lectures on shimura varieties

7/27/2019 Lectures on Shimura Varieties

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LECTURES ON SHIMURA VARIETIES

A. GENESTIER AND B.C. NGO

Abstract. The main goal of these lectures will be to explain therepresentability of moduli space abelian varieties with polarization,endomorphism and level structure, due to Mumford [GIT] and thedescription of the set of its points over a finite field, due to Kot-twitz [JAMS]. We also try to motivate the general definition of

Shimura varieties and their canonical models as in the article ofDeligne [Corvallis]. We will leave aside important topics like com-pactifications, bad reductions andp-adic uniformization of Shimuravarieties.

This is the set notes for the lectures on Shimura varieties givenin the Asia-French summer school organized at IHES on July 2006.It is based on the notes of a course given by A. Genestier and myselfin Universte Paris-Nord on 2002.

Date: September 2006, preliminary version.1


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Contents

1. Quotients of Siegels upper half space 31.1. Review on complex tori and abelian varieties 31.2. Quotient of the Siegel upper half space 71.3. Torsion points and level structures 82. Moduli space of polarized abelian schemes 92.1. Polarization of abelian schemes 92.2. Cohomology of line bundles on abelian varieties 122.3. An application of G.I.T 132.4. Spreading abelian scheme structure 152.5. Smoothness 162.6. Adelic description and Hecke operators 17

3. Shimura varieties of PEL type 203.1. Endomorphism of abelian varieties 203.2. Positive definite Hermitian form 233.3. Skew-Hermitian modules 233.4. Shimura varieties of type PEL 243.5. Adelic description 263.6. Complex points 274. Shimura varieties 284.1. Review on Hodge structures 284.2. Variation of Hodge structures 314.3. Reductive Shimura datum 324.4. Dynkin classification 354.5. Semi-simple Shimura datum 354.6. Shimura varieties 365. CM tori and canonical model 375.1. PEL moduli attached to a CM torus 375.2. Description of its special fibre 395.3. Shimura-Taniyama formula 435.4. Shimura varieties of tori 435.5. Canonical model 445.6. Integral models 44

6. Points of Siegel varieties over finite fields 446.1. Abelian varieties over finite fields up to isogeny 446.2. Conjugacy classes in reductive groups 456.3. Kottwitz triple (0, , ) 47References 50

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1. Quotients of Siegels upper half space

1.1. Review on complex tori and abelian varieties. LetV denotea complex vector space of dimension n and Ua lattice in V which isby definition a discrete subgroup ofV of rang 2n. The quotientX=V /UofV byUacting on Vby translation, is naturally equipped witha structure of compact complex manifold and a structure of abeliangroup.

Lemma 1.1.1. We have canonical isomorphisms fromHr(X,Z)to thegroup of alternatingr-form

r U Z.Proof. Since X = V /U with V contractible, H1(X, U) = Hom(U,Z).The cup-product defines a homomorphism

rH1(X,Z) Hr(X,Z)

which is an isomorphism since X is isomorphic with (S1)2n as real

manifolds whereS1= R/Z is the unit circle.

LetLbe a holomorphic line bundle over the compact complex varietyX. Its Chern class c1(L) H

2(X,Z) is an alternating 2-form on Uwhich can be made explicite as follows. By pulling back L to V bythe quotient morphism : V X, we get a trivial line bundle since

every holomorphic line bundle over a complex vector space is trivial.We choose an isomorphismL OV. For everyu U, the canonicalisomorphismuLLgives rise to an automorphism ofOV whichconsists in an invertible holomorphic function

eu (V, OV).

The collection of these invertible holomorphic functions for all u U,satisfies the cocycle equation

eu+u(z) =eu(z+ u)eu(z).

If we write au(z) =e2ifu(z) where fu(z) are holomorphic function welldefined up to a constant inZ, the above cocycle equation is equivalentto

F(u1, u2) =fu2(z+ u1) + fu1(z) fu1+u2(z) Z.

The Chern class

c1: H1(X, OX) H

2(X,Z)

sends the class ofL in H1(X, OX) on c1(L) H2(X,Z) whose corre-

sponding 2-form E :2 U Z is given by

(u1, u2)E(u1, u2) :=F(u1, u2) F(u2, u1).3


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Lemma 1.1.2. The Neron-Severi group NS(X), defined as the imageof c1 : H

1(X, OX) H2(X,Z) consists in the alternating2-formE :2 U Z satisfying the equationE(iu1, iu2) =E(u1, u2)

in whichEdenotes the alternating2-form extended to UZ R =V byR-linearity.

Proof. The short exact sequence

0 Z OX OX0

induces a long exact sequence which contains

H1(X, OX) H2(X,Z) H2(X, OX).

It follows that the Neron-Severi group is the kernel of the map H2(X,Z)H2(X, OX). This map is the composition of the obvious maps

H2(X,Z) H2(X,C) H2(X, OX).

The Hodge decomposition

Hm(X,C) =

p+q=m

Hp(X, qX)

where qX is the sheaf of holomorphic q-forms on X, can be madeexplicite [13, page 4]. For m= 1, we have

H1

(X,C) =VR R C =V

C V

C

where VC is the space ofC-linear maps V C, VC is the space of

conjugate C-linear maps and VR is the space ofR-linear maps V R. There is a canonical isomorphism H0(X, 1X) = V

C defined by

evaluating a holomorphic 1-form onXon the tangent space V ofXatthe origine. There is also a canonical isomorphism H1(X, OX) =V

C.

By taking2 of the both sides, the Hodge decomposition of H2(X,C)

can also be made explicite. We have H2(X, OX) =2 VC, H1(X, 1X) =

VC V

C and H0(X, 2X) =

2 VC . It follows that the map H2(X,Z)H2(X, OX) is the obvious map

UZ 2 VC . Its kernel are precisely

the integral 2-forms E on Uwhich satisfies the relation E(iu1, iu2) =E(u1, u2) after extension to V byR-linearity.

Let E:2 U Z be an integral alternating 2-form on U satisfying

E(iu1, iu2) = E(u1, u2) after extension to V by R-linearity. The real2-formEonVdefines a Hermitian form on the C-vector spaceV by

(x, y) =E(ix,y) + iE(x, y)

which in turns determines Eby the relation E= Im(). The Neron-Severi group NS(X) can be described in yet another way as the groupof Hermitian forms on the C-vector spaceVof which the imaginary

part takes integral values on U.4


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Theorem 1.1.3 (Appell-Humbert). The holomorphic line bundles onX=V /Uare in bijection with the pairs(, ) where is a Hermitian

form onVof which the imaginary part takes integral values onU and: US1is a map fromUto the unit circleS1satisfying the equation

(u1+ u2) =eiIm()(u1,u2)(u1)(u2).

For every(, )as above, the line bundleL(, )is given by the cocycle

eu(z) =(u)e(z,u)+ 12(u,u).

Let denote Pic(X) the abelian group of isomorphism classes of linebundle onX, Pic0(X) the subgroup of line bundle of which the Chernclass vanishes. We have an exact sequence :

0 Pic

0

(X) Pic(X) NS(X) 0.Let denote X = Pic0(X) whose elements are characters : U S1from U to the unit circle S1. Let V

R = HomR(V, R). There is a

homomorphism VR X sending v VR on the line bundle L(0, )

where : US1 is the character

(u) = exp(2iu, v).

This induces an isomorphism VR/U X where

U ={u VR such thatu U, u, u Z}.

We can identify the real vector space Vwith the spaceV

C of conjugate

C-linear application V C. This gives to X = VC/U a structureof complex torus which is called the dual complex torus of X. Withrespect to this complex structure, the universal line bundle over X Xgiven by Appell-Humbert formula is a holomorphic line bundle.

A Hermitian form onV induces a C-linear mapV V

C. If moreover

its imaginary part takes integral values in U, the linear map V V

C

takesU intoU and therefore induces a homomorphism X Xwhichis symmetric. In this way, we identify the Neron-Severi group NS(X)

with the group of symmetric homomorphisms from X to X i.e. :X Xsuch that= .

Let (, ) as in the theorem and H0(X, L(, )) be a globalsection ofL(, ). Pulled back toV, becomes a holomorphic functionon Vwhich satisfies the equation

(z+ u) =eu(z)(z) =(u)e(z,u)+ 12(u,u)(z).

Such function is called a theta-function with respect to the hermitianform and the multiplicator . The Hermitian form needs to bepositive definite for L(, ) to have a lot of sections, see [13, 3]

Theorem 1.1.4. The line bundleL(, ) is ample if and only if theHermitian formH is positive definite. In that case,

dimH0(X, L(, )) =det(E).5


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Consider the case where H is degenerate. Let Wbe the kernel ofHor ofE i.e.

W ={x V|E(x, y) = 0, y V}.Since E is integral on U U, W U is a lattice ofW. In particular,W/W U is compact. For any x X,u W U, we have

|(x + u)|= |(x)|

for all n N, H0(X, L(, )d). By the maximum principle, itfollows that is constant on the cosets ofXmodulo Wand thereforeL(, ) is not ample. Similar argument shows that ifHis not positivedefinite, L(H, ) can not be ample, see [13, p.26].

If the Hermitian form His positive definite, then the equality

dimH0(X, L(, )) =det(E)holds. In [13, p.27], Mumford shows how to construct a basis, well-defined up to a scalar, of the vector space H0(X, L(, )) after choosinga sublattice U Uof ranknwhich is Lagrangian with respect to thesymplectic formEand such that U =U RU. Based on the equalitydimH0(X, L(, )d) =

det(E), one can proveL(, )3 gives rise to

a projective embedding ofX for any positive definite Hermitian form. See Theorem 2.2.3 for a more complete statement.

Definition 1.1.5. (1) An abelian variety is a complex torus thatcan be embedded into a projective space.

(2) A polarization of an abelian varietyX=V /Uis an alternatingform :

2 U Z which is the Chern class of an ample linebundle.

With a suitable choice of a basis ofU, can be represented by amatrix

E=

0 DD 0

whereDis a diagonal matrixD = (d1, . . . , dn) whered1, . . . , dnare non-negative integers such thatd1|d2| . . . |dn. The formE is non-degenerate

if these integers are zero. We call D = (d1, . . . , dn) type of the polar-izationE. A polarization is called principalif its type is (1, . . . , 1).

Corollary 1.1.6 (Riemann). A complex torusX = V /U can be em-bedded as a closed complex submanifold into a projective space if andonly if the exists a positive definite hermitian form onV such thatthe restrictionImH onUis a 2-form with integral values.

Let us rewrite Riemanns theorem in term of matrices. We choose aC-basis e1, . . . , en for V and a Z-basis u1, . . . , u2n ofU. Let be then 2n-matrix = (ji) with ui =

nj=1 jiej for all i = 1, . . . , 2n.

is called the period matrix. Since 1, . . . , 2n form a R-basis ofV,

the matrix 2n 2n-matrix is invertible. The alternating form6


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E :2 U Z is represented by an alternating matrix, also denoted

by E is the Z-basis u1, . . . , u2n. The form : V V C given by

(x, y) = E(ix,y) + iE(x, y) is hermitian if and only if E1 t = 0.His positive definite if and only if the symmetric matrix iE1 t> 0is positive definite.

Corollary 1.1.7. The complex torusX= V /U with period matrixis an abelian variety if and only if there is a nondegenerate alternatingintegral2n 2n matrixEsuch that

(1) E1 t = 0,(2) iE1 t> 0.

1.2. Quotient of the Siegel upper half space. LetXbe an abelian

variety of dimension n over Cand letEbe a polarization ofXof typeD= (d1, . . . , dn). There exists a basisu1, . . . , un, v1, . . . , vnof H1(X,Z)with respect to which the matrix ofEtakes the form

E=

0 DD 0

A datum (X,E, (u, v)) is called polarized abelian variety of type Dwith symplectic basis. We want to describe the moduli of polarizedabelian variety of type D with symplectic basis.

The Lie algebraV ofX is an-dimensional C-vector space withU=

H1(X,Z) as a lattice. Choose a C-basis e1, . . . , en ofV. The vectorse1, . . . , en, ie1, . . . , i en form a R-basis of V. The isomorphism R :U R V is given by an invertible real 2n 2n-matrix

R =

11 1221 22

The complex n 2n-matrix = (1, 2) is related to R by therelations 1= 11+ i21 and 2 = 12+ i22.

Lemma 1.2.1. The set of polarized abelian variety of type D withsymplectic basis is canonically in bijection with the set ofGLC(V)orbits

of isomorphisms of real vector spacesR : U R V such that forallx, y V, we haveE(1R ix,

1R iy) = E(

1R x,

1R y) and that the

symmetric formE(1R ix, 1R y) is positive definite.

There are at least two methods to describe this quotient. The firstone is more concrete but the second one is more suitable for general-ization.

In each GLC(V) orbit, there exists a unique R such that 1R ei =

1di

vi for i = 1, . . . , n. Thus, the matrix R has thus the form

R = 11 D

21 0 7


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and has the form = (Z, D) with where

Z= 11+ i21 Mn(C)satisfying tZ=Zand im(Z)> 0.

Proposition 1.2.2. There is a canonical bijection from the set of po-larized abelian varieties of type D with symplectic basis to the Siegelupper half-space

Hn= {ZMn(C)|tZ=Z, im(Z)> 0}.

On the other hand, an isomorphism R : U R V defines acocharacter h : C GL(U R) by transporting the complex struc-ture ofV on U R. It follows from the relationE(1R ix,

1R iy) =

E(1R x,

1R y) that the restriction ofh to the unit circle S1 defines a

homomorphism h1 : S1 SpR(U, E). Moreover, the GLC(V)-orbit ofR : U R V is well determined by the induced homomorphismh1:S1 SpR(U, E).

Proposition 1.2.3. There is a canonical bijection from the set of po-larized abelian varieties of type D with symplectic basis to the set ofhomomorphism of real algebraic groupsh1 :S1 SpR(U, E) such thatthe following conditions are satisfied

(1) the complexification h1,C : Gm Sp(U C) gives rises to adecomposition into direct sum ofn-dimensional vector subspaces

U C = (UC)+ (U C)

of eigenvalues+1 and1;(2) the symmetric formE(h1(i)x, y) is positive definite.

This set is a homogenous space under the action ofSp(U R) actingby inner automorphisms.

Let SpD be Z-algebraic group of automorphism of the symplecticform Eof type D. The discrete group SpD(Z) acts simply transitivelyon the set of symplectic basis ofU Q.

Proposition 1.2.4. There is a canonical bijection between the set ofisomorphism classes of polarized abelian variety of typeD and the quo-tientSpD(Z)\Hn.

According to H. Cartan, there is a way to give an analytical structureto this quotient and then to prove that this quotient has indeed astructure of quasi-projective normal variety over C.

1.3. Torsion points and level structures. Let X = V /U be anabelian variety of dimension n. For every integer N, The group ofN-torsion points X[N] = {x X|N x = 0} can be identified withthe finite group N1U/U that is isomorphic to (Z/NZ)2n. Let E be

a polarization ofX of type D = (d1, . . . , dn) with (dn, N) = 1. The8


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alternating formE:2 U Zcan be extended to a non-degenerating

symplectic form on U Q. The Weil pairing

(, )exp(2iE(, ))

is a symplectic non-degenerate form

eN :X[N] X[N]N

where Nis the group ofN-th roots of unity, provided Nbe relativelyprime with dn. Let choose a primitiveN-th root of unity so that theWeil pairing takes values in Z/NZ.

Definition 1.3.1. LetN be an integer relatively prime to dn. A prin-cipalN-level structure of an abelian varietyXwith a polarizationE is

an isomorphisme from the symplectic moduleX[N] with the standardsymplectic module(Z/NZ)2n given by the matrix

J=

0 InIn 0

whereIn is the identityn n-matrix.

Let 1(N) be the subgroup of SpD(Z) of the automorphisms of (U, E)with trivial induced action on U/NU.

Proposition 1.3.2. There is a natural bijection between the set of

isomorphism classes of polarized abelian variety of type D equippedwith a principalN-level structure and the quotientA0n,N= A(N)\Hn.

For N 3, the group 1(N) does not contains torsion and actfreely on Siegel half-spaceHn. The quotientA

0n,Nis therefore a smooth

complex analytic space.

2. Moduli space of polarized abelian schemes

2.1. Polarization of abelian schemes.

Definition 2.1.1. An abelian scheme over a scheme S is a smooth

proper group scheme with connected fiber. As a group scheme, X isequipped with the following structures

(1) an unit sectioneX :S X(2) a multiplication morphismXSXX(3) an inverse morphismXX

such that the usual axioms for abstract groups hold.

Recall the following classical rigidity lemma.

Lemma 2.1.2. Let X and X two abelian schemes over S and :XX a morphism that sends unit section ofXon the unit section

ofX. Then is a homomorphism.9


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Proof. We will summarize the proof when S is a point. Consider themap :X XX given by

(x1, x2) =(x1x2)(x1)1(x2)

1.

We have (eX, x) = eX for all x X. For any affine neighborhoodU ofeX inX

, there exists an affine neighborhood U ofeXsuch that(U X) U. For every u U,maps the proper scheme u X into the affine U. It follows that the restricted to u X is constant.Since (ueX) = eX, (u, x) = eX for any x X. It follows that(u, x) =eX for anyu, x X since Xis irreducible.

Let us mention to useful consequences of the rigidity lemma. Firstly,the abelian scheme is necessarily commutative since the inverse mor-

phism X X is a homomorphism. Secondly, given the unit section,a smooth proper scheme can have at most one structure of abelianschemes. It suffices to apply the rigidity lemma for the identity ofX.

An isogeny: XX is a surjective homomorphism whose kernelker() is a finite group scheme overS. Letd be a positive integer. LetSbe a scheme whose all residual characteristic is relatively prime to d.Let : XX be a isogeny of degree d andK() be the kernel of.For every geometric point s S,K()s is a discrete group isomorphicto Z/d1Z Z/dnZwithd1| |dn andd1 . . . dn=d. The functionthat maps a point s |S|to the type ofK()s for any geometric pointsovers is a locally constant function. So it makes sense to talk about

the type of an isogeny of degree prime to all residual characteristic.Let X/S be an abelian scheme. Consider the functor PicX/S from

the category of S-schemes to the category of abelian groups whichassociates to every S-scheme T the group of isomorphism classes of(L, ) ou L is an invertible sheaf on XST and is a trivializationeXL OT along the unit section. See [2, p.234] for the followingtheorem.

Theorem 2.1.3. LetXbe a projective abelian scheme overS. Thenthe functor PicX/S is representable by a smooth separated S-schemewhich is locally of finite presentation overS.

The smooth scheme PicX/S equipped with the unit section corre-sponding to the trivial line bundle OX admits a neutral componentPic0X/Swhich is an abelian scheme over S.

Definition 2.1.4. LetX/S be a projective abelian scheme. The dual

abelian scheme X/S is the neutral component Pic0(X/S) of the Pi-card functor P icX/S. We call Poincare sheaf P the restriction of the

universal invertible sheaf onXSPicX/S to XSX.

For every abelian scheme X/S with dual abelian scheme X/S, the

dual abelian scheme ofX/S is X/S. For every homomorphism :

X X, we have a homomorphism : X X. If is an isogeny,10


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the same is true for . A homomorphism : X Xis saidsymmetricif the equality = holds.

Lemma 2.1.5. Let : X Y be an isogeny and let : Y X bethe dual isogeny. There is a canonical perfect pairing

ker() ker() Gm.

Proof. Let y ker() and let Ly be the corresponding line bundle onY with a trivialization along the unit section. Pulling it back to X,we get the trivial line bundle equipped with a trivialization on ker().This trivialization gives rises to a homomorphism ker() Gm whichdefines the desired pairing. It is not difficult to check that this pairing

is perfect, see [13, p.143].

LetL PicX/Sbe an invertible sheaf over Xwith trivialized neutralfibre Le= 1. For any point x X over s S, let Tx:Xs Xs be thetranslation byx. The invertible sheafTxL L

1 L1x has trivializedneutral fibre

(TxL L1 L1x )e= Lx L

1e L

1x = 1.

so that L defines a morphism L :XPicX/S. Since the fibres ofX

are connected,Lfactors through the dual abelian scheme Xand givesrise to a morphism

L: X X.

Since L send the unit section ofXon the unit section ofX so thatL is necessarily a homomorphism. Let denote K(L) the kernel ofL.

Lemma 2.1.6. For every line bundle L on X with a trivializationalong the unit section, the homomorphismL :X X is symmetric.If moreover, L= xP for a sectionx: S X, thenL= 0.

Proof. By construction, the homomorphismL: X Xrepresents theline bundlemL p1L

1 p2L1 onX Xwherem is the multiplica-

tion andp1, p2are projections, equipped with the obvious trivializationalong the unit section. The homomorphismLis symmetric as this linebundle.

IfL = OXwith the obvious trivialization along the unit section, itis immediate that L = 0. Now for anyL = x

P, L can be deformedcontinuously to the trivial line bundle and it follows that L = 0. Inorder to make the argument rigorous, one can form a family over Xand apply the rigidity lemma.

Definition 2.1.7. A line bundleL over an abelian schemeX equippedwith a trivialization along the unit section is called non-degenerated if

L: X X is an isogeny.11


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In the case where the base S is Spec(C) and X = V /U, L is nondegenerate if and only if the associated Hermitian form on V is non-

degenerate.Let L be a non-degenerate line bundle on X with a trivialization

along the unit section. The canonical pairing K(L) K(L) Gm,Sis then symplectic. Assume S connected with residual characteristicprime to the degree of L, there exists d1| . . . |ds such that for ev-ery geometric point s S, the abelian group K(L)s is isomorphicto (Z/d1Z Z/dnZ)

2. We call D = (d1, . . . , dn) the type of thepolarization

Definition 2.1.8. Let X/S be an abelian scheme. A polarization of

X/S is a symmetric isogeny : X X which locally for the etaletopology ofS, is of the formL for some ample line bundleL ofX/S.

In order to make this definition workable, we will need to recallbasic facts about cohomology of line bundles on abelian varieties. Seecorollary 2.2.4 in the next paragraph.

2.2. Cohomology of line bundles on abelian varieties. We aregoing to recollect known fact about cohomology of line bundles onabelian varieties. For the proofs, see [13, p.150]. Let Xbe an abelianvariety over a field k. Let denote

(L) =iZ

dimkHi(X, L)

the Euler characteristic ofL.

Theorem 2.2.1 (Riemann-Roch theorem). For all line bundle L onX, ifL= OX(D) for a divisorD, we have

(L) =(Dg)

g!

where(Dg) is theg-fold self-intersection number ofD.

Theorem 2.2.2 (Mumfords vanishing theorem). LetL be a line bun-dle on X such that K(L) is finite. There exists a unique integeri = i(L) with0 i n = dim(X) such thatHj(X, L) = 0 for j = iandHi(X, L)= 0. Moreover, L is ample if and only if i(L) = 0. Foreverym 1, i(Lm) =i(L).

Assume S= Spec(C), X = V /U with V = Lie(X) and U a latticeinV. Then the Chern class ofL corresponds to a Hermitian for Handthe integer i(L) is the number of negative eigenvalues ofH.

Theorem 2.2.3. For any ample line bundleL on an abelian variety

X, thenL2 is base-point free andLm is very ample ifm 3.12


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Since L is ample, i(L) = 0 and consequently, H0(X, L) =(L)> 0.There exists an effective divisor D such that L OX(D). SinceL :

X X is a homomorphism, the divisor Tx (D) +Tx(D) is linearlyequivalent to 2DandTx (D) + T

y (D) + T

xy(D) is linearly equivalent

to 3D. By moving x, y Xwe get a lot of divisors linearly equivalentand to 2D and to 3D. The proof is based on this fact and on theformula for the dimension of H0(X, Lm). For the detailed proof, see[13, p.163].

Corollary 2.2.4. LetXSan abelian scheme over a connected baseand letL be an invertible sheaf onX such thatK(L) is a finite groupscheme overS. If there exists a points Ssuch thatKs is ample onXs thenL is relatively ample forX/S.

Proof. Since Ls is ample, H0(Xs, Ls)= 0 and H

i(Xs, Ls) = 0 for everyi = 0. For t varying in s, the function t dimHi(Xt, Lt) is uppersemi-continuous and the function t(Xt, Lt) is constant. The onlyway for the Mumfords vanishing theorem to be satisfied is for allt S,H0(Xt, Lt) = 0 and Hi(Xt, Lt) = 0 for all i = 0. It follows that Lt isample. IfLt is ample, Lis relatively ample on Xover a neighborhoodoft in S.

2.3. An application of G.I.T. Let fix two positive integer n 1,N3 and a type D= (d1, . . . , dn) with d1| . . . |dn. LetA the functor

which associates to a scheme S the groupoid of polarized S-abelianschemes of type D : for every S, A(S) is the groupoid of (X,,)where

(1) Xis an abelian scheme over S ;

(2) : X X is a polarization of type D ;(3) is a symplectic similitude (Z/NZ)2n X[N] .

Theorem 2.3.1. The functor A defined above is representable by asmooth quasi-projective scheme.

Proof. Let X be an abelian scheme over S and X its dual abelianscheme. LetP be the Poincare line bundle overXSXequipped witha trivialization over the neutral section eXSidX : X XS X ofX. Let L() be the line bundle over Xobtained by pulling back thePoincare line bundle P

L() = (idX, )P

by the composite homomorphism

(idX, ) = (idX ) :XXSXXSX

where :XXSXis the diagonal. The line bundle L() gives

rise to a symmetric homomorphismL():X X.

Lemma 2.3.2. The equalityL() = 2 holds.13


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Proof. Locally for etale topology, we can assume = L for some linebundle over Xwhich is relatively ample. Then

L() = (idX )P = (L pr1L

1 pr2L1).

It follows that

L() = (2)L L2

where (2) : X X is the multiplication by 2. As for every N N,(N)L = N

2L, in particular (2)L = 4L, and thus we obtain thedesired equality L() = 2.

Since locally overS, = LwhereLis a relatively ample line bundle,L() is a relatively ample line bundle,L()3 is very ample. It follow

that its higher direct images by : XSvanishRiL

()3 = 0 for all i 1

and M=L() is a vector bundle of rank

m + 1 := 6nd

over S.

Definition 2.3.3. A linear rigidification of a polarized abelian scheme(X, ) is an isomorphism

: PmS PS(M)

whereM = L(). In other words, a linear rigidification of a polar-ized abelian scheme(X, ) is a trivialization of thePGL(m + 1)-torsorassociated to the vector bundleMof rankm + 1.

Let H be the functor that associates to any schemeSthe groupoid oftriples (X,,,) where (X,,) is an polarized abelian scheme overSof typeD and where is a linear rigidification. Forgetting, we geta morphism

H A

which is a PGL(m + 1)-torsor.The line bundle L()3 provides a projective embedding

X PS(M).

Using the linear rigidification , we can embed X into the standardprojective space

X PmS.

For every r N, the higher direct images vanish

RiL()r = 0 for alli >0

and L()r is a vector bundle of rank 6ndrn so that we have a mor-

phism of functor

f :Hn HilbQ(t),1(Pm)14


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where HilbQ(t),1(Pm) is the Hilbert scheme of 1-pointed subschemes ofPm with Hilbert polynomial Q(t) = 6ndtn : f sending (X,,) to the

image ofX in Pm which pointed by the unit ofX.

Proposition 2.3.4. The morphismf identifiesH with an open sub-functor ofHilbQ(t),1(Pm) which consist of pointed smooth subschemesofPm.

Proof. Since a smooth projective pointed variety Xhas at most oneabelian variety structure, the morphism f is injective. Following the-orem 2.4.1 of the next paragraph, any smooth projective morphismef : X S over a geometrically connected base S with a sectione: SXhas an abelian scheme structure if and only if one geometric

fiber Xs does.

Since a polarized abelian varieties with principal N-level structurehave no trivial automorphisms, PGL(m+ 1) acts freely on H. Wetake A as the quotient ofH by the free action of PGL(m+ 1). Theconstruction of this quotient as a scheme requires nevertheless a quitetechnical analysis of stability. IfNis big enough then X[N] X Pmis not contained in any hyperplane, furthermore no more than N2n/m+1 points from these N-torsion points can lie in the same hyperplane ofPm. In that case, (A,,,) is a stable point. In the general case, wecan add level structure and then perform a quotient by a finite group.

See [14, p.138] for a complete discussion.

2.4. Spreading abelian scheme structure. Let us now report on atheorem of Grothendieck [14, theorem 6.14].

Theorem 2.4.1. LetSbe a connected noetherian scheme. LetX S be a smooth projective morphism equipped with a section e : S X. Assume for one geometric points= Spec((s)), Xs is an abelianvariety over(s)with neutral point(s). ThenXis an abelian schemeoverSwith neutral section.

Let us consider first the infinitesimal version of this assertion.

Proposition 2.4.2. LetS= Spec(A) whereA is an Artin local ring.Letm be the maximal ideal ofA and letI be an ideal ofA such thatmI = 0. Let S0 = Spec(A/I). Let f : X S be a proper smoothscheme with a sectione : S X. Assume thatX0 = XSS0 is anabelian scheme with neutre section e0 = e|S0. Then X is an abelianscheme with neutral sectione.

Proof. Let k = A/m and X = XAk. Let 0 : X0 S0 X0 be themorphism 0(x, y) =x y and let : XkXXbe the restrictionofX0. The obstruction to extend0 to a morphism XSXX isan element

H1(X X, TXkI)15


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where TX is the tangent bundle ofXwhich is a trivial vector bundleof fibre Lie(X). Thus, by Kunneth formula

H1(X X, TXk I) = (Lie(X) k H1(X)) (H1(X) k Lie(X)) k I.

Considerg1, g2:X0 X0S0X0withg1(x) = (x, e) andg2(x) = (x, x).The endomorphisms ofX0, g1 0 = idX0 and g2 0 = (e f) haveobvious way to extend to X so that the obstruction classes 1 = g

1

and 2 = g2 must vanish. Since one can express in function of1

and 2 by Kunneth formula, vanishes too.The set of all extensions of 0 is a principal homogenous space

underH0(XkX ,

TXkI)).

Among these extensions, there exists a unique such that (e, e) =ewhich provides a group scheme structure on X/S.

We can extend the abelian scheme structure to an infinitesimal neigh-borhood ofs. This structure can be algebrized and then descend to aZariski neightborhood since the abelian scheme structure is unique ifit exists. It remains to prove the following lemma due to Koizumi.

Lemma 2.4.3. LetS= Spec(A) whereA is a discrete valuation ringwith generic point. Letf :XSbe a proper and smooth morphismwith a sectione :S X. Assume thatA is an abelian variety withneutral pointe(). ThenX is an abelian scheme with neutral section

.

Proof. Suppose A is henselian. Since X S is proper and smooth,inertia group I acts trivially on Hi(X,Q). By Grothendieck-Ogg-Shafarevichs criterion, there exists an abelian scheme A over S withA = X and A is the Nerons model ofA. By the universal prop-erty of Nerons model there exist a morphism X A extending theisomorphism :XA. LetL be a relatively ample invertible sheafon X/S. Choose a trivialization on the unit point ofX = A. ThenL with the trivialization on the unit section extends uniquely on Ato a line bundle L since Pic(A/S) satisfies the valuative criterion for

properness. It follows that Ls and Ls have the same Chern class. If have a fiber of positive dimension then the restriction to that fiberof Ls is trivial. In contrario, the restriction ofLs to that fiber isstill ample. This contradiction implies that all fiber of have dimen-sion zero. The finite birational morphism :XA is necessarily anisomorphism according to Zariskis main theorem.

2.5. Smoothness. In order to prove that Ais smooth, we will need toreview Grothendieck-Messings theory of deformation of abelian schemes.

Let S = Spec(R) be a thickening of S = Spec(R/I) with I2 = 0,or more generally, locally nilpotent and equipped with a structure of

divided power. According to Grothendieck and Messing, we can attach16


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to an abelian scheme A of dimensionn overSa locally freeOS-moduleof rank 2n

H1cris(A/S)S

such that

H1cris(A/S)SOSOS= H1dR(A/S).

We can associate with every abelian scheme A/Ssuch thatA SS=Aa sub-OS-module

A/SH1dR(A/S) = H

1cris(A)S

which is locally a direct factor of rank n and which satisfies

A/SOSOS=A/S.

Theorem 2.5.1(Grothendieck-Messing).The functor, defined as above,from the category of abelian schemesA/SwithA SS=A to the cat-egory sub-OS-modules H

1(A/S)Swhich are locally a direct factorsand which satisfy such that

OSOS=A/S

is an equivalence of categories.

See [10, p.151] for the proof of this theorem.Let S = Spec(R) be a thickening of S = Spec(R/I) with I2 = 0.

Let A be an abelian scheme over S and be a polarization of A oftype (d1, . . . , ds) with integer di relatively to residual characteristic ofS. The polarization induces an isogeny

: A A

where A

is the dual abelian scheme of A/S. Since the degree ofthe isogeny is relatively prime to residual characteristic, it induces anisomorphism

H1cris(A

/S)SH1cris(A/S)S

or a bilinear form on H1cris(A/S)Swhich is a symplectic form. The

module of relative differential A/Sis locally a direct factor of H1cris(A/S)Swith is isotropic with respect to the symplectic form . It is knownthat the Lagrangian grassmannian is smooth so that one can lift A/Sto a locally direct factor of H1cris(A/S)Swhich is isotropic. According toGrothendieck-Messing theorem, we got an a lifting ofA to an abelianschemeA/Swith a polarization that lifts .

2.6. Adelic description and Hecke operators. Let X and X beabelian varieties over a base S. A homomorphism : X X is anisogeny if one of the following conditions is satisfied

is surjective and ker() is a finite group scheme over S ;17


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there exists :X Xsuch that is the multiplicationbyN inXand is the multiplication by N inX for some

positive integer NA quasi-isogeny is an equivalence class of pair (, N) formed by aisogeny : X X and a positive integer N, (, N) (, N) ifand only ifN=N . Obviously, we think of the equivalence classe(, N) as N1.

Fixn, N,D as in 2.4. There is another description of the category Awhich is less intuitive but more convenient when we have to deal withlevel structures.

Let U be a free Z-module of rank 2n and let Ebe an alternatingform U U MUwith value in some rank one free Z-module MU.

Assume that the type of E is D. Let G be the group of symplecticsimilitudes of (U, MU) which associates to any ringR the groupeG(R)of pairs (g, c) GL(U R) R such that

E(gx,gy) =cE(x, y)

for everyx, y U R. ThusG is a group scheme defined over Z and isreductive away from the prime dividingD. For every prime p, definethe compact open subgroup K G(Q)

if|N, then KN, =G(Z) ; if |N, then K is the kernel of the homomorphism G(Z)

G(Z/NZ).

Let fix a prime p not dividing N nor D. Let Z(p) be the localizationofZ obtained by inverting all prime different from p.

Consider the schemes S whose residual characteristic are 0 or p.Consider the groupoid A(S) defined as follows

(1) objets ofA are triples (X,,) where Xis an abelian scheme over S; : X X is a Z(p) multiple of a polarization of degree

prime top, such that such that for every prime , for everys S the symplectic form induced by on H1(Xs,Q) issimilar to U Q;

for every prime =p, is aK-orbit of symplectic simili-tudes from H1(Xs,Q) to UQ which is invariant under1(S, s). We assume that for almost all prime, this K-orbit corresponds to the auto-dual lattice H1(Xs,Z).

(2) a homomorphism HomA((X,,), (X, , )) is a quasi-isogeny :XX such that () and differs by a scalarin Q and () =.

Consider the functor A A which associates to (X,,) A(S)the triple (X,,) A(S) where the are defined as follows. Letsbea geometric point ofS. Let be a prime not dividingNandD. Giving

a symplectic similitude from H1(Xs,Q) toUQ up to action ofKis18


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equivalent to give an auto-dual lattice of H1(Xs,Q). The K-orbit isstable under1(S, s) if and only the auto-dual lattice is invariant under

1(S, s). We pick the obvious choice H1(Xs,Z) as auto-dual lattice ofH1(Xs,Q) which is invariant under 1(S, s). If dividesD, we want a1(S, s)-invariant lattice such that the restriction of the Weil symplecticpairing is of typeD. Again, H1(Xs,Z) fulfills this property. IfdividesN, Given a symplectic similitude from H1(Xs,Q) to UQup to actionofK is equivalent to given an auto-dual lattice of H1(Xs,Q) and arigidification of the pro--part ofN torsions points ofXs. But this isprovided by the level structure in the moduli problem A.

Proposition 2.6.1. The above functor is an equivalence of categories.

Proof. As defined, it is obviously faithful. It is fully faithful be-cause a quasi-isogeny : X X which induces an isomorphism : H1(X

,Z) H1(X,Z), is necessarily an isomorphism of abelianschemes. By assumption carry on a rational multiple . But both and are polarizations of same type, must carry on . Thisprove that the functor is fully faithful.

The essential surjectivity derives from the fact that we can modify anabelian schemesXequipped with level structure , by a quasi-isogeny: XX so that the isomorphisme

U Q H1(X,Qp) H1(X,Q)

identifies U Zp with H1(X,Zp). There is a unique way to choose a

rigidification ofX[N] in compatible way with p for p|N. Since thein symplectic form Eon U is of the type (D, D) the polarization onX is also of this type.

Let us now describe the points ofA with value in C. Consider anobjet of (X,,) A(C) equipped with a symplectic basis of H1(X,Z).In this case, since is aZ(p)-multiple of a polarization ofX, it is givenby an element of

hn ={ZMn(C)| tZ=Z, im(Z)> 0}

For all = p, defines an element ofG(Qp)/Kp. At p, the integralTate module H1(X,Z) defines an element ofG(Qp)/G(Zp). It followsthat

An,N=G(Q)\[hn G(Af)/KN].

The advantage of the prime description of the moduli problem isthat we can replace the principal compact open subgroups KNby anycompact open subgroup K =

Kp G(Af) such that Kp = G(Zp)

for almost all p. In the general case, the proof of the representability

is reduced to the principal case.19


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3. Shimura varieties of PEL type

3.1. Endomorphism of abelian varieties. Let Xbe an abelian va-riety of dimension n over an algebraically closed field k. Let End(X)the ring of endomorphisms of X and EndQ(X) = End(X) Q. Ifk= C,X=V /Uthen we have two faithful representations

a : End(X) EndC(V) and r : End(X) EndZ(U).

It follows that End(X) is a torsion free abelian group of finite type.Over arbitrary field k, we need to introduce Tate modules. Let bea prime different from characteristic ofk then for every m, the kernelX[m] of the multiplication in X is isomorphic to (Z/mZ)2n.

Definition 3.1.1. The Tate moduleT(X) is the limit

T(X) = lim X[n]

of the inverse system given by the multiplication by : X[n+1] X[n].AsZ-module, T(X) = Z

2n . We noteV=T Z Q.

We can identify Tate module T(X) with first etale homology H1(X,Z)which is by definition is the dual of H1(X,Z). Similarly, V(X) =H1(X,Q).

Theorem 3.1.2. For any abelian varietiesX, Y overk,Hom(X, Y) isa finitely generated abelian group, and the natural map

Hom(X, Y) Z HomZ(T(X), T(Y))is injective.

See [13, p.176] for the proof.

Definition 3.1.3. An abelian variety is called simple if it does notadmit strict abelian subvariety.

Proposition 3.1.4. If X is a simple abelian variety, EndQ(X) is adivision algebra.

Proof. Letf :XXbe a non-zero endomorphism ofX. The identitycomponent of its kernel is a strict abelian subvariety of

Xwhich must

be zero. Thus the whole kernel of fmust be a finite group and theimage offmust be X for dimensional reason. It follows that f is anisogeny and therefore invertible in EndQ(X) and therefore EndQ(X) isa division algebra.

Theorem 3.1.5 (Poincare). Every abelian varietyXis isogenous to aproduct of simple abelian varieties.

Proof. Let Y be an abelian subvariety of X. We want to prove theexistence of a quasi-supplement ofY inXthat is a subabelian varietyZofXsuch that the homomorphismY ZXis an isogeny. Let X

be the dual abelian variety and : X Ybe the dual homomorphism20


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to the inclusion Y X. LetL be an ample line bundle over X andL :X Xthe isogeny attached to L. By restriction to Y, we get a

homomorphism E|Y :Y Ywhich is surjective since L|Y is stillan ample line bundle. Therefore the kernel Z of the homomorphism E :X Y is a quasi-complement ofY inX.

AssumeXto be isogenous to

i Xmii where theXiare mutually non-

isogenous abelian varieties andmi N. Then EndQ(X) =

i Mmi(Di)where Mmi(Di) is the algebra ofmi mi-matrices over the skew-fieldDi = EndQ(Xi).

Corollary 3.1.6. EndQ(X)is a finite-dimensional semi-simple algebraoverQ.

Proof. IfX is isogenous to Xd11 Xdrr then

EndQ(X) =Md1(D1) Mdr(Dr)

where Di = EndQ(Xi) are division algebras finite-dimensional over Q.

We have a function

deg : End(X) N

defined by the following rule : deg(f) is the degree of the isogeny f iff

is an isogeny and deg(f) = 0 iff is not an isogeny. Using the formuladeg(mf) = m2ndeg(f) for all f End(X), m Z and n = dim(X),we can extend this function to EndQ(X)

deg : EndQ Q+.

For every prime = char(k), we have a representation of the endo-morphism algebra

: EndQ(X) End(V).

These representations for different are related by the function degree.

Theorem 3.1.7. For everyfEndQ(X), we have

deg(f) = det (f) anddeg(n.1X f) =P(n)

whereP(t) = det(t (f)) is the characteristic polynomial of(f).In particular, tr((f)) is a rational number which is independent of.

Let: X Xbe a polarization ofX. One attach to an involutionon the semi-simple Q-algebra EndQ(X).

1

1Oue convention is that an involution of a non-commutative ring satisfies the

relation (xy)

=y

x

.21


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Definition 3.1.8. The Rosati involution onEndQ(X) associated with is the involution defined by the following formula

f f =1f

for everyfEndQ(X).

The polarization : X X induces an alternating form X[m] X[m] m for every m. By passing to the limit on m, we get asymplectic form

E:V(X) V(X) Q(1).

By definition f is the adjoint off for this symplectic form

E(fx,y) =E(x, fy).

Theorem 3.1.9. The Rosati involution is positive. For every f EndQ(X), tr((f f

)) is a positive rational number.

Proof. Let = L for some ample line bundle L. One can prove theformula

tr(f f) =

2n(Ln1.f(L))

(Ln) .

Since L is ample, the cup-products (Ln1.f(L)) (resp. Ln) is thenumber of intersection of an effective divisor f(L) (resp. L) withn1 generic hyperplans of |L|. Since L is ample, these intersectionnumbers are positive integers.

Let Xbe abelian variety equipped with a polarization . The semi-simple Q-algebra B= EndQ(X) is equipped with

(1) a complex representation a and a rational representation rsatisfying rQ C =a a.

(2) an involution b b such that for all b B {0}, we havetrr(bb

)> 0.

Suppose thatB is a simple algebra of centerF. ThenFis a numberfield equipped with a positive involution b b restricted from B.

There are three possibilities(1) the involution is trivial on F then F is a totally real number

field (involution of first kind). In this case,B QR is a productofMn(R) or is a product ofMn(H) where H is the algebra ofHamiltonian quaternions equipped with their respective posi-tive involutions (case C and D).

(2) the involution is non trivial onF, its fixed points forms a totallyreal number field F0 and F is a totally imaginary quadraticextension ofF0(involution of second kind). In this case,B QRis a product of Mn(C) equipped with its positive involution

(case A).22


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3.2. Positive definite Hermitian form. Let Bbe a finite-dimensionalsemisimple algebra over R with an involution. A Hermitian form on

a B-moduleV is a symmetric form V V R such that (bv,w) =(v, bw). It is positively definite if (v, v)> 0 for all v V.

Lemma 3.2.1. The following assertions are equivalent

(1) There exists a faithfulB -moduleV such thattr(xx, V)> 0 forallx B {0}.

(2) The above is true for every faithfulB-moduleV(3) trB/R(xx

)> 0 for all nonzero x B .

3.3. Skew-Hermitian modules. Summing up what has been said inthe last two sections, the endomorphisms of a polarized abelian variety,

after tensoring with Q is a finite-dimensional semi-simple Q-algebraequipped with a positive involution. For every prime = char(k),this algebra has a representation on the Tate module V(X) which isequipped with a symplectic form. We are going now to look at thisstructure in more axiomatic way.

Let k be a field. LetB be a finite-dimensional semisimple k-algebraequipped with an involution . Let 1, . . . , r be a basis of B as k-vector space. For any finite-dimensional B-module V we can define apolynomial detV k[x1, . . . , xr] by the formula

detV= det(x11+ xrr, V kk[x1, . . . , xr])

Lemma 3.3.1. Two finite-dimensional B-modules V andU are iso-morphic if and only ifdetV= detU.

Proof. Ifk is an algebraically closed field, B is a product of matrixalgebras over k. The lemma follows from the classification of modulesover a matrix algebra. Now let kbe an arbitrary field andkits algebraicclosure. Automorphism of a B-module is of the form GLm(F) whereF is the center ofB, has trivial Galois cohomology. This allows us todescend from the algebraically closure ofk to k.

Definition 3.3.2. A skew-HermitianB-module is aB-moduleUwhichis equipped with a symplectic form

U UMU

with value in a 1-dimensional k-vector space MU such that (bx,y) =(x, by) for anyx, y V.

The groupG(U) of automorphism of a skew-HermitianB -module Uis pair (g, c) where g GLB(U) and c Gm,k such that (gx,gy) =c(x, y) for any x, y U.

Ifk is an algebraically closed field, two skew-Hermitian modules Vand U are isomorphic if and only if detV = detU. In general, theset of skew-Hermitian modules V with detV = detU is classified by

H1(k, G(U)).23


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Let k = R, B is a finite-dimensional semi-simple algebra over Requipped with an involution and U is a skew-Hermitian B-module.

Let h : C EndB(UR) such that (h(z)v, w) = (v, h((z)w) and suchthat the symmetric bilinear form (v, h(i)w) is positive definite.

Lemma 3.3.3. Let h, h : C EndB(UR) be two such homomor-phisms. Suppose that the two BR C-modulesU induced byh andhare isomorphic, thenh andh are conjugate by an element ofG(R).

Let B be is a finite-dimensional simple Q-algebra equipped with aninvolution and let UQ be skew-Hermitian module UQ UQ MUQ .An integral structure is an order OB ofB and a free abelian group Uequipped with multiplication byOB and an alternating formU U

MUof which the generic fibre is the skew Hermitian module UQ.

3.4. Shimura varieties of type PEL. Let fix a prime p. We willdescribe the PEL moduli problem over a discrete valuation ring withresidual characteristic punder the assumption that the PEL datum isunramified at p.

Definition 3.4.1. A rational PE-structure (polarization and endomor-phism) is a collection of data as follows

(1) B is a finite-dimensional simpleQ-algebra, assume thatBQp isa product of matrix algebra over unramified extensions ofQp;

(2) is a positive involution ofB;(3) UQ is skew-HermitianB-module;(4) h: C EndB(UR) such that(h(z)v, w) = (v, h(z)w) and such

that the symmetric bilinear form(v, h(i)w) is positive definite.

The homomorphismhinduces a decomposition UQ Q C =U1 U2whereh(z) acts onU1byzand onU2byz. Let choose a basis1, . . . , rof the Z-module OB which is free of finite rank. Let X1, . . . , X t beindeterminates. The determinant polynomial

det1 = det(x11+ + xrr, V1 C[x1, . . . , xr])

is a homogenous polynomial of degree dimC U1. The subfield ofCgen-erated by the coefficients of the polynomial f(X1, . . . , X t) is a numberfield which is independent of the choice of the basis 1, . . . , t. Theabove number fieldEis called the reflex fieldof the PE-structure. It isequivalent to define E as the definition field of the isomorphism classof the BC-moduleU1.

Definition 3.4.2. An integral PE structure consists in a rational PEstructure equipped with the following extra data

(5) OB is a order ofB stable under which is maximal atp.

(6) Uis anOB-integral structure of the skew-Hermitian moduleUQ.24


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Suppose that 1, . . . , br is a Z-basis ofOB, then the coefficients ofthe determinant polynomial detU1 lie in O= OEZ Z(p).

Let fix an integer N3. Consider the moduli problem Bof abelianschemes with PE-structure and with principal N-level structure. Thefunctor Bassociates to any O-scheme Sthe category B(S) whose ob-jects are

(A,,,)

where

(1) Ais an abelian scheme over S

(2) : A Ais a polarization(3) : OB End(A) such that the Rosati involution induced by

restricts to the involution ofOB and such that

det(1X1+ rxr, Lie(A)) = detU1

such that for all prime = p such that for all geometric points of S, T(As) equipped with the action of OB and with thealternating form induced by is similar to U Z.

(4) is a similitude from A[N] equipped with the symplectic formand the action ofOB and U/NUthat can be lifted to an iso-morphism H1(A,A

pf) with UZ A

pf.

Theorem 3.4.3. The functor which associates to aE-schemeSto theset of isomorphism classes B(S) is smooth representable by a quasi-

projective scheme overOE Z(p).Proof. For =p, the isomorphism class of the skew-Hermitian moduleT(As) is locally constant with respect to s so that we can forget thecondition on this isomorphism class in representability problem.

By forgetting endomorphisms, we have a morphism B A. It isequivalent to have and to have actions of1, . . . , r satisfying certainconditions. Therefore, it suffices to prove thatB A is representableby a projective morphism for what it is enough to prove the followinglemma.

Lemma 3.4.4. Let A be a projective abelian scheme over a locally

noetherian schemeS. Then the functor that associate to anyS-schemeT the setEnd(AT) is representable by a disjoint union of of projectivescheme overS.

Proof. A graph of an endomorphism b ofA is a closed subscheme of ofA SA so that the functor of endomorphisms for a subfunctor of someHilbert scheme. Lets check that this subfunctor is representable by alocally closed subscheme of the Hilbert scheme.

Let Z A SA a closed subscheme flat over a connected base S.Lets check that the condition s Ssuch thatZs is a graph is an opencondition. Suppose that pA : Zs As is an isomorphisme over a point

s S. By flatness, the relative dimension ofZoverSis equal to that of25


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A. For everys S, everya A, the intersectionZs{a}Asis eitherof dimension bigger than 0 either consists in exactly one point since the

intersection number is constant under deformation. This implies thatthe morphisme p1 :ZA is birationnal projective morphism. Thereis an open subset U ofA over which p1 : Z A is an isomorphism.Since : A S is proper, A(A U) is closed. Its complementSA(AU) which is open, is the set ofs Sover whichp1 : Zs Asis an isomorphism.

Let Z ASA be a graph of a morphism f : A A. f is ahomomorphism if and only iffsends the unit on the unit so that thecondition s Ssuch that fs is a homomorphism is a closed condition.So the functor which attach to a S-scheme T the set EndT(AT) is

representable by a locally closed subscheme of a Hilbert scheme.In order to prove that this subfunctor is represented by a closedsubscheme of the Hilbert scheme, it is enough to verify the valuativecriterion.

Let S= Spec(R) be the spectrum of a discrete valuation ring withgeneric point . Let A ba a S-abelian scheme and f : A A bean endomorphism. Then f can be extended in a unique way to anendomorphism f :A A by Weils extension lemma.

Theorem 3.4.5 (Weil). LetG be a smooth group scheme overS. LetX be smooth scheme overS and U X is a open subscheme whosecomplement Y = XU has codimension 2. Then and morphismf : U G can be extended to X. In particular, if G is an abelianscheme, we can always extend.

3.5. Adelic description. Let G be the Q-reductive group defined asthe automorphism group of the skew-Hermitian module UQ. For everyQ-algebra R, let

G(R) ={(g, c)GLB(U)(R) R|(gx,gy) =c(x, y)}.

For all = p we have a compact open subgroup K G(Q) whichconsists in g G(Q) such that g(U Z) = (U Z) and whichsatisfies an extra condition in the case |Nthat the action induced byg on (U Z)/N(U Z) is trivial .

Lemma 3.5.1. There exists a unique smooth group schemeGK overZ such thatGKZ Q = G Q Q andGK(Z) =K.

Consider the functor B which associates to any E-scheme the cate-gory B(S) : objects of this category are

(A,,,)

where

(1) Ais a S-abelian schemes over S,

(2) : A Ais a Z(p)-multiple of a polarization,26


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(3) : OB End(A) such that the Rosati involution induced by restricts to the involution ofOB and such that

det(1X1+ tXt, Lie(A)) =f(X1, . . . , X t)

(4) fix a geometric point s ofS, for every prime =p, is a K-orbit of isomorphisms from V(As) to Q compatible withsymplectic forms and action ofOB and stable under the actionof1(S, s)

Morphisms of from (A,,,) to (A,,,) is a quasi-isogeny : A A of degree prime to p carrying to a scalar (in Q) multiple of

and carrying on .

Proposition 3.5.2. The obvious functorB B is an equivalence of

categories.

The proof is the same as in the Siegel case.

3.6. Complex points. An isomorphism class of objet (A,,,) B(C) gives rises to

(1) a skew-Hermitian B -module H1(A,Q),(2) for every prime , a Q-similitude H1(A,Q)UQ as skew-

Hermitian B Q-modules, defined up to the action ofK.

For = p, this is required in the moduli problem. For the prime p,for everyb B , tr(b, H1(A,Qp)) = tr(b, Q) because both are equal

with tr(b, H1(A,Q)) for any =p. It follows that the skew-Hermitianmodules tr(b, H1(A,Qp)) and tr(b, Qp) are isomorphic after basechange to a finite extension ofQp and therefore the isomorphism classof the skew-Hermitian module defines an element of p H

1(Qp, G).Now, in the groupodB(C) the arrows are given by prime to p isogeny,H1(A,Zp) is a well-defined self-dual lattice stable by multiplication byOB. It follows that the classp H

1(Qp, G) mentioned above comesfrom a class in H1(Zp, GZp) where GZp is the reductive group schemeover Zp which extend GQp. In the case Gso GZp has connected fibres,this implies the vanishing ofp. Kottwitz gave a further argument inthe case where Gis not connected.

The first datum gives rise to a class

H1(Q, G)

and the second datum implies that the images of in H1(Q, G) van-ishes. We have

ker1(Q, G) = ker(H1(Q, G)

H1(Q, G)).

According to Borel and Serre, ker1(Q, G) is a finite set. For every ker1(Q, G), fix a skew-Hermitian B-module V() whose class inker1(Q, G) isand fix a Q-similitudeV

() Q withVQ as skew-

Hermitian B Q-module and also a similitude over R.27


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Let denote B()(C) be the subset ofB()(C) of (A,,,) such thatH1(A,Q) is isomorphic to V

(). Let (A,,,) B()(C) and let be

an isomorphism of skew-Hermitian B-modules from H1(A,Q) to V().The set of quintuple (A,,,,) can be described as follows

(1) Then defines an element G(Af)/K.(2) The complex structure on Lie(A) =VQR defines a homomor-

phismh : C EndB(VR) such thath(z) is the adjoint operatorofh(z) for the symplectic form on VR. Since is a polariza-tion, (v, h(i)w) is positive or negative definite. Moreover theisomorphism B C-moduleV is specified by the determinantcondition on the tangent space. It follows that h lies in a G(R)-conjugacy classeX.

Therefore the set of quintuples isXG(Af)/K. Two different triv-ializationsand differs by an automorphism of the skew-HermitianB-module V(). This group is the inner formG() ofG obtained by theimage of H1(Q, G) in H1(Q, Gad). In conclusion we get

B()(C) =G()(Q)\[X G(Af)/K]

and

B(C) =

ker1(Q,G)

B()(C).

4. Shimura varieties

4.1. Review on Hodge structures. See [Deligne, Travaux de Grif-fiths]. Let Q be a subring ofR : we think specifically about the casesQ = Z,Q or R. A Q-Hodge structure will be called respectively inte-gral, rational or real Hodge structure.

Definition 4.1.1. A Q-Hodge structure is a projective Q-module Vequipped with a bigraduation ofVC = V Q C

VC =p,q

Hp,q

such that Hp,q

and Hq,p

are complex conjugate i.e. the semi-linearautomorphism ofVC =V Q C given byv zv z, satisfies therelation(Hp,q) =Hq,p for everyp, q Z.

The integers hp,q = dimC(Hp,q) are called Hodge numbers. We have

hp,q = hq,p. If there exists an integer n such that Hp,q = 0 unlessp+ q = n then the Hodge structure is said to be pure of weight n.When the Hodge structure is pure of weight n, the Hodge filtrationFpV =

rp V

rs determines the Hodge structure by the relation Vpq =

FpV FqV.Let S = ResC/RGm be the real algebraic torus defined as the Weil

restriction from C to R of Gm,C. We have a norm homomorphism28


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C R whose kernel is the unit circle S1. Similarly, we have anexact sequence of real tori

1 S1 S Gm,R 1.

We have an inclusion R C whose cokernel can be representedby the homomorphism C S1 given by z z/z. We have thecorresponding exact sequence of real tori

1 Gm,R S S1 1.

The inclusionw: Gm,R S is called the weight homomorphism.

Lemma 4.1.2. Let G = GL(V) the linear group defined over Q. AHodge structure onV is equivalent to a homomorphismh: S GR =

GQ R. The Hodge structure is pure of weight n if the restrictionof x to Gm,R S factors through the centerGm,R = Z(GR) and thehomomorphismGm,R Z(GR) is given byttn.

Proof. A bi-graduationVC =

p,qVp,q is the same as a homomorphism

hC : G2m,C GC. The complex conjugation ofVC exchange the factorsVp,q and Vq,p if and only if hC descends to a homomorphism of realalgebraic groupsh: S GR.

Definition 4.1.3. A polarization of a Hodge structure (VQ, Vp,q) of

weightn, is a bilinear formK onVK such that the induced form

onVR is invariant underh(S1

) and such that the form(x, h(i)y) issymmetric and positive definite.

It follows from the identity h(i)2 = (1)n that the bilinear form(x, y) is symmetric ifn is even and alternate ifn is odd :

(x, y) = (1)n(x, h(i)2y) = (1)n(h(i)y, h(i)x) = (1)n(y, x).

Example. An abelian varieties induces a typical Hodge structure.Let X=V /Ube an abelian variety. Let G beGL(UQ) as algebraicgroup defined over Q. The complex structure V on the real vectorspace U R =V induces a homomorphism of real algebraic groups

: S GR

so that Uis equipped with a structure of integral Hodge structure ofweight 1. A polarization ofXis symplectic form Eon V, taking in-tegral values onUsuch thatE(ix, iy) =E(x, y) and such that E(x,iy)is a positive definite symmetric form.

Let V be a projective Q-module of finite rank. A Hodge structureonVinduces Hodge structures on tensor products Vm (V)n. Letfix a finite set of tensors (si)

si Vmi (V)ni.

Let G GL(V) be the stabilizer of these tensors.29


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Lemma 4.1.4. There is a bijection between the Hodge structures onVfor which the tensorssi are of type(0, 0) and the set of homomorphism

S GR.

Proof. A homomorphism h : S GL(V)R factors through GR if andonly if the image h(S) fix all tensors si. This is equivalent to say thatthese tensors are of type (0, 0) for the induced Hodge structures.

There is a related notion of Mumford-Tate group. Let G be a Q-group and: S1 GRa The Mumford-Tate group ofXis the smallestalgebraic subgroup Hg() ofG, defined over Q such that : S1 GRfactors through Hg(). Originally, Mumford called Hg() the Hodgegroup.

Definition 4.1.5. LetG be an algebraic group overQ. Let: S1 GR be a homomorphism of real algebraic groups. The Mumford-Tategroup of (G, ) is the smallest algebraic subgroup H = Hg() of Gdefined overQ such that factors throughHR.

Let Q[G] be the ring of algebraic functions over G and R[G] =Q[G] Q R. The group S1 acts on R[G] through the homomorphism. Let R[G]=1 be the subring of functions fixed by(S1) and considerthe subring

Q[G] R[G]=1

ofQ[G]. For everyv Q[G]R[G]=1, letGvbe the stabilizer subgroupofG at v. SinceGv is defined over Q and factors through Gv,R, wehave the inclusion HGv. In particular, v Q[G]H. It follows that

Q[G] R[G]=1 = Q[G]H.

This property does not however characterize H. In general, for anysubgroup H ofG, we have an obvious inclusion

HH =

vQ[G]H

Gv.

which might be strict. IfH=H then we say that H is an observable

subgroup ofG. To prove that this is indeed the case for the Mumford-Tate group of an abelian variety, we will need the following generallemma.

Lemma 4.1.6. LetH be a reductive subgroup of a reductive group GthenHis observable.

Proof. Assume the base field k = C. According to Chevalley, see[Borel], for every subgroup H of G, there exists a representation :G GL(V) and a vector v V such that H is the stabilizer of theline kv. Since H is reductive, there exists a complement U ofkv inV. Let kv V be the line orthogonal to Uwith some generator v.

Then H is the stabilizer of the vector v v V V.30


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Let G = GL(UQ) and : S1 GR be a homomorphism such that(i) induces a Cartan involution on GR. Let Cbe the smallest tensor

subcategory stable by subquotient of the category of Hodge structurethat contains (UQ, ). There is the forgetful functor FibC :C VectQ.

Lemma 4.1.7. H is the automorphism group of the functorFib.

Proof. LetVbe a representation ofGdefined over Q equipped with theHodge structure defined by . Let Ube a subvector space ofV com-patible with the Hodge structure. ThenHmust stabilizeU. It followsthat Hacts naturally on FibC i.e. we have a natural homomorphismHAut(FibC).

Proposition 4.1.8. The Mumford-Tate group of a polarizable Hodgestructure is a reductive group.

Proof. Since we are working over a fields of characteristic zero, H isreductive if and only if the category of representations ofH is semi-simple. Using the Cartan involution, we can exhibit a positive definitebilinear form on V. This implies that every subquotient of Vm (V)n is a subobject.

4.2. Variation of Hodge structures. Let Sbe a complex analyticvariety. The letterQ denote a ring contained in R which could be Z,Qor R.

Definition 4.2.1. A variation of Hodge structures (VHS) on S ofweightn consists in the following data

(1) a local system projectiveQ-modulesV(2) a decreasing filtration FpV on the vector bundle V = V Q

OS such that the Griffiths transversality is satisfied i.e. for allintegeri

(FpV) Fp1V 1S

where : V V 1S is the connectionv f v df forwhichV Q C is the local system of horizontal sections

(3) for everys S, the filtration induces onVs a pure Hodge struc-ture of weightn.

There are obvious notion of the dual VHS and tensor product ofVHS. The Leibnitz formula (v v) =(v) v +v (v) assurethat Griffiths transversality is satisfied for the tensor product.

Typical examples of polarized VHS are provided by cohomology ofsmooth projective morphism. Let f : X Sbe a smooth projectivemorphism over a complex analytic variety S. Then Hn = RnfQ isa local system of Q-vector spaces. Since Hn Q OS is equal to deRham cohomology HndR = R

nfX/S where

X/S is the relative de

Rham complex. Since the spectral sequence degenerates on E2, the31


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abutments HndR are equipped with a decreasing filtration by subvectorbundle Fp(HndR) with

(Fp/Fp+1)HndR= Rqf

pX/S

withp + q= n. The connexion satisfies the Griffiths transversality.By Hodges decomposition, we have instead a direct sum

HndR(Xs) =pq

Hp,q

with Hp,q = Hq(Xs, pXs

) and Hp,q = Hq,p so that all axioms of VHSare satisfied.

If we choose a projective embedding, X PdS, the line bundle OPd(1)defines a class

cH0(S, R2fQ).

By hard Lefschetz theorem, the cup product by cdn induces an iso-morphism

RnfQ R2dnfQ defined by c

dn

so that by Poincare duality we get a polarization on RnfQ.

4.3. Reductive Shimura datum. The torus S= ResC/RGm plays aparticular role in the formalism of Shimura varieties shaped by Delignein [4], [5].

Definition 4.3.1. A Shimura datum is a pair (G, X) consisting of areductive group G overQ and aG(R)-conjugacy classXof homomor-phismsh: S GR satisfying the following properties

(SD1) Forh X, only the charactersz/z, 1,z/z occur in the repre-sentation ofS onLie(G);

(SD2) adh(i) is a Cartan involution ofGad i.e. if the real Lie group{g G(C)| ad(h(i))(g) =g} is compact.

The action S, restricted to Gm,R is trivial on Lie(G) so that h : S GR sends Gm,R into the center ZR ofGR. The induced homomorphismw = h|Gm,R : Gm,R ZR is independent of the choice ofh X. Wecall w the weight homomorphism.

Base change to C, we have S R C = Gm Gm where the factorsare ordered in the way that S(R) S(C) is the map z (z, z).Let : Gm,C SC the homomorphism defined by z (z, 1). If h: S GL(V) is a Hodge structure then h=hC : GC GL(VC)determines its Hodge filtration.

Siegel case. Abelian varietyA = V /Uis equipped with a polarizationEwhich is a non-degenerate symplectic form on UQ = UQ. Let GSpthe group of symplectic similitudes

GSp(UQ, E) ={(g, c) GL(UQ) Gm,Q|E(gx,gy) =cE(x, y)}.32


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The scalar c is called the similitude factor. Base changed to R, weget the group of symplectic similitudes of the real symplectic space

(UR, E). The complex vector space structure on V = UR induces ahomomorphism

h: S GSp(UR, E).

In this case, X is the set of complex structures on UR such thatE(h(i)x, h(i)y) = E(x, y) and E(x, h(i)y) is a positive definite sym-metric form.

PEL case. Suppose B is a simple Q-algebra of center F equippedwith a positive involution . LetF0 Fbe the fixed field by . LetGbe the group of symplectic similitudes of a skew-Hermitian B-module

VG= {(g, c) GLB(V) Gm,Q|(gx,gy) =c(x, y)}.

Let G1 be the subgroup ofG defined by

G1(R) ={x (CQ R|xx = 1}

for any Q-algebra R. We have an exact sequence

1 G1 G Gm 1.

The group G1 is a scalar restriction of a group G0 defined over F0.Since simple R-algebra with positive involution must be Mn(C),

Mn(R) or Mn(H) with their standard involutions there will be threecases to be considered.

(1) Case (A) : If [F :F0] = 2, then F0 is a totally real field and Fis a totally imaginary extension. Over R,B Q Ris product of[F0: Q] copies ofMn(C). G1 = ResF0/QG0 whereG0is an innerform of the quasi-split unitary group attached to the quadraticextension F/F0.

(2) Case (C) : IfF =F0 thenF is a totally real field. and B Risisomorphic to a product of [F0 : Q] copies ofMn(R) equippedwith their positive involution. In this case, G1 = ResF0/QG0

where G0 is an inner form of a quasi-split symplectic groupover F0.

(3) Case (D) :B R is isomorphic to a product of [F0: Q] copies ofMn(H) equipped with positive involutions. The simplest caseis B = H, V is a skew-Hermitian quaternionic vector space.In this case, G1 = ResF0/QG0 where G0 is an even orthogonalgroup.

Tori case. In the case whereG = Tis a torus over Q, both conditions(SD1) and (SD2) are obvious since the adjoint representation is trivial.The conjugacy class ofh : S TR contains just one element since T is

commutative.33


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Deligne proved the following statement in [5, prop. 1.1.14] whichprovides a justification for to the not so natural notion of Shimura

datum.

Proposition 4.3.2. Let (G, X) be a Shimura datum. ThenX has aunique structure of a complex manifold such that for every representa-tion : G GL(V), (V, h)hX is a variation of Hodge structurewhich is polarizable.

Proof. Let : G GL(V) be a faithful representation ofG. Sincewh ZG, the weight filtration of Vh is independent of h. Since theweight graduation is fixed, the Hodge structure is determined by theHodge filtration. It follows that the morphism to the Grassmannian

:XGr(VC)

which sends h on the Hodge filtration attached to h, is injective. Weneed to prove that this morphism identifies Xwith the complex sub-variety of Gr(VC). It suffices to prove that

d : ThXT(h)Gr(VC)

identifiesThXwith a complex vector subspace of Gr(VC).Let g be the Lie algebra of G and ad : G GL(g) the adjoint

representation. Let Gh be the centralizer ofh, and gh its Lie algebra.We have gh =gR g0,0 for the Hodge structure on g induced by h. Itfollows that the tangent space to the real analytic variety Xat his

ThX=gR/g0,0C gR.

Let Wbe a pure Hodge structure of weight 0. Consider the R-linearmorphisme

WR/WR W0,0C WC/F

0WC

which is injective. Since both vector spaces have the same dimensionover R, it is also surjective. It follows that WR/WR W

0,0C admits a

canonical complex structure.Since the above isomorphism is functorial on the pure Hodge struc-

tures of weight 0, we have a commutative diagram

gR/g0,0C gR

End(VR)/End(VR) End(VC)0,0

gC/F0gC End(VC)/F

0End(VC)

which proves that the image of ThX = gR/g0,0C in T(h)Gr(VC) =

End(VC)/F0End(VC) is a complex subvector space.34


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The Griffiths transversality ofVOXfollows from the same diagram.There is a commutative triangle of vector bundles

T X

End(V OX)

End(V OX)/F0End(V OX)

where the horizontal arrow is the derivation in V OX. The Griffithstransversality of V OX is satisfied if and only if the image of thederivation is contained inF1End(V OX). But this follows from thefact that

gC = F1gC

and the map T X End(V OX)/F0End(V OX) factors through(gC OX)/F

0(gC OX).

4.4. Dynkin classification. Let (G, X) be a SD-datum. Over C, wehave a conjugacy class of cocharacter

ad : Gm,C GadC.

The complex adjoint semi-simple group Gad is isomorphism to a prod-uct of complex adjoint simple groupsGad =

i Gi. The simple complex

adjoint groups are classified by their Dynkin diagrams. The axiomSD1implies that ad induces an action ofGm,C on gi of which the set of

weights is {1, 0, 1}. Such cocharacters are called minuscules. Minus-cule coweights are some of the fundamental coweights and therefore canbe specified by special nodes in the Dynkin diagram. Every Dynkindiagram have at least one special node except three of them those thatare named F4, G2, E8. We can classify DS-data over the complexnumbers with helps of Dynkin diagram.

4.5. Semi-simple Shimura datum.

Definition 4.5.1. A semi-simple Shimura datum is a pair (G, X+)consisting of a semi-simple algebraic group G overQ and a G(R)+-

conjugacy class of homomorphismh

1

:S1

GR satisfying the axioms(SD1) and(SD2). HereG(R)+ denotes the neutral component ofG(R)for the real topology.

Let (G, X) be a reductive Shimura datum. LetGad be the adjointgroup ofG. Everyh Xinduces a homomorphismh1 : S1 Gad. TheGad(R)+-conjugacy class X+ of h1 is isomorphic with the connectedcomponent ofh inX.

The spaces X+ are exactly the so-called Hermitian symmetric do-mains with symmetry group G(R)+.

Theorem 4.5.2 (Baily-Borel). Let be a torsion free arithmetic sub-

group of G(R)+. The quotient \X+ has a canonical realization as35


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Zariski open subset of a complex projective algebraic variety. In par-ticular, it has a canonical structure of complex algebraic variety.

These quotients \X+ as complex algebraic variety, are called con-nected Shimura variety. The terminology is a bit confusing, becausethey are not Shimura varieties which are connected but the connectedcomponents of Shimura varieties.

4.6. Shimura varieties. Let (G, X) be a Shimura datum. For a com-pact open subgroup K ofG(Af), consider the double coset space

ShK(G, X) =G(Q)\[X G(Af)/K]

in whichG(Q) acts onX and G(Af) on the left andKacts on G(Af)on the right.

Lemma 4.6.1. LetG(Q)+ = G(Q) G(R)+. LetX+ be a connectedcomponent ofX. Then there is a homeomorphism

G(Q)\[X G(Af)/K] =

\X+

whereruns over a finite set of representatives ofG(Q)+\G(Af)/Kand =K

1 G(Q).

Proof. Consider the map

\X+ G(Q)+\[X+ G(Af)/K]

sending the class ofx X+ on the class of (x, ) X G(Af) whichis bijective by the very definition of the finite set and of the discretegroups .

It follows from the theorem of real approximation that the map

G(Q)+\[X+ G(Af)/K] G(Q)\[X G(Af)/K]

is a bijection.

Lemma 4.6.2 (Real approximation). For any connected group G overQ, G(Q) is dense inG(R).

See [16, p.415].

Remarks.

(1) The group G(Af) acts on the inverse limit G(Q)\[X G(Af].On Shimura varieties of finite level, there is an action of Heckealgebras by correspondences.

(2) In order to have an arithmetic significance, Shimura varietiesmust have models over a number field. According to the the-ory of canonical model, there exists a number field called thereflex field Edepending only on the SD-datum over which theShimura variety has a model which can be characterized by

certain properties.36


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(3) The connected components of Shimura varieties have canonicalmodels over abelian extensions of the reflex E which depend

not only on the SD-datum but also on the level structure.(4) Strictly speaking, the moduli of abelian varieties with PEL is

not a Shimura varieties but a disjoint union of Shimura varieties.The union is taken over the set ker1(Q, G). For each class ker1(Q, G), we have a Q-group G() which is isomorphic to Gover Qp and over R but which might not be isomorphic to Gover Q.

(5) The Langlands correspondence has been proved in many partic-ular cases by studing the commuting action of Hecke operatorsand of Galois groups of the reflex field on the cohomology of

Shimura varieties.5. CM tori and canonical model

5.1. PEL moduli attached to a CM torus. Let F be a totallyimaginary quadratic extension of a totally real number field F0of degreef0 over Q. We have [F : Q] = 2f0. Such a field Fis called a CM field.Let Fdenote the non-trivial element of Gal(F/F0). This involutionacts on the set HomQ(F,Q) of cardinal 2f0.

Definition 5.1.1. A CM-type of F is a subset HomQ(F,Q) ofcardinalf0 such that

() = and () = HomQ(F,Q).

A CM type is a pair(F, )constituting of a CM fieldFand a CM type ofF.

Let (F, ) be a CM type. The absolute Galois group Gal(Q/Q) actson HomQ(F,Q). Let Ebe the fixed field of the open subgroup

Gal(Q/E) ={ Gal(Q/Q)|() = }.

For every b F,

(b)E

and conversely Ecan be characterized as the subfield ofQ generatedby the sums

(b) for b F.

Let OFbe an order ofF. Let be the finite set of primes where OF isramified over Z. By construction, the scheme ZF= Spec(OF[p1]p)is a finite etale over Spec(Z) . By construction the reflex field Eis also unramified away from and let ZE= Spec(OF[p1]p). Thenwe have a canonical isomorphism

ZF ZE= (ZF0 ZE) (ZF0 ZE)()

where (ZF0 ZE) and (ZF0 ZE)()are two copies of (ZF0 ZE)()

with ZF0 = Spec(OF0[p1]p).37


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To complete the PE-structure, we will take U to be the Q-vectorspace F. The Hermitian form on Uwith be give by

(b1, b2) = trF/Q(cb1(b2))

for some element c F such that (c) =c. The reductive groupGassociated to this PE-structure is a Q-torusTequipped with a cochar-acterh: S Twhich can be made explicite as follows.

LetT = ResF/QGm. The CM-type induces an isomorphism R-algebras and of tori

FR C =

C andT(R) =

C.

According to this identification, h: STR is the diagonal homomor-phismC

C.

The complex conjugation induces an involution onT. The normNF/F0 given by x x(x) induces a homomorphism ResF/QGm ResF0/QGm.

The torusT is defined as the pullback of the diagonal subtorus GmResF0/QGm. In particular

T(Q) ={x F|x(x) Q}.

The character h : S TR factors through Tand defines a characterh: S T. As usual hdefines a character

: Gm,C TC

defines at level of points

C

Hom(F,Q)

C

is identity on the component and is trivial on the component(). The reflex field E is the field of definition of.

Let p / an unramified prime ofOF. Choose an open compact

subgroup Kp T(Apf) and take Kp= T(Zp).

We consider the functor Sh(T, h) which associates to a ZE-schemeSthe set of isomorphism classes of

(A,,,)

where

Ais an abelian scheme of relative dimension f0 over S ; : OF End(A) an action ofFonAsuch that for everyb F,

for every geometric point sofS

tr(b, Lie(As)) =

(a);

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is a polarization ofA whose Rosati involution induces on Fthe complex conjugation ;

is a level structure.

Proposition 5.1.2. Sh(T, h) is a finite etale scheme overZE.

Proof. Since Sh(T, h) is quasi-projective over ZE, it suffices to checkthe valuative criterion for properness and the unique lifting propertyof etale morphism.

Let S = Spec(R) be a spectrum of a discrete valuation ring withgeneric point Spec(K) and with closed point Spec(k). Pick a pointxKSh(T, h)(K) with

xK= (AK, K, K, K).

The Galois group Gal(K/K) acts on the F Q-module H1(AKK,Q). It follows that Gal(K/K) acts semisimply. After replacingKby a finite extension K, R by its normalization R in K, AK ac-quires a good reduction i.e. there exists an abelian scheme over R

such that whose generic fiber is AK. The endomorphisms extend byWeils extension theorem. The polarization needs a little more care.The symmetric homomorphismK :AK AKextends to a symmetrichomomorphism : A A. After finite etale base change ofS, thereexists an invertible sheafL on A such that = L. By assumptionLK is an ample invertible sheaf over AK. is an isogeny, L is nondegenerate on generic and on special fibre. Mumfords vanishing theo-rem implies that H0(XK, LK)= 0. By upper semi-continuity property,H0(Xs, Ls) = 0. But since Ls is non-degenerate, Mumfordvanishingtheorem says that Ls is ample.

This proves that Sh(T, h) is proper. Let S= Spec(R) whereR is alocal artinian OE-algebra with residual field k and S= Spec(R) withR= R/I,I2 = 0. Let denote s = Spec(k) the closed point ofSand S.Letx Sh(T, h)(S) withx = (A,,,). We have the exact sequence

0 A H1dR(A) Lie(A) 0

with compatible action ofOFZ OE. AsOZFZE -module, As is sup-

ported by (ZF0 ZE)) and Lie(A) is supported by ZF0 ZE() sothat the above exact sequence splits. This extends to a canonical splitof the cristalline cohomology H1cris(A/S)S. According to Grothendieck-Messing, this splitting induces a lift of the abelian scheme A/Sto anabelian scheme A/S. The additional structures,, by functorialityof Grothendieck-Messings theory.

5.2. Description of its special fibre. We will keep the notationsof the previous paragraph. Let pick a place v of the reflex field Ewhich does not lie over the finite set of primes where OFis ramified.OE is unramified ovec Z at the place v. We want to describe the set

ShK(T, h)(Fp) equipped with the operator of Frobenius Frobv.39


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Theorem 5.2.1. There is a natural bijection

ShK(T, h)(Fp) =

T(Q)\Yp

Yp

where

(1) runs over the set of isogeny classes compatible with action ofOE(p)

(2) Yp =T(Apf)/Kp

(3) Yp=T(Qp)/T(Zp)(4) for every T(Apf) we have(x

p, xp) = (xp, xp)

(5) the FrobeniusFrobv acts by the formula

(xp, xp)(xp, NEv/Qp((p

1))xp)

Proof. Let x0 = (A0, 0, 0, 0) ShK(T, h)(Fp). Let X be the setpair (x, ) where x= (A,,,)ShK(T, h)(Fp) and

: A0 A

is a quasi-isogeny which is compatible with the actions of OF andtransform 0 onto a rational multiple of.

We will need to prove the following two assumptions :

(1) X=Yp Ypwith the prescribed action of Hecke operators andof Frobenius ;

(2) the group of quasi-isogenies ofA0compatible with0and trans-

forms 0 into a rational multiple, is T(Q).

Quasi-isogeny of degree relatively prime to p. Let Yp the subset ofX where we impose the degree of the quasi-isogeny to be relativelyprime to p. Consider the prime description of the moduli problem.A point (A,,,) is a abelian variety up to isogeny, is a rationalmultiple of a polarization, is the multiplication by OF on A and is an isomorphism from H1(A,Q) and U compatible with andtransform on a rational multiple of symplectic form on U, givenmodulo a open compact subgroup K. By this description, an isogenyof degree prime to p compatible with and preserving the Q-line of

the polarization, is given by an element g T(Apf). The polarization gdefines an isomorphism in the category B if and only ifg = . Thus

Yp =T(Apf)/Kp

with obvious action of Hecke operators and trivial action of Frobv.

Quasi-isogeny of degree power ofp. Let Yp the subset ofXwhere weimpose the degree of the quasi-isogeny to be a power ofp. We will usecovariant Dieudonne theory to describe the set Yp with action of theFrobenius operator.

Let W(Fp) be the ring of Witt vectors with coefficients in Fp. Let L

be the field of fractions ofW(Fp) and we will write OLinstead ofW(Fp).40


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The Frobenius automorphism : x xp ofFpinduces by functorialityan automorphism on the Witt vectors. For every abelian variety A

over Fp, H1cris(A/OL) is a free OL-module of rank 2n equipped withan operator which is -linear. Let D(A) = Hcris1 (A/OL) denotesthe dual OL-module of H1cris(A/OL), where acts in

1-linear way.Furthermore, there is a canonical isomorphism

Lie(A) =D(A)/D(A).

Let L be the field of fractions ofOL. A quasi-isogeny : A0 Ainduces an isomorphism D(A0) FpL D(A) FpL compatible withthe multiplication by OF and preserving the Q-line of the polariza-tions. The following proposition is an immediate consequence of theDieudonne theory.

Proposition 5.2.2. Let H = D(A0)Fp L. The above constructiondefines a bijection betweenYp and the set of latticesD Hsuch that

(1) pD D D,(2) stable under the action ofOB and which satisfies the relation

tr(b, D/V D) =

(b) for allb OB,(3) D is autodual up to a scalar inQp.

Moreover, the Frobenius operator onYpthat transforms the quasi-isogeny : A0 A on the quasi-isogeny : A0 A

A acts on theabove set of lattices by sendingD on1D.

Since Sh(T, h) is etale, there exists an unique lifting

x Sh(T, h)(OL)

ofx0 = (A0, 0, 0, 0)Sh(T, h)(Fp). By assumption,

D(A0) = HdR1 (

A)

is a free OFOL-module of rank 1 equipped with a pairing given by anelementc (OFp)

=1. The1-linear operator onH=D(A0)FpLis of the form

=t(1 1)

for an element tT(L).Lemma 5.2.3. The elementt lies in the coset(p)T(OL).

Proof. H is a free OF L-module of rank 1 where

OF L=

Hom(OF,Fp)

L

is a product of 2f0 copies ofL. By ignoring the autoduality condition,t can be represented by an element

t= (t) Hom(OF,Fp)L

41


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It follows from the assumption

pD0 D0 D0that for all Hom(OF,Fp) we have

0 valp(t) 1.

Remember that the CM type induces a decomposition Hom(OF,Fp) = () and that

tr(b, D0/D0) =

(b)

for all b OF. It follows that

valp(t) = 0 if /1 if By the definition of, it follows that t (p)T(OL).

Description ofYp continued. A latticeD stable under the action ofOFand autodual up to a scalar, can be uniquely written under the form

D= mD0

for m T(L)/T(OL). The condition pD D D and the tracecondition on the tangent space is equivalent to m1t(m) (p)T(OL)and thus m lies in the groups of-fixed points in T(L)/T(O)L.

m [T(L)/T(OL)]

Now there is a bijection between the cosets mT(L)/T(OL) fixed by and the cosets T(Qp)/T(Zp) by considering the exact sequence

1 T(Zp) T(Qp) [T(L)/T(OL] H1(, T(OL))

where the last cohomology group vanishes by Langs theorem. It followsthat

Yp=T(Qp)/T(Zp)

and acts on it as (p).In H, Frobv(1

r) acts as r so that

Frobv(1 r) = ((p)(1 1))r

= (p1)((p1)) . . . r1((p1))(1 r).

thus the Frobenius Frobv acts on Yp Yp by the formula

(xp, xp)(xp, NEv/Qp((p

1))xp).

Auto-isogenies. For every prime = p, H1(A0,Q) is a free FQ Q-module of rank one. It follows that

EndQ(A0, 0) Q Q=FQ Q.

It follows that EndQ(A0, 0) = F. The auto-isogenies ofA0 form thegroup F and those who transports the polarization 0 on a rational

multiple of0 form by definition the subgroup T(Q) F. 42


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5.3. Shimura-Taniyama formula. Let (F, ) be a CM-type. LetOF be an order ofF which is maximal almost everywhere. Let p be

a prime whereOF is unr

lectures on shimura varieties

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