property 3.6.1 (a) we get that [u−j6 , u−j6] ⊆ u−j1 . This implies that U−

j16

Z(U−j6); thus (j1, j6) ∈ J2. Let t(1)j6,j1 := t(1)j2 + t(1)j3 . The action of t(1)j6,j1 on

182

u−j2 ⊕ u−j3→u−j6/u−j1

is a W (k)-linear automorphism. Thus Dy,j6,j1 has a formal Lie group

structure Fz,j6,j1 isomorphic to the formal Lie group of D+j2,W (k) ⊕D+

j3,W (k), cf. Theorem9.8.3.2.

(d) As U−j1

6 Z(U−j6) = Z(U−

G(ϕ)) and as the condition 9.8.2.3 (i) holds for j1, from

Theorem 9.8.4.1 we get that we have a natural action of Fz,j1 on Dy.

(e) We have (j2, j3, j1) ∈ J3 as 0 6= [u−j2 , u−j3] ⊆ u−j1 . From Subsection 9.8.6 we get the

existence of a natural bilinear commutating morphism cj2,j3,j1 : Fz,j2 ×Spf(W (k)) Fz,j3 →Fz,j1 .

(f) Each choice of coordinates for Dy,j1 , Dy,j2 , and Dy,j3 can be used for p ≥ 2 (resp.for p ≥ 3) to define a natural formal Lie group structure on Dy isomorphic to the first(resp. to the second) formal Lie group of C0 ⊗ k. We refer to future work for the study of:

(f.i) good coordinates for Dy,j1 , Dy,j2 , and Dy,j3 , and

(f.ii) the variations of these formal Lie group structures on Dy (in terms of the choice

of coordinates and in terms of the choice of the torus T in the beginning of this Subsub-section).

183

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