heights of kudla-rapoport divisors and derivatives of l ... · 1.1. motivation: heights of heegner...

HEIGHTS OF KUDLA-RAPOPORT DIVISORS AND DERIVATIVES OF

L-FUNCTIONS

JAN HENDRIK BRUINIER, BENJAMIN HOWARD, AND TONGHAI YANG

Abstract. We study special cycles on integral models of Shimura varieties associated

with unitary similitude groups of signature (n− 1, 1). We construct an arithmetic theta

lift from harmonic Maass forms of weight 2 − n to the first arithmetic Chow groupof a toroidal compactification of the integral model of the unitary Shimura variety, by

associating to a harmonic Maass form f a suitable linear combination of Kudla-Rapoportdivisors, equipped with the Green function given by the regularized theta lift of f . Our

main result expresses the height pairing of this arithmetic Kudla-Rapoport divisor with a

CM cycle in terms of a Rankin-Selberg convolution L-function of the cusp form of weightn corresponding to f and the theta function of a positive definite hermitian lattice of

rank n− 1. When specialized to the case n = 2, this result can be viewed as a variant of

the Gross-Zagier formula for Shimura curves associated to unitary groups of signature(1, 1). We also prove that the generating series of the height pairings of arithmetic Kudla-

Rapoport divisors with a fixed CM cycle is an elliptic modular form of weight n. These

results rely on (among other things) a new method for computing improper arithmeticintersections.

Contents

1. Introduction 12. Preliminaries 103. Shimura varieties and their divisors 134. Green functions on toroidal compactifications 175. Integral models of Shimura varieties 286. CM values of automorphic Green functions 437. Metrized line bundles and derivatives of L-functions 518. Arithmetic cycles classes and derivatives of L-functions 709. Scalar valued modular forms 75References 85

1. Introduction

Let k ⊂ C be an imaginary quadratic field of odd discriminant dk, let dk be the differentof k, and abbreviate D = |dk|. Let χk be the quadratic Dirichlet character determined bythe extension k/Q.

Date: February 27, 2013.

2000 Mathematics Subject Classification. 14G35, 14G40, 11G18, 11F27.The first author is partially supported by DFG grant BR-2163/2-2. The second author is partially

supported by NSF grant DMS-1201480. The third author is partially supported by a NSF grant DMS-

1200380 and a Chinese grant.

1

2 JAN H. BRUINIER, BENJAMIN HOWARD, AND TONGHAI YANG

1.1. Motivation: heights of Heegner points. To motivate the results of this paper,we first recall the results of Gross and Zagier [GZ] relating the heights of Heegner pointson modular curves to central derivatives of Rankin-Selberg convolution L-functions. Fix anormalized new eigenform

g(τ) ∈ S2(Γ0(N)),

and assume that N and k satisfy the usual Heegner hypothesis: every prime divisor ofN splits in k. This allows us to fix an ideal n ⊂ Ok satisfying Ok/n ∼= Z/NZ. For anyfractional Ok-ideal a, the cyclic N -isogeny of elliptic curves

ya = [C/a→ C/n−1a]

defines a Heegner point on X0(N)(H), where H is the Hilbert class field of k. If we definea weight 2 cuspform

ΘHeeg(τ) =

∞∑m=1

Tm(yOk−∞)qm

valued in J0(N)(H), where the Tm are Hecke operators, then the geometric theta lifting

ΘHeeg(g) = 〈ΘHeeg, g〉Pet ∈ J0(N)(H)⊗ C

is essentially the projection of the divisor yOk− ∞ to the g-isotypic component of the

Jacobian J0(N).After endowing the fractional ideal a with the self-dual hermitian form 〈x, y〉 = N(a)−1xy,

we may construct the weight one theta series

θa(τ) =∑x∈a

q〈x,x〉 ∈M1(Γ0(D), χk).

The Rankin-Selberg convolution L-function L(g, θa, s) satisfies a functional equation forcingit to vanish at s = 1, and the Gross-Zagier theorem implies[

ΘHeeg(g) : ya −∞]NT

= c · L′(g, θa, 1).

Here c is some explicit nonzero constant, and the pairing on the left is the Neron-Tateheight.

The goal of this paper is to obtain similar results when g is replaced by a cusp form ofhigher weight, the fractional ideal a is replaced by a hermitian lattice of higher rank, andthe Heegner points on modular curves are replaced by special cycles on Shimura varietiesassociated to groups of unitary similitudes.

1.2. Speculation in higher weight. Fix an n ≥ 2, and any cusp form

g(τ) =∑m

a(m)qm ∈ Sn(Γ0(D), χnk).

We do not require that g be an eigenform. Let Λ be a positive definite self-dual hermitianlattice of rank n− 1; that is, a projective Ok-module of rank n− 1 endowed with a positivedefinite hermitian form 〈·, ·〉 inducing an isomorphism Λ ∼= HomOk

(Λ,Ok). The coefficientof qm in the weight n− 1 theta series

θscΛ (τ) =

∑x∈Λ

q〈x,x〉 ∈Mn−1(Γ0(D), χn−1k )

is the representation number

RscΛ (m) = |x ∈ Λ : 〈x, x〉 = m|

HEIGHTS OF KUDLA-RAPOPORT DIVISORS 3

(the superscript sc is a reminder that this is a scalar valued theta series; throughout mostof the paper we work with vector valued theta series.) From this data we may construct theRankin-Selberg convolution L-function

(1.2.1) L(g, θscΛ , s) = Γ

(s2

+ n− 1) ∑m≥1

a(m)RscΛ (m)

(4πm)s2 +n−1

.

Our normalization is different from that of Gross and Zagier; in this normalization the pointof interest is at s = 0. Note that our cusp form g has level D, and so we are far away fromthe Heegner hypothesis. In particular, there is no reason to expect the Rankin-Selberg L-function to vanish at s = 0. All the same, it makes sense to ask if the behavior at s = 0 isrelated to the Arakelov intersection multiplicity of some cycles on a Shimura variety.

For a pair of nonnegative integers (p, q), denote by M(p,q) the moduli space of principallypolarized abelian varieties A→ S over k-schemes, equipped with an action of Ok satisfyingthe signature (p, q) condition: every a ∈ Ok acts on Lie(A) with characteristic polynomial(T − a)p(T − a)q. We require also that the Rosati involution on End(A) ⊗ Q restrict tocomplex conjugation on the image of Ok. The moduli space M(p,q) is a Deligne-Mumfordstack, smooth over k of dimension pq. It is a disjoint union of Shimura varieties associated tounitary similitude groups. The theory of integral models of these stacks remains incomplete,but we only need two special cases:

(1) there is a smooth stack M(p,0) over Ok with generic fiber M(p,0),(2) there is a regular and flat stack M(p,1) over Ok with generic fiber M(p,1).

In both cases the integral model is defined by imposing a slightly refined version of thesignature condition. The flatness and regularity of M(p,1) are due to Pappas and Kramer.See Section 5.1 for details. The product

M = M(1,0) ×OkM(n−1,1)

is an n-dimensional regular algebraic stack, flat over Ok. The complex fiber M(C) is discon-nected, and its connected components are indexed by the finitely many isomorphism classesof self-dual hermitian lattices of signature (n− 1, 1). After [Lan] and [Ho3], the stack M hasa canonical toroidal compactification M∗, whose boundary is a smooth divisor.

The stack M comes equipped with a family of Kudla-Rapoport divisors Z(m, r), introducedin [KR1] and studied further in [KR2], [Ho2], and [Ho3]. Here r is an Ok-ideal dividing dk,and m ∈ N(r)−1Z is positive. These divisors correspond, roughly speaking, to embeddings

GU(n− 2, 1) −→ GU(n− 1, 1).

The precise definition is as follows. Given a pair (A0, A) ∈ M(S), the Ok-module

L(A0, A) = HomOk(A0, A)

carries a natural positive definite hermitian form. The stack Z(m, r) is defined as the modulispace of triples (A0, A, λ) in which (A0, A) is a point of M, and λ ∈ r−1L(A0, A) has hermitiannorm m. There is an additional vanishing condition imposed on the action of

√dk ·λ on Lie

algebras, for which we refer the reader to Definition 5.1.5 and Remark 5.1.7.On the open Shimura variety M there is a natural way to construct Green functions for

the Kudla-Rapoport divisors, based on the ideas of [Br1] (when n = 2, there is some mildambiguity in the normalization of the Green function, but we ignore this subtlety in theintroduction). By analyzing the behavior of the Green function near the boundary of thetoroidal compactification, we show that one can add a prescribed linear combination of


boundary components to Z(m, r) in order to obtain the total arithmetic Kudla-Rapoportdivisors

Ztotal(m, r) ∈ CH1

C(M∗).

The right hand side is the arithmetic Chow group with complex coefficients, defined, asin the work of Gillet-Soule [SABK], as the space of rational equivalence classes of divisorsendowed with Green functions. In fact, we must use the more general arithmetic Chowgroups defined by Burgos-Kramer-Kuhn [BKK], which allow for Green functions with log-log singularities along the boundary. Our Green functions are constructed as regularizedtheta lifts of harmonic Maass forms, as in [Br1], [BF], and [BY], and so are different fromthe Kudla-style Green functions used in [Ho2] and [Ho3]. Extend the definition to m = 0by setting

Ztotal(0, r) = T∗,

where the right hand side is the metrized cotautological bundle on M∗, defined in Section 7.2on the open Shimura variety, and extended to M∗ in Section 8.1.

Restricting to r = Ok, one may form the formal generating series

Θ(τ) =∑m≥0

Ztotal(m,Ok) · qm ∈ CH1

C(M∗)[[q]].(1.2.2)

Conjecturally, this formal generating series is a modular form of the same weight, level, andnebentypus as g, in the sense that∑

m≥0

ρ(Ztotal(m,Ok)

)· qm ∈Mn(Γ0(D), χnk)

for any linear functional ρ : CH1

C(M∗) → C. If this is true, then it makes sense to form thePetersson inner product

(1.2.3) Θconj(g) = 〈Θ, g〉Pet ∈ CH1

C(M∗).

This conjectural arithmetic cycle class, whose construction is based on the ideas of Kudla[Ku4], is the arithmetic theta lift of g.

The moduli space M contains another cycle, which corresponds, roughly speaking, to anembedding of a compact unitary group

GU(n− 1, 0) −→ GU(n− 1, 1).

More precisely, we first define a stack

Y = M(1,0) ×OkM(0,1) ×Ok

M(n−1,0).

It is smooth of relative dimension 0 over Ok, and the morphism Y→ M defined by

(A0, A1, B) 7→ (A0, A1 ×B)

allows us to view Y as a 1-dimensional cycle on M. To every geometric point (A0, A1, B)there is an associated self-dual hermitian Ok-module HomOk

(A1, B) of signature (n− 1, 0),whose isomorphism class is constant on each connected component of Y. Let YΛ ⊂ Y be theunion of all connected components on which HomOk

(A1, B) ∼= Λ.Our aim is an equality of the form

(1.2.4) [Θconj(g) : YΛ]?= L′(g, θsc

Λ , 0),

where the pairing on the left is the arithmetic intersection, defined as the composition

CH1

C(M∗) −→ CH1

C(YΛ)deg−−→ C.


Before one attempts to prove such a formula, there are two obstructions to even finding thecorrect statement. On the right hand side, the construction of (1.2.1) is a bit too naive,and must be modified to ensure that it has a nice functional equation forcing it to vanishat s = 0. Much more seriously, on the left hand side we do not know the modularity of the

generating series Θ(τ), and so the existence of the class Θconj(g) remains mere speculation.To fix the problem on the right, we introduce vector valued modular forms. There is

an induction map g 7→ ~g from Sn(Γ0(D), χnk) to a certain space of cusp forms valued in afinite-dimensional representation of SL2(Z). Similarly, the above theta series θsc

Λ has a vectorvalued counterpart θΛ. If one imitates the construction of the convolution L-function (1.2.1)with these vector valued forms, the resulting L-function has a simple functional equation ins 7→ −s, which forces it to vanish at s = 0, see (6.4.2).

To fix problem on the left, we introduce a space of vector valued harmonic Maass formsof weight 2 − n, on which one can define, unconditionally, an arithmetic theta lift. Thisspace of harmonic Maass forms is then related to the space of holomorphic cusp forms ofweight n via a differential operator ξ. In the next subsection, we explain these ideas in moredetail, and state the main results.

1.3. Statement of the main results. To a hermitian module V over the adele ring Akthere is an associated invariant inv(V ) ∈ ±1, defined as a product of local invariants. Ifinv(V ) = 1 then V is coherent, in the sense that V arises as the adelization of a hermitianspace over k. Otherwise, V is incoherent. In Section 2.1 we define the notion of a hermitian

(kR, Ok)-module L . Essentially, L is an integral structure on a hermitian Ak-module. Itconsists of an archimedean part L∞, which is a hermitian space over kR = k ⊗Q R, and a

finite part Lf , which is a hermitian Ok-module. As explained in Section 5.1, to each point

of the moduli space M there is associated a hermitian (kR, Ok)-module, whose isomorphismclass is constant on the connected components of M. Thus we obtain a decomposition

M =⊔L

ML

where L runs over all incoherent self-dual hermitian (kR, Ok)-modules of signature (n, 0).Similarly, the stack YΛ admits a decomposition

YΛ =⊔L0

Y(L0,Λ),

where L0 runs over all incoherent self-dual hermitian (kR, Ok)-modules of signature (1, 0).From now on we fix one such L0, and set L = L0 ⊕ Λ; for the meaning of the direct sum,see Remark 5.2.3. The morphism YΛ → M restricts to a morphism Y(L0,Λ) → ML , and wedenote by ZL (m, r) the restriction of the Kudla-Rapoport divisor Z(m, r) to ML .

The hermitian form on Lf determines a Q/Z-valued quadratic form on the finite discrim-

inant group d−1k Lf/Lf . If we denote by SL the space of complex valued functions on this

finite quadratic space, there is a Weil representation ωL : SL2(Z)→ Aut(SL ). Let Sn(ωL )be the space of weight n cusp forms for SL2(Z) with values in SL , transforming accordingto the complex conjugate representation ωL . In Section 9.1 we define an induction map

Sn(Γ0(D), χnk)→ Sn(ωL )∆,

where ∆ is the automorphism group of the finite group d−1k Lf/Lf with its Q/Z valued

quadratic form. This map is denoted g 7→ ~g. The vector valued modular form ~g has a


q-expansion

~g(τ) =∑

m∈Q>0

~a(m)qm

with ~a(m) ∈ SL .Let SΛ be the space of complex valued functions on d−1

k Λ/Λ. There is a vector valuedtheta series

θΛ(τ) =∑

m∈Q>0

RΛ(m)qm

taking values in the dual space S∨Λ , whose m-th Fourier coefficient RΛ(m) : SΛ → C is therepresentation number

RΛ(m,ϕ) =∑

λ∈d−1k Λ

〈λ,λ〉=m

ϕ(λ).

We consider the vector variant

(1.3.1) L(~g, θΛ, s) = Γ(s

2+ n− 1

) ∑m∈Q>0

~a(m), RΛ(m)

(4πm)

s2 +n−1

of (1.2.1), where the pairing ·, · is the tautological pairing between SL and S∨L , andRΛ(m) is viewed as an element of S∨L using the natural surjection SL → SΛ. Unlike(1.2.1), the L-function (1.3.1) satisfies a simple functional equation in s 7→ −s, which forcesL(~g, θΛ, 0) = 0.

Now we turn to the left hand side of (1.2.4). Let H2−n(ωL ) be the space of harmonicMaass forms for SL2(Z) of weight 2 − n with values in the vector space SL . There is anarithmetic theta lift of harmonic Maass forms

(1.3.2) H2−n(ωL )∆ −→ CH1

C(M∗L ),

denoted f 7→ ΘL (f), whose definition is roughly as follows. There is a theta lift fromfunctions on the upper half plane to functions on the Shimura variety ML (C). If one attemptsto lift an element f ∈ H2−n(ωL ), the theta integral diverges due to the growth of f atthe cusp. There is a natural way to regularize the divergent integral in order to obtaina function ΦL (f) on ML , but the regularization process introduces singularities into thefunction ΦL (f), see [Bo1] and [BF]. These singularities turn out to be of logarithmic type.In fact ΦL (f) is a Green function for a certain divisor ZL (f)(C) on ML (C), which can bewritten in an explicit way as a linear combination of the complex Kudla-Rapoport divisorsZL (m, r)(C). As the complex Kudla-Rapoport divisors have canonical extensions to theintegral model ML , we obtain an extension of ZL (f)(C) to the integral model as well. Theresult is a divisor ZL (f) on ML together with a Green function ΦL (f). The arithmetic thetalift of f is then defined by first adding boundary components with appropriate multiplicitiesin order to define a compactified arithmetic divisor

ZtotalL (f) ∈ CH

1(M∗L ),

and then adding a certain multiple (depending on the constant term of f) of the metrized

cotautological bundle T∗ to obtain

ΘL (f) ∈ CH1(M∗L ).

There is a surjective complex-conjugate-linear differential operator

ξ : H2−n(ωL ) −→ Sn(ωL ),


introduced in [BF]. We conjecture that (1.3.2) factors through ξ, resulting in a commutativediagram

H2−n(ωL )∆

ξ

ΘL

&&

Sn(Γ0(D), χnk)g 7→~g // Sn(ωL )∆

?// CH

1

C(M∗L ).

Assuming the existence of the dotted arrow, the composition along the bottom row nec-essarily takes g to (the restriction to M∗L of) the conjectural arithmetic theta lift (1.2.3).Thus we find an obvious way to give an unconditional construction of the class (1.2.3): pick

any f ∈ H2−n(ωL )∆ satisfying ξ(f) = ~g, and form the class ΘL (f). The downside to thisroundabout definition is that the construction a priori may depend on the choice of f lifting

~g. The independence of ΘL (f) of this choice is equivalent to the modularity of the vectorvalued analogue of the generating series (1.2.2). See Section 8.3 for the precise conjectures.

The following is our main result. It is stated in the text as Theorem 8.2.2.

Theorem A. For any g ∈ Sn(Γ0(D), χnk), and any f ∈ H2−n(ωL )∆ satisfying ξ(f) = ~g,we have

(1.3.3) [ΘL (f) : Y(L0,Λ)] = −degC Y(L0,Λ) · L′(~g, θΛ, 0).

The constant appearing on the right is

degC Y(L0,Λ) =∑

y∈Y(L0,Λ)(C)

1

|Aut(y)|.

An explicit formula for this constant is given in Remark 6.1.4.

Note that the right hand side of (1.3.3) is visibly independent of the choice of f lifting~g. The following result, which provides evidence for the modularity of (1.2.2), is a formalconsequence of this observation together with the modularity criterion of Borcherds [Bo2,Theorem 3.1]. See Theorem 8.3.4 for a slightly stronger statement.

Theorem B. The formal q-expansion∑m≥0

[ZtotalL (m,Ok) : Y(L0,Λ)] · qm

defines an element of Mn(Γ0(D), χnk).

It remains to explain the relation between the two L-functions (1.2.1) and (1.3.1). This isthe subject of Section 9. The general relation is rather complicated, but the formulas simplifyconsiderably when n is even and g is a new eigenform, allowing us to restate Theorem A interms of the original L-function (1.2.1).

If h is an element of the unitary similitude group GUΛ(Q) of Λ, then the Ok-module

Λh = (Λ⊗ZQ)∩hΛ has a natural self-dual hermitian form. The assignment h 7→ Λh definesa map from the 0-dimensional Shimura variety

XΛ = GUΛ(Q)\GUΛ(Q)/GUΛ(Z)

to the set of isomorphism classes of self-dual hermitian Ok-modules of signature (n− 1, 0).

The group GUΛ(Q) also acts on incoherent self-dual hermitian (kR, Ok)-modules of signature


(1, 0) through the determinant. It is easily seen that L0,h ⊕ Λh ∼= L (Lemma 9.1.2), andtherefore we may form a new CM cycle Y(L0,h,Λh) on M∗L . Let

η : XΛ −→ C.

be an algebraic automorphic form for GUΛ, and assume that η transforms under the naturalaction of the class group CL(k) on XΛ by some character χη : CL(k) → C×. We obtain acycle

ηY(L0,Λ) =∑h∈XΛ

η(h) · Y(L0,h,Λh)

on M∗L with complex coefficients, and a corresponding scalar valued theta function

θscη,Λ =

∑h∈XΛ

η(h)

|Aut(Λh)|· θsc

Λh∈Mn−1(Γ0(D), χn−1

k ).

Theorem C. If n is even and g is new eigenform, then

[ΘL (f) : ηY(L0,Λ)] = − h2k

2o(dk)−1w2k

· dds

[L(g, θsc

η,Λ, s) ·∏p|D

(1 + χη(p)εp(g)γp(L )p

s2

)]∣∣s=0

.

for any f ∈ H2−n(ωL )∆ with ξ(f) = ~g. Here p is the unique prime of k above p, εp(g) ∈±1 denotes the eigenvalue for g of the Atkin-Lehner involution at p, and

γp(L ) = (−1)(p−1)n

4 χk,p(det(Lp)).

The constants hk, wk, and o(dk) are the class number of k, the number of roots of unity ink. and the number of prime divisors of dk.

There is a similar theorem when n is odd, but the result is less elegant. We refer thereader to Theorem 9.3.1 for the general case.

Remark 1.3.1. There are corresponding statements for the complementary case when L is

a coherent positive definite (k∞, Ok)-module of rank n. Then we may choose an Ok-latticeL whose completion is L , and use the 0-dimensional Shimura variety

XL = GUL(Q)\GUL(Q)/GUL(Z)

as a substitute for ML . In this case the sign in the functional equation of L(~g, θΛ, s) is 1.Moreover, the central value at s = 0 can be computed by integrating the theta lift of g toXL over a certain cycle of CM points determined by Λ. We intend to elaborate on this ina separate paper.

1.4. Outline of the paper. Section 2 is a rapid review of definitions. We first recall theinvariants attached to hermitian spaces, and introduce the notion of coherent and incoherent

hermitian (kR, Ok)-modules. Then we recall various standard definitions and propertiesof Siegel theta functions, vector-valued modular forms, harmonic Maass forms, and thedifferential operator ξ introduced in [BF].

In Section 3 we introduce the complex fiber of the unitary Shimura variety on which wewill be working, and its special divisors. In particular we attach a divisor Z(f) to a harmonicMaass form, and explain how the mechanism of regularized theta lifts equips this divisorwith a natural Green function Φ(f). We show that Φ(f) is (sub-)harmonic and satisfies anintegrability condition. These properties, together with the logarithmic singularity alongZ(f), characterize it uniquely up to an additive constant. One of the minor miracles ofthe construction of Φ(f) is that, despite having a logarithmic singularity along Z(f), it is


defined at every point of the complex Shimura variety. Expressed differently, the smoothfunction Φ(f), initially defined on the complement of Z(f), has a natural (discontinuous)extension to all points. The behavior of Φ(f) at the points of Z(f), described in Corollary3.3.2, plays an essential role in the calculation of the improper intersections needed to proveTheorem A.

Section 4 first recalls the construction of the canonical smooth toroidal compactificationof the complex Shimura variety. What is new is the analysis of the behavior of the Greenfunction Φ(f) near the boundary of the compactification. We find that one can compactifythe divisor Z(f) by adding a prescribed linear combination of boundary components, in sucha way that Φ(f) is a Green function for the compactified divisor (at least up to negligiblelog-log error terms). The multiplicities of the boundary components are given by regularizedintegrals of f against theta functions of positive definite hermitian lattices of rank n−2, seeTheorem 4.3.3. A similar result was proved in [Ho3], but for different Green functions. Themethod used here, based on the calculation of Fourier-Jacobi expansions (Theorem 4.4.2),is unrelated. Because these results are of independent interest, in Sections 3 and 4 we workon Shimura varieties with arbitrary level structure; this is more generality than is needed inthe rest of the paper.

Section 5 introduces the integral model M of the Shimura variety, the Kudla-Rapoportdivisors Z(m, r), and the cycle of CM points YΛ. Section 5.3 contains the first steps towardthe intersection theory calculation lying at the heart of Theorem A. Roughly speaking, weshow that the intersection Z(m, r)∩YΛ decomposes as a disjoint union of stacks X(m1,m2, r)indexed by pairs of nonnegative integers (m1,m2) with m1 + m2 = m. If m1 > 0 thenthe stack X(m1,m2, r) has dimension 0 and is supported in a single nonzero characteristic.Theorem 5.3.2 gives an explicit formula for the degree of this 0-dimensional stack. Thecalculation of the lengths of its local rings ultimately reduces to the calculations performedby Gross [Gr] as part of the proof of the original Gross-Zagier theorem. In contrast, the stackX(0,m, r) has dimension 1, and appears because of improper intersection between Z(m, r)and YΛ.

In Section 6 we describe the complex uniformizations of the stacks M, Z(m, r) and YΛ,and then apply the general theory of Section 3 in order to attach an arithmetic cycle class(Z(f),Φ(f)) on M to a harmonic Maass form f of weight 2−n. Particular choices fm,r yieldGreen functions for the divisors Z(m, r). The main result of Section 6, Theorem 6.4.1, is aformula for the value of the Green function Φ(f) on the CM cycle YΛ(C). When combinedwith the finite intersection calculation of Theorem 5.3.2, this archimedean calculation (with

f = fm,r) is sufficient to compute the intersection multiplicity [Ztotal(m, r) : YΛ], at leastwhen the cycles in question intersect properly. Unfortunately, this happens only very rarely.

Most of Section 7 is devoted to the calculation of improper intersections. Calculationsof improper intersection have been done in other situations elsewhere in the literature (forexample in [GZ], [KRY2], [Ho1]), but our methods are new, and are the most original contri-bution of this paper. In both [KRY2] and [Ho1] the method of computation is to decomposethe cycles into irreducible components and painstakingly intersect these components one ata time, using some form of adjunction formula, in order to obtain an explicit formula for theintersection. Our method also uses a kind of adjunction formula, but we do not know, andnever need to know, what the irreducible components of our cycles look like. In particular,we do not need to compute the multiplicities of vertical components of the divisors Z(m, r),and in fact we don’t know whether or not such vertical components exist. Instead of explicit


calculation, we use deformation theory to show that a certain metrized line bundle

Z♥(m, r) = Z(m, r)⊗ T−RΛ(m,r)

on M acquires a canonical nonzero section σm,r when restricted to YΛ. To compute the

intersection multiplicity [Z♥(m, r) : YΛ] it suffices to compute the degree of the 0-cyclediv(σm,r) on YΛ, and the norm ||σm,r||y at each y ∈ YΛ. The divisor div(σm,r) turns outto be exactly the divisor obtained by intersecting Z(m, r) ∩ YΛ and then throwing away allcomponents of the intersection having dimension > 0. In other words, it is the proper partof the intersection, and was computed in Theorem 5.3.2. The norm ||σm,r||y turns out to bethe value of the Green function for Z(m, r) at y (even when y lies on Z(m, r), the singularityof the Green function!), which was computed in Theorem 6.4.1. Thus we are able to compute

[Z(m, r) : YΛ] = [Z♥(m, r) : YΛ] +RΛ(m, r)[T : YΛ]

by computing only proper intersections, the CM values of Green functions, and the in-tersection [T : YΛ]. The Chowla-Selberg formula allows us to give an explicit formula for

[T : YΛ], and putting these formulas together relates [Z(m, r) : YΛ] to the derivative ofL(ξ(fm,r), θΛ, s). The precise formula is found in Theorem 7.3.1, the preliminary form ofour main result.

Section 8 puts all of the pieces together to prove Theorems A and B. Section 9 explainsthe precise relation between the L-functions (1.2.1) and (1.3.1), and completes the proof ofTheorem C.

1.5. Notation and terminology. We write H for the complex upper half plane. For acomplex number z we put e(z) = e2πiz. As usual, we denote by A the ring of adeles of Qand write Af for the finite adeles.

The quadratic imaginary field k and its embedding k → C are fixed throughout thepaper, and dk and dk denote the different and discriminant of k. In Sections 2 through 4we make no restriction on dk, but beginning in Section 5, and continuing until the end ofthe paper, we assume that dk is odd. We write Ok, Ak and Ak,f for the ring of integers,adeles, and finite adeles of k, respectively. The class number of k is hk, and wk = |µ(k)| isthe number of roots of unity in k. Denote by o(dk) the number of distinct prime divisors ofdk, and by

χk : A× −→ ±1

the quadratic character determined by the extension k/Q. For any m ∈ Q>0 define

(1.5.1) ρ(m) = |b ⊂ Ok : N(b) = m|.

Obviously ρ(m) = 0 unless m ∈ Z>0.Abbreviate kR = k⊗QR. Of course the fixed embedding of k into C identifies kR ∼= C, but

there will be moments (in Section 6.1, for example) when we consider a finite dimensionalkR-module with a varying family of complex structures, and so it will be useful sometimesto not make this identification.

2. Preliminaries

This section contains some basic definitions and notation concerning hermitian spaces,theta series, and vector valued modular forms.


2.1. Invariants of hermitian spaces. A hermitian Ok-module is a projective Ok-moduleL of finite rank equipped with a hermitian form 〈·, ·〉 : L×L→ Ok. Our convention is thathermitian forms are Ok-linear in the first variable and Ok-conjugate-linear in the secondvariable. All hermitian forms are assumed to be nondegenerate. For an Ok-ideal r | dk,every vector x ∈ r−1L satisfies

(2.1.1) 〈x, x〉 ∈ N(r)−1Z,

and Q(x) = 〈x, x〉 defines a d−1k Z/Z-valued quadratic form on d−1

k L/L. A hermitian Ok-module L is self-dual if it satisfies

L = x ∈ L⊗Z Q : 〈x, L〉 ⊂ Ok.

We can similarly talk about self-dual hermitian Ok-modules, and hermitian spaces over k,over its completions, and over Ak.

If A0 and A are hermitian Ok-modules with hermitian forms hA0and hA, the Ok-module

(2.1.2) L(A0,A) = HomOk(A0,A)

carries a hermitian form 〈·, ·〉 determined by the relation

〈f, g〉 · hA0(x, y) = hA(f(x), g(y))

for all x, y ∈ A0. If A0 and A are self-dual then so is L(A0,A). Of course a similar discussion

holds for hermitian Ok-modules.A hermitian space V over Ak has an archimedean part V∞ and a nonarchimedean part

Vf =∏p Vp, which are hermitian spaces over kR and Ak,f , respectively. The archimedean

part is uniquely determined by its signature, while each factor Vp is uniquely determinedby its dimension and the local invariant

invp(V ) = χk,p(det(Vp)) ∈ ±1.

Of course the invariant is also defined for p = ∞, but carries less information than thesignature. The invariant of V is the product of local invariants:

inv(V ) =∏p≤∞

invp(V ).

If inv(V ) = 1 then there is a hermitian space V over k, unique up to isomorphism, satisfyingV ∼= V ⊗Q A. In this case we say that V is coherent. If instead inv(V ) = −1 then no suchV exists, and we say that V is incoherent.

We will need a notion of a hermitian space over Ak with an integral structure.

Definition 2.1.1. A hermitian (kR, Ok)-module is a hermitian space V over Ak together

with a finitely generated Ok-submodule Lf ⊂ Vf of maximal rank on which the hermitian

form is Ok-valued. Equivalently, a hermitian (kR, Ok)-module is a pair L = (L∞,Lf ),in which L∞ is a hermitian space over kR, and Lf =

∏p Lp is a hermitian space over

Ok of the same rank as L∞. One recovers the first definition from the second by settingV∞ = L∞ and Vf = Lf ⊗Z Af .

(1) The signature of a hermitian (kR, Ok)-module L is the signature of L∞,

(2) L is self-dual if Lf is a self-dual hermitian Ok-module in the above sense,(3) L is coherent (or incoherent) if V is.


Obviously, every hermitian Ok-module L gives rise to a coherent hermitian (kR, Ok)-

module L determined by L∞ = L⊗Z R and Lf = L⊗Z Z. Conversely, for each hermitian

(kR, Ok)-module L there is a finite collection of hermitian Ok-modules that give rise to it.This finite collection is the genus of L , and is denoted

gen(L ) =

isomorphism classes ofhermitian Ok-modules L

:L∞ ∼= L⊗Z RLf∼= L⊗Z Z

.

The genus is nonempty if and only if L is coherent, and any two L,L′ ∈ gen(L ) satisfyL⊗Z Q ∼= L′ ⊗Z Q as hermitian spaces over k.

Remark 2.1.2. Given a hermitian space V over Ak and a rational prime p nonsplit in k,there is a nearby hermitian space V (p) over Ak, uniquely determined (up to isomorphism)by the properties

(1) V (p)` ∼= V` for every place ` 6= p,(2) V (p)p 6∼= Vp.

In other words, V (p) is obtained from V by changing the local invariant at p, and so

inv(V (p)) = −inv(V ).

If instead we take p =∞ then there is no single notion of V (∞). However, in the applicationswe have in mind V will be positive definite, and V (∞) will be obtained from V by switchingthe signature from (n, 0) to (n− 1, 1).

2.2. Theta functions and vector valued modular forms. Let (M,Q) be an even inte-gral lattice, that is, a free Z-module of finite rank equipped with a non-degenerate Z-valuedquadratic form Q. For simplicity we assume here that the rank of M is even. We denotethe signature of M by (b+, b−). Let M ′ be the dual lattice of M . The quadratic form Qinduces a Q/Z-valued quadratic form on the discriminant group M ′/M .

Let ω be the restriction to SL2(Z) of the Weil representation of SL2(Q) (associated with

the standard additive character of A/Q) on the Schwartz-Bruhat functions on M ⊗Z Q. Therestriction of ω to SL2(Z) takes the subspace SM of Schwartz-Bruhat functions which are

supported on M ′ and invariant under translations by M to itself. We obtain a representationωM : SL2(Z)→ Aut(SM ). Throughout we identify SM with the space of functions M ′/M →C. Let S∨M be the dual space of SM , and denote by

·, · : SM × S∨M −→ C

the tautological C-bilinear pairing. The group SL2(Z) acts on S∨M through the dual repre-

sentation ω∨M , given by ω∨M (γ)(f) = f ω−1M (γ) for f ∈ S∨M . On the space SM we also have

the conjugate representation ωM given by

ωM (γ)(ϕ) = ωM (γ)(ϕ)

for ϕ ∈ SM . Note that ωM is the representation denoted ρM in [Bo1], [Br1], [BF]. Thesame construction can also be applied in slightly greater generality. For instance, in later

applications we will use it when M is a quadratic module over Z.Let Gr(M) be the Grassmannian of negative definite b−-dimensional subspaces of M⊗ZR.

For z ∈ Gr(M) and λ ∈ M ⊗Z R, we denote by λz and λz⊥ the orthogonal projection of λto z and z⊥, respectively. If ϕ ∈ SM and τ ∈ H, we let

(2.2.1) ΘM (τ, z, ϕ) = vb−/2

∑λ∈M ′

ϕ(λ)e(Q(λz⊥)τ +Q(λz)τ

)


be the associated Siegel theta function. For γ ∈ SL2(Z) it satisfies the transformation law

ΘM (γτ, z, ϕ) = (cτ + d)b+−b−

2 ΘM (τ, z, ωM (γ)ϕ).

Following [Ku3], we view the Siegel theta function as a function

H×Gr(M) −→ S∨M , (τ, z) 7→ ΘM (τ, z).

The above transformation law implies that ΘM (τ, z) is a (non-holomorphic) modular formof weight (b+ − b−)/2 for the group SL2(Z) with values in S∨M .

Let k ∈ Z, and let σ be a finite dimensional representation of SL2(Z) on a complex vectorspace Vσ, which factors through a finite quotient of SL2(Z). We denote by Hk(σ) the vectorspace of harmonic Maass forms1 of weight k for the group SL2(Z) with representation σ as in[BY]. We write M !

k(σ), Mk(σ), and Sk(σ) for the subspaces of weakly holomorphic modularforms, holomorphic modular forms, and cusp forms, respectively. The natural action of theorthogonal group of M on the space SM commutes with the action of SL2(Z). Hence thereis an induced action on the above spaces of vector valued modular forms.

A harmonic Maass form f ∈ Hk(σ) has a Fourier expansion of the form

f(τ) =∑m∈Qm≥m0

c+(m)qm +∑m∈Qm<0

c−(m)Γ(1− k, 4π|m|v)qm(2.2.2)

with Fourier coefficients c±(m) ∈ Vσ. Here v = Im(τ), q = e2πiτ , and Γ(s, x) =∫∞xe−tts−1dt

denotes the incomplete gamma function. The coefficients are supported on rational numberswith bounded denominator. The first summand on the right hand side of (2.2.2) is denotedby f+ and is called the holomorphic part of f , the second summand is denoted by f− andis called the non-holomorphic part.

Recall from [BF] that there is a conjugate-linear differential operator ξk : Hk(ωM ) →S2−k(ωM ) given by

ξk(f)(τ) = 2ivk∂f

∂τ.(2.2.3)

The kernel of ξk is equal to M !k(ωM ). According to [BF, Corollary 3.8], we have the exact

sequence

0 // M !k(ωM ) // Hk(ωM )

ξk // S2−k(ωM ) // 0 .(2.2.4)

If f ∈ Hk(ωM ) with Fourier coefficients c±(m) ∈ SM as in (2.2.2) and if µ ∈M ′/M , we putc±(m,µ) = c±(m)(µ) ∈ C.

3. Shimura varieties and their divisors

In this section we consider the analytic theory of Shimura varieties associated to hermitianspaces of signature (n − 1, 1) over imaginary quadratic fields. We also study their specialdivisors and automorphic Green functions for special divisors.

Let V be a hermitian space over k equipped with a hermitian form 〈·, ·〉. Throughoutwe let VR = V ⊗Q R and assume that the signature of V is (n − 1, 1). We write 〈·, ·〉Q forthe symmetric bilinear form 〈x, y〉Q = trk/Q〈x, y〉. The associated quadratic form over Q is

Q(x) = 12 〈x, x〉Q = 〈x, x〉. Note that the Weil representations of SL2 ⊂ U(1, 1) associated

to the quadratic form over Q and the hermitian form are the same.

1For ease of notation we drop the adjective ‘weak’ in harmonic weak Maass form.


3.1. Hermitian spaces and unitary Shimura varieties. We realize the hermitian sym-metric space associated to the unitary group U(V ) as the Grassmannian D of negativekR-lines in VR. It can be viewed as an open subset of the projective space P(VR) of thecomplex vector space VR. The domain D is not a tube domain unless n = 2, in which case itis isomorphic to the complex upper half plane H. In general, D has a realization as a Siegeldomain as follows.

Let ` ∈ V be a nonzero isotropic vector and let ˜∈ V be isotropic such that 〈`, ˜〉 = 1.The orthogonal complement

W = `⊥ ∩ ˜⊥

is a positive definite hermitian space over k of dimension n−2, and we have V = W⊕k`⊕k ˜.If z ∈ D, then 〈z, `〉 6= 0. Hence z has a unique basis vector of the form

z + τ√dk`+ ˜

with z ∈ WR and τ ∈ C. We denote this vector by the pair (z, τ). The condition that therestriction of the hermitian form to z is negative definite is equivalent to requiring that

N(z, τ) = −〈(z, τ), (z, τ)〉 = 2√|dk| Im(τ)− 〈z, z〉(3.1.1)

is positive. Consequently, D is isomorphic to

H`,˜ = (z, τ) ∈WR ×H : 2√|dk| Im(τ) > 〈z, z〉.

For z ∈ D and λ ∈ VR we let λz⊥ and λz be the orthogonal projections of λ to z⊥ and z,respectively. Then the majorant

〈λ, µ〉z = 〈λz⊥ , µz⊥〉 − 〈λz, µz〉

associated to z defines a positive definite hermitian form on VR. If 0 6= z0 ∈ z, we have

〈λ, λ〉z = 〈λ, λ〉+ 2|〈λ, z0〉|2

|〈z0, z0〉|.

The hermitian domain D carries over it a tautological bundle L, whose fiber at the pointz ∈ D is the negative line z. The hermitian form on VR induces a hermitian metric on L. Itsfirst Chern form Ω is U(V )(R)-invariant and positive. It corresponds to an invariant Kahlermetric on D and gives rise to an invariant volume form dµ(z) = Ωn−1. In the coordinatesof H`,˜ we have

Ω = −ddc log N(z, τ).(3.1.2)

Let L ⊂ V be an Ok-lattice, that is, a finitely generated Ok-submodule such that V =L⊗Ok

k and such that the restriction of 〈·, ·〉 to L takes values in d−1k . With the quadratic

form Q(x) = 〈x, x〉, we may also view L as a lattice over Z. Throughout we assume that Lis even as a lattice over Z, that is, 〈x, x〉 ∈ Z for all x ∈ L. This condition is automaticallyfulfilled if the hermitian form on L takes values in Ok. Let

L′ = x ∈ V : 〈x, y〉Q ∈ Z for all y ∈ L,L′Ok

= x ∈ V : 〈x, y〉 ∈ Ok for all y ∈ L

be the Z-dual and the Ok-dual of L, respectively. We have that L′ = d−1k L′Ok

⊃ L.Let Γ be a finite index subgroup of the unitary group U(L) of L. The quotient

XΓ = Γ\D.


is a complex orbifold of dimension n − 1. It can be viewed as the complex points of aconnected component of a Shimura variety. It is compact if and only if V is anisotropic. Inparticular, if n > 2, then XΓ is non-compact. We define the volume of XΓ by vol(XΓ) =∫XΓ

Ωn−1 and the degree of a divisor Z on XΓ by

deg(Z) =

∫Z

Ωn−2.(3.1.3)

3.2. Special divisors. For any vector λ ∈ V of positive norm we put

D(λ) = z ∈ D : 〈z, λ〉 = 0.Let SL be the complex vector space of functions L′/L → C. In the spirit of [Ku2], forϕ ∈ SL and m ∈ Q>0 we define the special divisor

Z(m,ϕ) =∑λ∈L′〈λ,λ〉=m

ϕ(λ)D(λ).

We write Z(m) for the element of

HomC(SL,DivC(D)) ∼= DivC(D)⊗C S∨L

given by ϕ 7→ Z(m,ϕ). If ϕ is invariant under Γ, then Z(m,ϕ) is a Γ-invariant divisor anddescends to a divisor on the quotient XΓ, which we will also denote by Z(m,ϕ).

3.3. Regularized theta lifts. In this subsection we define automorphic Green functionsfor special divisors as regularized theta lifts of harmonic Maass forms.

We use τ as a standard variable in the upper half plane H, and put u = Re(τ), v = Im(τ).The Ok-lattice L together with the symmetric bilinear form 〈·, ·〉Q is an even Z-lattice ofsignature (2n−2, 2). Let ωL be the corresponding Weil representation on SL as in Section 2.For z ∈ D, the Siegel theta function ΘL(τ, z) is a non-holomorphic modular form of weightn− 2 for SL2(Z) with representation ω∨L.

Let f ∈ H2−n(ωL), and denote its Fourier coefficients by c±(m) ∈ SL as in (2.2.2). Thepairing f,ΘL(τ, z) is a function on H, which is invariant under SL2(Z). Following [Bo1]and [BF], we consider the regularized theta lift

Φ(z, f) =

∫ reg

SL2(Z)\Hf,ΘL(τ, z)dµ(τ)(3.3.1)

of f , where dµ(τ) = du dvv2 is the invariant measure. The integral is regularized by taking

the constant term in the Laurent expansion at s = 0 of the meromorphic continuation of

Φ(z, f, s) = limT→∞

∫FTf,ΘL(τ, z)v−sdµ(τ).(3.3.2)

Here FT denotes the standard fundamental domain for SL2(Z) truncated at height T . IfRe(s) > 0, the limit exists and defines a smooth function in z on all of D, which is invariantunder the action of Γ if f is invariant under Γ. It has a meromorphic continuation in s toC, see [Bo1], [BF]. The function Φ(z, f) is defined on all of D (!), but it is only smooth onthe complement of the divisor

Z(f) =∑m>0

c+(−m), Z(m).(3.3.3)

To describe the behavior near this divisor, we extend the incomplete Gamma functionΓ(0, t) =

∫∞te−v dvv to a function on R≥0 by defining it as the constant term in the Laurent


expansion at s = 0 of the meromorphic continuation of Γ(s, t) =∫∞te−vvs dvv . Hence we

have

Γ(0, t) =

Γ(0, t), if t > 0,

0, if t = 0.

The following result is a slight strengthening of [Bo1, Theorem 6.2] in our setting.

Theorem 3.3.1. For any z0 ∈ D there exists a neighborhood U ⊂ D such that the function

Φ(z, f)−∑

λ∈L′∩z⊥0

c+(−〈λ, λ〉, λ)Γ(0, 4π|〈λz, λz〉|)

is smooth on U . Here c+(m,λ) stands for the value c+(m)(λ) of c+(m) at λ+ L.

Proof. We begin by noticing that L′ ∩ z⊥0 is a positive definite Ok-module of rank ≤ n− 1.Hence the sum on the right hand side is finite.

Arguing as in the proof of [Bo1, Theorem 6.2] (see also [Br1, Theorem 2.12]), we see thatthere exists a small neighborhood U ⊂ D of z0 on which the function

Φ(z, f)−∑

λ∈L′∩z⊥0

c+(−〈λ, λ〉, λ) CTs=0

[ ∫ ∞v=1

e4π〈λz,λz〉vv−s−1 dv

]is smooth. Here CTs=0[·] denotes the constant term in the Laurent expansion in s at 0.

Inserting the definition of Γ(0, t) we obtain the assertion.

Corollary 3.3.2. For any z0 ∈ D we have

Φ(z0, f) = limz→z0z/∈Z(f)

[Φ(z, f) +

∑λ∈L′∩z⊥0λ 6=0

c+(−〈λ, λ〉, λ)(log(4π|〈λz, λz〉|)− Γ′(1))

].

Proof. Using the fact that Γ(0, t) = − log(t) + Γ′(1) + o(t) as t → ∞, the corollary followsfrom Theorem 3.3.1.

By a Green function for a divisor D on a complex manifold X we mean a smooth functionG on X rD with the property that for every point z0 ∈ X there is a neighborhood U anda local equation φ for D on U such that G+ log |φ|2 extends to a smooth function on all ofU . Using this definition, we may rephrase Theorem 3.3.1 and the corollary by saying thatΦ(z, f) is a Green function for Z(f). In fact, the difference of log |〈λz, λz〉| and log |φλ|2 forany local equation φλ = 0 of D(λ) extends to a smooth function. In the next section wewill study the growth of Φ(z, f) at the boundary of a toroidal compactifaction of XΓ andshow that it can also be considered as a Green function for a suitably ‘compactified’ divisorthere.

Corollary 3.3.2 will be used in Section 7.5 together with Theorem 6.4.1 to compute theheight pairing of a hermitian line bundle corresponding to an arithmetic Kudla-Rapoportdivisor with a CM cycle.

Proposition 3.3.3. Let ∆D be the U(V )(R)-invariant Laplacian on D. There exists anon-zero real constant c (which only depends on the normalization of ∆D and which isindependent of f), such that

∆DΦ(z, f) = c · degZ(f).

Proof. This can be proved in the same way as [Br1, Theorem 4.7].


Proposition 3.3.4. Let f ∈ H2−n(ωL) be Γ-invariant. If n > 2, then the Green func-tion Φ(z, f) belongs to Lp(XΓ,Ω

n−1) for every p < 2. If n > 3, then Φ(z, f) belongs toL2(XΓ,Ω

n−1).

We will prove this Proposition at the end of Section 4.3.

Theorem 3.3.5. Assume that n > 2 and that f ∈ H2−n(ωL) is Γ-invariant. Let G be asmooth real valued function on XΓ r Z(f) with the properties:

(i) G is a Green function for Z(f),(ii) ∆DG = constant,(iii) G ∈ L1+ε(XΓ,Ω

n−1) for some ε > 0.

Then G(z) differs from Φ(z, f) by a constant.

Proof. The difference G(z) − Φ(z, f) is a smooth subharmonic function on the completeRiemann manifold XΓ which is contained in L1+ε(XΓ,Ω

n−1). By a result of Yau, such afunction must be constant (see e.g. [Br1, Corollary 4.22]).

For n = 2 one can obtain a similar characterization by also requiring growth conditionsat the cusps of XΓ (if there are any).

4. Green functions on toroidal compactifications

As in Section 3, let V be a hermitian space over k of signature (n− 1, 1). In the presentsection we study the behavior of the automorphic Green function Φ(z, f) at the boundaryof a toroidal compactification. We show that it is a log-log Green function in the sense of[BKK] for the divisor Z(f) +B(f), where B(f) is a certain linear combination of boundarydivisors. We show that the multiplicities of the boundary divisors are given by regularizedtheta lifts of the harmonic Maass form f to positive definite quadratic spaces of signature(n− 2, 0). To this end we compute the Fourier-Jacobi expansion of Φ(z, f) and analyze thedifferent terms at the boundary.

4.1. The toroidal compactification. The complex space XΓ = Γ\D can be compactifiedas follows. Let Iso(V ) be the set of isotropic one-dimensional subspaces I ⊂ V . The groupΓ acts on Iso(V ) with finitely many orbits. The rational boundary point corresponding toI ∈ Iso(V ), is the point IR = I ⊗Q R ∈ P(VR). It lies in the closure of D in P(VR). TheBaily-Borel compactification of XΓ is obtained by equipping the quotient

Γ\(D ∪ IR : I ∈ Iso(V )

)with the Baily-Borel topology and complex structure. However, the boundary points areusually singular.

In contrast, here we work with a canonical toroidal compactification of XΓ, which wenow describe, see also [Hof, Chapter 1.1.5] and [Ho3, Section 3.3]. It can be viewed as aresolution of the singularities at the boundary points of the Baily-Borel compactification.

Let I ∈ Iso(V ) be a one-dimensional isotropic subspace. Let ` ∈ I be a generator, and

let ˜∈ V be isotropic such that 〈`, ˜〉 = 1. For ε > 0 we put

Uε(`) = z ∈ D : − 〈z, z〉|〈z, `〉|2

>1

ε.

In the coordinates of H`,˜ we have

Uε(`) ∼= (z, τ) ∈ H`,˜ : N(z, τ) > 1/ε.


The stabilizer U(V )` of ` acts on this subset. Let Γ` = Γ ∩U(V )`. If ε is sufficiently small,then

Γ`\Uε(`) −→ XΓ(4.1.1)

is an open immersion. The center of U(V )` is given by the subgroup of translations Ta fora ∈ Q, where

Ta(λ) = λ+ a〈λ, `〉√dk`

for λ ∈ V . It is isomorphic to the additive group over Q. The action of the translations onH`,˜ is given by Ta(z, τ) = (z, τ + a). The center of Γ` is of the form

Γ`,T = Ta : a ∈ rZ(4.1.2)

for a unique r ∈ Q>0, which is sometimes called the width of the cusp IR. If we putqr = e2πiτ/r, then (z, τ) 7→ (z, qr) defines an isomorphism from Γ`,T \Uε(`) to

Vε(`) =

(z, qr) ∈ Cn−2 × C : 0 < |qr| < exp

(− π

r√|dk|

(〈z, z〉+ 1/ε)

).

Hence Γ`,T \Uε(`) can be viewed as a punctured disc bundle over Cn−2. Adding the originto every disc gives the disc bundle

Vε(`) =

(z, qr) ∈ Cn−2 × C : |qr| < exp

(− π

r√|dk|

(〈z, z〉+ 1/ε)

).

The action of Γ` on Vε(`) extends to an action on Vε(`), which leaves the boundary divisorqr = 0 invariant, and which is free if Γ is sufficiently small. We obtain an open immersionof orbifolds

Γ`\Uε(`) −→ (Γ`/Γ`,T ) \Vε(`).(4.1.3)

It can be used to glue the right hand side to XΓ to obtain a partial compactification, whichis smooth if Γ is sufficiently small. For a point (z0, 0) ∈ Vε(`) and δ > 0, we put

Bδ(z0, 0) =

(z, qr) ∈ Vε(`) : 〈z− z0, z− z0〉 < δ, |qr| < δ.(4.1.4)

Then the images of the Bδ(z0, 0) for δ > 0 under the natural map to (Γ`/Γ`,T ) \Vε(`) definea basis of open neighborhoods of the boundary point given by (z0, 0).

We let X∗Γ be the compactification of XΓ obtained by gluing the right hand side of(4.1.3) to XΓ for every Γ-class of Iso(V ). We denote by BI the boundary divisor of X∗Γcorresponding to I ∈ Iso(V ).

The behavior of the special divisor Z(m,ϕ) near the boundary can be described as follows.Let I ∈ Iso(V ) and let ` ∈ I be a generator. Let 0 < ε < 1

2m be small enough such that(4.1.1) defines an open immersion. Then Lemma 4.1.1 below implies that the pullback ofZ(m,ϕ) to Uε(`) is given by the local special divisor

Z`(m,ϕ) =∑

λ∈L′∩`⊥〈λ,λ〉=m

ϕ(λ)D(λ).

Lemma 4.1.1. If z0 is a generator of z ∈ D and λ ∈ V ⊗Q R, we have

〈λ, λ〉z ≥|〈λ, `〉|2|〈z0, z0〉|

2|〈z0, `〉|2.


Proof. The right hand side is independent of the choice of the generator z0. Hence we mayassume 〈z0, `〉 = 1. Moreover, both sides of the inequality remain unchanged if we act on λand z0 with elements of the stabilizer of ` in U(V )(R). Therefore, it suffices to consider the

case z0 = τ√dk`+ ˜. The remaining computation we leave to the reader.

4.2. Regularized integrals. Let k ∈ Z≥0. Let (M,Q) be an even integral lattice as inSection 2. Following [Bo1], for f ∈ H−k(ωM ) and g ∈ Mk(ω∨M ) we define a regularizedPetersson pairing by

(f, g)reg =

∫ reg

SL2(Z)\Hf(τ), g(τ)dµ(τ)(4.2.1)

= limT→∞

∫FTf(τ), g(τ)dµ(τ).

In Section 4.3 such integrals will occur as multiplicities of the boundary components, whereg will be the theta function of a positive definite hermitian lattice given by a quotient of L.

In the special case when k = 0 and g is constant, this integral is evaluated in [Bo1,Theorem 9.1]. Here we describe how the integral can be computed when k > 0. We denotethe Fourier expansion of g by

g(τ) =∑m≥0

b(m)qm,

with coefficients b(m) ∈ S∨M . We let ϑ = q ddq be the Ramanujan theta operator on q-series.

Recall that the image under ϑ of a holomorphic modular form g of weight k is in generalnot a modular form. However, the function

ϑ(g) = ϑ(g)− k

12gE2

is a holomorphic modular form of weight k + 2. Here E2(τ) = 1 − 24∑m≥1 σ1(m)qm

denotes the non-modular Eisenstein series of weight 2 for SL2(Z). If Rk = 2i ∂∂τ + kv denotes

the Maass raising operator and E∗2 (τ) = E2(τ) − 3πv the non-holomorphic (but modular)

Eisenstein series of weight 2, we also have

ϑ(g) = − 1

4πRk(g)− k

12gE∗2 .(4.2.2)

If h(q) ∈ C((q)) is a (formal) Laurent series in q, we denote by CT[h] its constant term.

Theorem 4.2.1. Let f ∈ H−k(ωM ) and g ∈Mk(ω∨M ) be as above.

(1) If k > 0, then

(f, g)reg =4π

kCT[f+, ϑ(g)] =

4π

k

∑m>0

m · c+(−m), b(m).

(2) If k = 0 (so that g is constant), then putting σ1(0) = − 124 we have

(f, g)reg =π

3CT[f+, gE2)] = −8π

∑m≥0

σ1(m) · c+(−m), g.

Proof. (1) We use the identity ∂(E∗2dτ) = − 3πdµ(τ) to obtain

(f, g)reg = −π3

∫ reg

SL2(Z)\Hf(τ), ∂(gE∗2dτ).(4.2.3)


In view of (4.2.2), we have

∂(gE∗2dτ) = − 3

kπ∂(Rk(g)dτ).

Putting this into (4.2.3), we get

(f, g)reg =1

k

∫ reg

SL2(Z)\Hf(τ), ∂(Rk(g)dτ)

=1

klimT→∞

∫FT

df(τ), Rk(g)dτ − 1

k

∫SL2(Z)\H

(∂f), Rk(g)dτ

= −1

klimT→∞

∫ 1

0

f(u+ Ti), Rk(g)(u+ Ti)du

+1

k

∫SL2(Z)\H

ξ−k(f), Rk(g)vk+2dµ(τ).

The second summand on the right hand side is a Petersson scalar product which is easilyseen to vanish. The first summand is equal to

4π

kCT[f+, ϑ(g)].

This concludes the proof of the k > 0 case.(2) If k = 0, and f ∈ M !

0(ωL), the assertion follows from [Bo1, Theorem 9.2]. If f ∈H0(ωL) it can be proved in the same way.

4.3. Automorphic Green functions at the boundary. Let I ∈ Iso(V ) be an isotropick-line. Then a = I ∩ L is a projective Ok-module of rank 1. The Ok-module

D = L ∩ a⊥/a

is positive definite of rank n − 2. Let ` ∈ a be a primitive (that is, Q` ∩ a = Z`) isotropic

vector. We write a = a0` with a fractional ideal a0 ⊂ k, and we let ˜∈ V be isotropic suchthat 〈˜, `〉 = 1.

The lattice D can be realized as a sublattice of L as follows. The lattice a∗ = L′Ok∩ I⊥

is a projective Ok-module of rank n− 1. The quotient L′Ok/a∗ is an Ok-module of rank 1,

which is projective since it is torsion free. Hence there is a projective Ok-module b ⊂ L′Ok

of rank 1 such that L′Ok= a∗ ⊕ b. We have 〈a, L′Ok

〉 = 〈a, b〉 = Ok and 〈b, L〉 = Ok. Weput

E = L ∩ a⊥ ∩ b⊥.

Lemma 4.3.1. With a defined as above,

(1) L ∩ a⊥ = E ⊕ a and D ∼= E;(2) if L is Ok-self-dual then L = E ⊕ a⊕ b;(3) if L is Ok-self-dual and dk is odd then in (ii) we may chose b to be isotropic.

Remark 4.3.2. Assume that Γ = U(L) is the full unitary group of L. Let n = 〈L, a〉 ⊂ Ok,and denote by Na the positive generator of (dkn)−1 ∩ Q. Then the quantity r in (4.1.2) isequal to N(a0) ·Na. In particular, if dk is odd and L is Ok-self-dual then r = N(a0).

Let f ∈ H2−n(ωL). In analogy with [Bo1, Theorem 5.3], the harmonic Maass form finduces an SD-valued harmonic Maass form fD ∈ H2−n(ωD). It is characterized by its


values on ν ∈ D′/D as follows:

fD(τ)(ν) =∑

µ∈L′/Lµ|L∩a⊥=ν

f(τ)(µ).(4.3.1)

Here µ | L∩ a⊥ denotes the restriction of µ ∈ Hom(L,Z) to L∩ a⊥, and we consider ν ∈ D′as an element of Hom(L ∩ a⊥,Z) via the quotient map L ∩ a⊥ → D.

Let ε > 0 such that (4.1.1) is an open immersion. For a boundary point (z0, 0) ∈ Vε(`)and δ > 0, we consider the Green function Φ(z, f) in the open neighborhood Bδ(z0, 0)defined in (4.1.4). The pullback of the special divisor Z(f) to Bδ(z0, 0) is given by the linearcombination of local special divisors

Z`(f) =∑m>0

c+(−m), Z`(m)

.

Note that Z`(m) is invariant under the subgroup of translations Γ`,T ⊂ Γ`. The support of

Z`(m) on Vε(`) is the union of the sets (z, qr) : 〈z + ˜, λ〉 = 0 for λ ∈ L′ ∩ `⊥/Γ`,T with〈λ, λ〉 = m.

Theorem 4.3.3. Let f ∈ H2−n(ωL) and denote its Fourier coefficients by c±(m). Let

(z0, 0) ∈ Vε(`) be a boundary point. The set

Sf = λ ∈ L′ ∩ `⊥ : 〈λ, λ〉 > 0, c+(−〈λ, λ〉, λ) 6= 0 and 〈z0 + ˜, λ〉 = 0

is finite. If δ > 0 is sufficiently small, then the function

Φ(z, f) +rΦD(fD)

2πN(a0)log |qr|+ c+(0, 0) log |log |qr||+ 2

∑λ∈Sf

c+(−〈λ, λ〉, λ) log |〈z + ˜, λ〉|

has a continuation to a continuous function on Bδ(z0, 0). It is smooth on the complementof the boundary divisor qr = 0, and its images under the differentials ∂, ∂, ∂∂ have log-loggrowth along the divisor qr = 0 in the sense of [BBK, Definition 1.2]. Here ΦD(fD) =(fD,ΘD)reg is the regularized Petersson pairing of fD and the theta function ΘD as definedin (4.2.1).

We postpone the proof of the theorem to Section 4.4.

Remark 4.3.4. Let c±D(m) ∈ SD be the coefficients of fD, and write ΘD(τ) =∑m≥0RD(m)qm,

where the representation numbers RD(m) ∈ S∨D are given by

RD(m)(ϕ) =∑λ∈D′

Q(λ)=m

ϕ(λ)

for ϕ ∈ SD. If n > 2, then according to Theorem 4.2.1 we have

ΦD(fD) =4π

n− 2CT[f+

D , ϑ(ΘD)] =4π

n− 2

∑m>0

m · c+D(−m), RD(m).

If n = 2, then D is trivial, and we have

ΦD(fD) =π

3CT[f+

D · E2] = −8π∑m≥0

c+D(−m)σ1(m).


We now associate a boundary divisor to the harmonic Maass form f ∈ H2−n(ωL). Wedefine the multiplicity of the boundary divisor BI with respect to f by

multBI (f) =rΦD(fD)

4πN(a0).(4.3.2)

If the principal part of f has rational coefficients, then according to Remark 4.3.4, thismultiplicity is rational. In the special case that dk is odd, L is Ok-self-dual, and Γ = U(L),we have in view of Remark 4.3.2 that

multBI (f) =1

4πΦD(fD).(4.3.3)

We define the boundary divisor associated with f by

B(f) =∑

I∈Iso(V )/Γ

multBI (f) ·BI .(4.3.4)

Theorem 4.3.3 implies the following corollary.

Corollary 4.3.5. The function Φ(z, f) is a logarithmic Green function on X∗Γ for the divisorZ(f) +B(f) with possible additional log-log growth along the boundary divisors BI .

4.4. The Fourier-Jacobi expansion. Here we compute the Fourier-Jacobi expansion ofthe automorphic Green function Φ(z, f) using [Hof], [Bo1] and [Br1], and we provide theproofs of Theorem 4.3.3 and Proposition 3.3.4.

The natural embedding of D into the Grassmannian of negative definite 2-dimensionaloriented real subspaces of VR is compatible with the actions of the unitary group U(V, 〈·, ·〉)and the orthogonal group O(V, 〈·, ·〉Q). We may calculate the theta lift of f ∈ H2−n(ωL)to XΓ by lifting to the orthogonal group O(V, 〈·, ·〉Q) and then pulling back to the unitarygroup.

We continue to use the setup of Section 4.3. In addition we introduce the followingnotation. We fix `′ ∈ L′ such that 〈`′, `〉Q = 1. We denote by N the positive integer whichgenerates the ideal 〈L, `〉Q ⊂ Z. We write `⊥,Q for the orthogonal complement of ` withrespect to the bilinear form 〈·, ·〉Q, and put K = (L ∩ `⊥,Q)/Z`. Then K is an even latticeover Z of signature (2n− 3, 1).

Remark 4.4.1. If b ⊂ L′Okwith 〈a, b〉 = Ok as in Section 4.3, we may choose `′ ∈ d−1

k b and˜∈ k`′ ⊕ k`. When L = L′Ok

and dk is odd, then according to Lemma 4.3.1 we may choose

b such that it is isotropic. Then we can we can take ˜∈ k`′.

For x ∈ VR we put x2 = 〈x, x〉Q and |x| =√|x2|. Let (z, τ) ∈ H`,˜ and let z be the

corresponding point in D. We have

`2z = − 2

N(z, τ),

where the quantity N(z, τ) = −〈(z, τ), (z, τ)〉 is positive. We also view z as a two-dimensional(oriented) real subspace of VR. The vector `z spans a one-dimensional real subspace of z,whose orthogonal complement in z with respect to 〈·, ·〉Q we denote by w, so that z = w⊕R`z.The real line w is generated by the vector w0(z) = −i(z, τ) = −i(z + τ

√dk`+ ˜), which we

use to define an orientation on w. Hence we obtain a map

D −→ Gr+(K)(4.4.1)


to the Grassmannian Gr+(K) of oriented negative lines in K ⊗Z R. If λ ∈ K ⊗Z R, we have〈−i(z, τ), λ〉Q = 2 Im〈(z, τ), λ〉. The orthogonal projection of λ to w is given by

|λw||`z|

= | Im〈(z, τ), λ〉|.

We also define the vector

µ = −`′ + `z2`2z

+`z⊥

2`2z⊥.

in L ∩ `⊥,Q. It is easily checked that

〈µ, λ〉Q = Re〈(z, τ), λ〉.For w ∈ Gr+(K) and λ ∈ K ⊗Z R, we write 〈w, λ〉Q > 0 if 〈w0, λ〉Q > 0 for a vector w0 ∈ wdefining the orientation.

Let f ∈ H2−n(ωL). Similarly as in (4.3.1), according to [Bo1, Theorem 5.3], the har-monic Maass form f induces an SK-valued harmonic Maass form fK ∈ H2−n(ωK). It ischaracterized by its values on ν ∈ K ′/K as follows:

fK(τ)(ν) =∑

µ∈L′/Lµ|L∩`⊥,Q=ν

f(τ)(µ).(4.4.2)

Here µ | L ∩ `⊥,Q denotes the restriction of µ ∈ Hom(L,Z) to L ∩ `⊥,Q, and we considerν ∈ K ′ as an element of Hom(L ∩ `⊥,Q,Z) via the quotient map L ∩ `⊥,Q → K.

Finally, following [Br1, (3.25)], we define a special function for A,B ∈ R by

Vn(A,B) =

∞∫0

Γ(n− 1, A2y)e−B2y−1/yy−3/2 dy.

According to [Br1, p. 74] we have

Vn(A,B) = 2(n− 2)!

n−2∑r=0

A2r

r!(A2 +B2)1/4−r/2Kr−1/2(2

√A2 +B2).

The following result is now an immediate consequence of [Br1, Theorem 3.9].

Theorem 4.4.2. Let f ∈ H2−n(ωL) and denote its Fourier coefficients by c±(m) ∈ SL. Letz ∈ D r Z(f) with |`2z| < 1

2m0, where m0 = maxm ∈ Q : c+(−m) 6= 0. Then the Green

function Φ(z, f) is equal to

1√2|`z|

ΦK(w, fK) + Cf + c+(0, 0) log |`2z|

− 2∑

λ∈K′r0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c+(−〈λ, λ〉, ν) log(1− e

(〈ν, `′〉Q + 〈λ, µ〉Q + i|λw|/|`z|

))

+2√π

∑λ∈K′〈λ,λ〉>0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c−(−〈λ, λ〉, ν)∑j≥1

1

je(j〈ν, `′〉Q + j〈λ, µ〉Q

)Vn(πj|λ||`z|

,πj|λw||`z|

),

where

Cf = −c+(0, 0) (log(2π) + Γ′(1))− 2∑

a∈Z/NZa 66=0

c+(0, a`/N) log |1− e(a/N)|.


Here ΦK(w, fK) denotes the function on Gr+(K) given by the regularized theta lift of fKfor the orthogonal group of K as in [Br1, Chapter 3.1]. We view it as a function on D viathe map (4.4.1). Finally, log(z) stands for the principle branch of the complex logarithm.

Remark 4.4.3. If f ∈ M !2−n(ωL) is weakly holomorphic, then according to [Hof, Theo-

rem 4.2.1] there exists a meromorphic modular form Ψ(z, f) of weight c+(0, 0)/2 for thegroup Γ (with a multiplier system of finite order) such that −2 log ‖Ψ(z, f)‖2 = Φ(z, f) anddiv(Ψ(z, f)) = 1

2Z(f). Here ‖ · ‖ denotes the suitably normalized Petersson metric. Theabove Fourier expansion of Φ(z, f) leads to the Borcherds product expansion

Ψ(z, f) = e(〈(z, τ), %W 〉

) ∏λ∈K′〈W,λ〉Q>0

∏ν∈L′/L

ν|L∩`⊥,Q=λ

(1− e(〈ν, `′〉Q + 〈(z, τ), λ〉)

)c+(−〈λ,λ〉,ν),

which converges for N(z, τ) > 4m0. Here W ⊂ Gr+(K) denotes a Weyl chamber corre-sponding to f (that is, a connected component of the complement of the singular locus ofΦK(w, f)), and %W ⊂ K⊗ZQ denotes the corresponding Weyl vector. Moreover 〈W,λ〉Q > 0means that 〈w, λ〉Q > 0 for w ∈W , see [Hof, Section 4.1.2].

We now turn to the proof of Theorem 4.3.3. We begin with two technical lemmas. Thefirst one gives an estimate for the majorant of the lattice K. For 0 < C < 1 we define

SC = (z, τ) ∈ H`,˜ : C · 2√|dk| Im(τ) > 〈z, z〉.

For λ ∈ K ⊗Z R and (z, τ) ∈ H`,˜ we define

h((z, τ), λ) = N(z, τ)〈λ, λ〉+ 2(Im〈(z, τ), λ〉)2.

Lemma 4.4.4. Let 0 < C < 1. There exists an ε > 0 such that for any (z, τ) ∈ SC and

any λ = λD − a√dk`− b√

dk˜∈ K ⊗Z R (where λD ∈ D ⊗Z R and a, b ∈ R), we have

h((z, τ), λ) ≥ ε(a2|dk|+ b2 Im(τ)2 + N(z, τ)〈λD, λD〉

).

Proof. This result can be viewed as a lower bound for the majorant 〈λw⊥ , λw⊥〉Q−〈λw, λw〉Qassociated to the negative line w = Rw0(z) ∈ Gr+(K). It directly follows from [Br1, Lemma4.13]. Note that in the proof of this lemma, of the equalities defining Rt we only need that

|q(YD)| < By1y2 with B = t4

t4+1 and t > 0.

Corollary 4.4.5. Let 0 < C < 1. There exists an ε > 0 such that for any (z, τ) ∈ SC and

any λ = λD − a√dk`− b√

dk˜∈ K ⊗Z R (where λD ∈ D ⊗Z R and a, b ∈ R) with 〈λ, λ〉 ≤ 0,

we have(Im〈(z, τ), λ〉)2 ≥ ε

(a2|dk|+ b2 Im(τ)2 + N(z, τ)〈λD, λD〉

).

The following lemma is a useful variant of the corollary.

Lemma 4.4.6. Let A ≥ 0 and 0 ≤ B < 1. Assume that Im(τ) > (|z|+2A)2

4|dk|(1−B)2 . Then we

haveIm〈(z, τ), λ〉 −A|λ| ≥ B (a|dk|+ b Im(τ))

for all λ = λD − a√dk`− b√

dk˜∈ K ⊗Z R with b ≥ 0 and 〈λ, λ〉 ≤ 0.

Proof. We have

Im〈(z, τ), λ〉 = Im〈z, λD〉+ a√|dk|+ b Im(τ)

≥ − 1

2|λD| · |z|+ a

√|dk|+ b Im(τ).


Since 0 ≥ 〈λ, λ〉 = 〈λD, λD〉 − 2ab, we also have |λ|2 ≤ 4ab and |λD|2 ≤ 4ab. Consequently,

Im〈(z, τ), λ〉 −A|λ| ≥ a√|dk|+ b Im(τ)−

√ab · (|z|+ 2A)

≥ B (a|dk|+ b Im(τ))

+ (1−B) (a|dk|+ b Im(τ))−√ab · (|z|+ 2A).

The quantity in the latter line can be interpreted as a binary quadratic form in√a and

√b,

which is positive definite if Im(τ) > (|z|+2A)2

4|dk|(1−B)2 . This implies the assertion.

Proof of Theorem 4.3.3. It is easily seen that Sf is finite. To obtain the claimed analyticproperties of Φ(z, f) on Bδ(z0, 0), we consider the different terms of the Fourier expansiongiven in Theorem 4.4.2.

1. We begin with the term∑λ∈K′〈λ,λ〉>0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c−(−〈λ, λ〉, ν)∑j≥1

1

je(j〈ν, `′〉Q + j〈λ, µ〉Q

)Vn(πj|λ||`z|

,πj|λw||`z|

).

According to [Br1, equality (3.26)], the function Vn(A,B) is bounded by a constant multiple

of e−√A2+B2

. Moreover, for λ ∈ K ′ with 〈λ, λ〉 > 0 we have

2λ2

|`2z|+ 2|λ2w||`2z|

>λ2

|`2z|+ 2|λ2w||`2z|

= h((z, τ), λ).

If we write λ = λD − a√dk` − b√

dk˜ (where λD ∈ D ⊗Z Q and a, b ∈ Q), then in view of

Lemma 4.4.4 there exists an ε′ > 0 such that

Vn(πj|λ||`z|

,πj|λw||`z|

) exp

(−ε′j

√a2|dk|+ b2 Im(τ)2 + N(z, τ)〈λD, λD〉

).

Since the coefficients c−(m,µ) have only polynomial growth as m→ −∞, we find that theabove sum over λ ∈ K ′ converges uniformly on Bδ(z0, 0) to a function which is bounded

by O(exp(−ε′′√− log |qr|)) as qr → 0 for some ε′′ > 0. Hence this sum converges to

a continuous function on Bδ(z0, 0) which vanishes along the divisor qr = 0. Analogousestimates hold for all iterated partial derivatives with respect to (z, τ). Using the fact that

dτ = r2πi

dqrqr

, we obtain that the differentials ∂, ∂, ∂∂ of this function have log-log growth

along qr = 0.2. For the term c+(0, 0) log |`2z|, we notice that

log |`2z| = − log(N(z, τ)/2)

= − log(√|dk| Im(τ)− 〈z, z〉/2

)= − log (− log |qr|)− log

(r√|dk|

2π+〈z, z〉

2 log |qr|

).

The second summand on the right hand side extends to a continuous function on Bδ(z0, 0)whose differentials have log-log growth along the boundary divisor qr = 0.


3. Next, we consider the term∑λ∈K′r0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c+(−〈λ, λ〉, ν) log(1− e

(〈ν, `′〉Q + 〈λ, µ〉Q + i|λw|/|`z|

))

=∑

λ∈K′r0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c+(−〈λ, λ〉, ν) log(1− e

(〈ν, `′〉Q + Re〈(z, τ), λ〉+ i| Im〈(z, τ), λ〉|

)).

There exists a constant C > 0 such that c+(m, ν) = O(eC√m) for m→∞. Hence it follows

from Corollary 4.4.5 and Lemma 4.4.6, that the sum over λ ∈ K ′ with 〈λ, λ〉 < 0 converges

uniformly on Bδ(z0, 0) to a function which is bounded by O(exp(−ε′′√− log |qr|)) as qr → 0

for some ε′′ > 0. Observe that 〈λ, λ〉 < 0 implies that 〈λ, `〉 6= 0.Moreover, Lemma 4.4.4 implies that, if δ is sufficiently small, the sum over λ ∈ K ′ with

〈λ, λ〉 ≥ 0 and 〈λ, `〉 6= 0 converges uniformly on Bδ(z0, 0) to a function which is bounded

by O(exp(−ε′′√− log |qr|)) as qr → 0 for some ε′′ > 0. Analogous estimates hold for all

iterated partial derivatives with respect to (z, τ). Hence, up to a continuous function withlog-log growth differentials, the above sum is equal to∑

λ∈K′r0〈λ,`〉=0

∑ν∈L′/L

ν|L∩`⊥,Q=λ

c+(−〈λ, λ〉, ν) log(1− e

(〈ν, `′〉Q + 〈λ, µ〉Q + i|λw|/|`z|

))(4.4.3)

=∑

λ∈L′∩`⊥/Z`λ6=0

c+(−〈λ, λ〉, λ) log(

1− e(

Re〈z + ˜, λ〉+ i| Im〈z + ˜, λ〉|)).

Notice that the this sum does not depend on τ . We let Tf be the finite set

Tf = λ ∈ L′ ∩ `⊥/Z` : 〈λ, λ〉 > 0, c+(−〈λ, λ〉, λ) 6= 0 and Im〈z0 + ˜, λ〉 = 0,(4.4.4)

and we let Tf be a fixed system of representatives for Tf/±1. If δ is sufficiently small,then on the the right hand side of (4.4.3), the sum over those λ which do not belong to Tfdefines a smooth function on Bδ(z0, 0). Hence, up to a smooth function, (4.4.3) is equal to∑λ∈Tf

c+(−〈λ, λ〉, λ) log(

1− e(

Re〈z + ˜, λ〉+ i| Im〈z + ˜, λ〉|))

=∑λ∈Tf

c+(−〈λ, λ〉, λ) log∣∣∣1− e(〈z + ˜, λ〉

)∣∣∣2 + 4π∑λ∈Tf

Im〈z+˜,λ〉<0

c+(−〈λ, λ〉, λ) Im〈z + ˜, λ〉.

We find that (4.4.3) is the sum of a smooth function on Bδ(z0, 0) and∑λ∈Sf

c+(−〈λ, λ〉, λ) log |〈z + ˜, λ〉|+ 4π∑λ∈Tf

Im〈z+˜,λ〉<0

c+(−〈λ, λ〉, λ) Im〈z + ˜, λ〉.(4.4.5)

4. It remains to consider the quantity 1√2|`z|

ΦK(w, fK). Let `K ∈ (k` ∩ L)/Z` = a/Z`be a primitive vector. Then a = Z`K +Z`. If we write `K = a` with a ∈ k, we have a = a0`with a0 = Za+ Z ⊂ k and

2 Im(a)√|dk|

= N(a0).(4.4.6)


The positive definite lattice K∩`⊥,QK /Z`K is isomorphic to the Ok-lattice D = L∩a⊥/a. Weuse the Fourier expansion given in [Br1, Chapter 3.1] with respect to the primitive isotropicvector `K , to describe the behavior on Bδ(z0, 0). The vector

w1 = −i (z, τ)√2 N(z, τ)

is the unique positively oriented vector in the real line w of length −1. For λ ∈ D ⊗Z R wehave

〈w1, λ〉Q =

√2 Im〈z, λ〉√

N(z, τ),

〈w1, `K〉Q =

√2 Im(a)√N(z, τ)

.

Let `′K ∈ K ′ such that 〈`′K , `K〉Q = 1. According to [Br1, p. 68], we have in our presentnotation that

ΦK(w1, fK) =1√

2〈w1, `K〉QΦD(fD)

+ 4√

2π〈w1, `K〉Q∑λ∈D′

∑ν∈L′/Lν|L∩`⊥=λ

c+(−〈λ, λ〉, ν)B2

(〈w1, λ〉Q〈w1, `K〉Q

+ 〈ν, `′K〉Q)

+ 4√

2

(π

〈w1, `K〉Q

)n−2 ∑λ∈D′r0

∑ν∈L′/Lν|L∩`⊥=λ

c−(−〈λ, λ〉, ν)|λ|n−1

×∑j≥1

jn−3e

(j〈w1, λ〉Q〈w1, `K〉Q

+ j〈ν, `′K〉Q)Kn−1

(2πj|λ|〈w1, `K〉Q

).(4.4.7)

Here B2(x) denotes the 1-periodic function on R which agrees on 0 ≤ x < 1 with the secondBernoulli polynomial B2(x) = x2 − x + 1/6, and Kν(x) denotes the K-Bessel function.Because of the exponential decay of the K-Bessel function, we find that

1√2|`z|

ΦK(w, fK) =

√N(z, τ)

2ΦK(w, fK)

=N(z, τ)

4 Im(a)ΦD(fD) + 4π Im(a)

∑λ∈L′∩a⊥/a

c+(−〈λ, λ〉, λ)B2

(Im〈z + ˜, λ〉

Im(a)

)+ s(z, τ),

where s(z, τ) is a continuous function on Bδ(z0, 0) with log-log growth differentials. If δ issufficiently small, then the second summand on the right hand side is the sum of a smoothfunction on Bδ(z0, 0) and

8π∑λ∈Tf

Im〈z+˜,λ〉<0

c+(−〈λ, λ〉, λ) Im〈z + ˜, λ〉.

Note that this term is the negative of the contribution coming from the second quantityin (4.4.5). We obtain that up to a continuous function on Bδ(z0, 0) with log-log growth


differentials, the term 1√2|`z|

ΦK(w, fK) is equal to

−rΦD(fD)

2πN(a0)log |qr|+ 8π

∑λ∈Tf

Im〈z+˜,λ〉<0

c+(−〈λ, λ〉, λ) Im〈z + ˜, λ〉.(4.4.8)

Here we have also used (4.4.6).5. Adding together all the contributions, we find that if δ is sufficiently small, then

Φ(z, f) +rΦD(fD)

2πN(a0)log |qr|+ c+(0, 0) log |log |qr||+ 2

∑λ∈Sf

c+(−〈λ, λ〉, λ) log |〈z + ˜, λ〉|

has a continuation to a continuous function on Bδ(z0, 0). It is smooth on the complementof the boundary divisor qr = 0, and its images under the differentials ∂, ∂, ∂∂ have log-loggrowth along the divisor qr = 0.

Proof of Proposition 3.3.4. We only prove that for n > 3 the Green function Φ(z, f) belongsto L2(X∗Γ,Ω

n−1) = L2(XΓ,Ωn−1). The other assertion can be proved analogously. Since

X∗Γ is compact, it suffices to show this locally for a small neighborhood of any point of X∗Γ.Since Φ(z, f) has only logarithmic singularities outside the boundary, and since Ωn−1 issmooth outside the boundary, this is clear outside the boundary points.

Therefore it suffices to show that for any primitive isotropic vector ` ∈ L and any bound-ary point (z0, 0) ∈ Vε(`) the function Φ(z, f) is square integrable with respect to the measureΩn−1 in a small neighborhood Bδ(z0, 0).

It is easily seen that there exists a non-zero constant c such that

Ωn−1 = c ·N(z, τ)−ndz dz dτ dτ

= − r2c

4π2·

(√|dk|rπ

log |qr| − 〈z, z〉

)−ndz dz

dqr dqr|qr|2

.

Here we have put dz = dz1 · · · dzn−2. Hence, according to Theorem 4.3.3, it suffices to showthat log |qr| is square integrable on Bδ(z0, 0) with respect to the measure Ωn−1. Since n > 3,this is now easily seen.

5. Integral models of Shimura varieties

In this section we introduce the arithmetic Shimura variety M on which we will be doingintersection theory, and introduce certain cycles on M in dimension one and codimensionone.

For the remainder of the article we assume that dk is odd. This hypothesis will be used inseveral places, but the primary reason for this assumption is that when dk is even, Theorem5.1.4 below is not known (or necessarily expected) to be true. It is Theorem 5.1.4 that tellsus that the Shimura variety M is regular, and regularity is a prerequisite for a having a goodintersection theory.

5.1. The stack M and the Kudla-Rapoport divisors. Let M(m,0) be the algebraic stack

over Ok whose functor of points assigns to an Ok-scheme S the groupoid2 of triples (A,ψ, i),in which

• A is an abelian scheme over S of relative dimension m,• ψ : A→ A∨ is a principal polarization,

2Recall that a groupoid is a category in which all morphisms are invertible.


• i : Ok → End(A) is an action of Ok on A.

We insist that the polarization ψ be Ok-linear, in the sense that ψ i(x) = i(x)∨ ψ forevery x ∈ Ok. We further insist that the action of Ok satisfy the signature (m, 0) condition:the induced action of Ok on the OS-module Lie(A) is through the structure morphismOk → OS . We usually just write A ∈ M(m,0)(S) for an S-valued point, and suppress ψ andi from the notation.

Remark 5.1.1. The stack M(1,0) is simply the moduli space of elliptic curves with complexmultiplication by Ok, with the action of Ok on Lie algebras normalized in a prescribed way.

Remark 5.1.2. The stack M(0,m) is defined in exactly the same way as M(m,0), except thatthe signature condition is replaced by the signature (0,m) condition: the induced action ofOk on Lie(A) is through the complex conjugate of the structure morphism Ok → OS . Thestack M(0,m) is obtained from M(m,0) by pullback through complex conjugation Spec(Ok)→Spec(Ok) on the base.

The following result is a restatement of [Ho3, Proposition 2.1.2].

Proposition 5.1.3. The stack M(m,0) is smooth and proper of relative dimension 0 over Ok.In particular, it is regular and flat over Ok. The same results hold for M(0,m).

Let M(m,1) be the algebraic stack over Ok whose functor of points assigns to an Ok-schemeS the groupoid of quadruples (A,ψ, i,F) in which

• A is an abelian scheme over S of relative dimension m+ 1,• ψ : A→ A∨ is a principal polarization of A,• i : Ok → End(A) is an action of Ok on A,• F ⊂ Lie(A) is an Ok-stable OS-submodule, which is locally an OS-module direct

summand of rank m.

We again insist that ψ be Ok-linear, and that the subsheaf F satisfy Kramer’s signature(m, 1) condition: the action of Ok on F is through the structure morphism Ok → OS ,while the action of Ok on the line bundle Lie(A)/F is through the complex conjugate ofthe structure morphism. When no confusion will arise, we suppress ψ, i, and F from thenotation, and write S-valued points as A ∈ M(m,1)(S).

The following theorem follows from results of Kramer [Kr] and Pappas [Pa].

Theorem 5.1.4 (Kramer, Pappas). The stack M(m,1) is regular, flat over Ok of relativedimension m, and smooth over Ok[1/dk].

For a fixed n ≥ 2, we next define divisors on the n-dimensional regular algebraic stack

M = M(1,0) ×OkM(n−1,1),

following Kudla-Rapoport [KR2, Section 2]. Let S be a connected Ok-scheme. For any pair(A0, A) ∈ M(S) we may form the Ok-module

(5.1.1) L(A0, A) = HomOk(A0, A).

This module is equipped with the positive definite hermitian form

〈x, y〉 = ψ−10 y∨ ψ x,

where the composition on the right is viewed as an element of Ok ∼= EndOk(A0).

Definition 5.1.5. For each positive m ∈ Q and each r | dk, define the Kudla-Rapoportdivisor Z(m, r) to be the algebraic stack over Ok whose functor of points assigns to everyconnected Ok-scheme S the groupoid of all triples (A0, A, λ) in which


• (A0, A) ∈ M(S),• λ ∈ r−1L(A0, A) satisfies 〈λ, λ〉 = m,

and the morphism δkλ : A0 → A induces the trivial map

(5.1.2) δkλ : Lie(A0)→ Lie(A)/F ,where δk is any Ok-module generator of dk.

Remark 5.1.6. Of course (2.1.1) implies that Z(m, r) = ∅ unless m ∈ N(r)−1Z.

Remark 5.1.7. The vanishing of (5.1.2) is automatic if N(r) ∈ O×S . In general, if N is anyOS-module with a commuting action of Ok, and if αn = αn for all α ∈ Ok and n ∈ N , thenδkN = 0. If N(r) ∈ O×S then any λ ∈ r−1L(A0, A) induces an Ok-linear map

λ : Lie(A0)→ Lie(A)/F .Recalling that Ok acts on the Lie(A0) through the structure morphism Ok → OS , and actson the quotient Lie(A)/F through the complex conjugate, we may take N to be the imageof this map to see that (5.1.2) is trivial.

Remark 5.1.8. When r = Ok, our Kudla-Rapoport divisors are related to the Z(m) of[KR2] as follows. In [loc. cit.] there is a stack M(n − 1, 1) similar to our M(n−1,1), butdefined by imposing Pappas’s wedge conditions on Lie(A), instead of imposing the subsheafF ⊂ Lie(A). Forgetting the subsheaf F defines a morphism M(n−1,1) →M(n− 1, 1), whichinduces a morphism from M to the stack M of [loc. cit.]. Our Z(m,Ok) is the pullback ofZ(m) to M.

When r 6= Ok, our Z(m, r) is not the pullback of any divisor on M, because one needsthe subsheaf F in order to formulate the vanishing condition (5.1.2).

The forgetful map Z(m, r) → M is finite, unramified, and representable, as in [KR2,Proposition 2.10]. Using the following proposition, we may view Z(m, r) as a divisor on M inthe usual way (the meaning of this is explained in detail in Section 7.1).

Proposition 5.1.9. The stack Z(m, r) is a local complete intersection of dimension n− 1.

Proof. When r = Ok this is [Ho3, Proposition 3.2.3]. The methods will be used again inSection 7.4, and so we give the argument in the general case. Let F be an algebraicallyclosed field, let

z = (A0, A, λ) ∈ Z(m, r)(F)

be a geometric point, and lety = (A0, A) ∈ M(F)

be the point below z. Denote by Rz the completed etale local ring of Z(m, r) at z, and byRy the completed etale local ring of M at y. By the regularity of M, it suffices to show thatthe kernel I of the natural surjection Ry → Rz is generated by a single nonzero element.

Set Rz = Ry/Imy, where my ⊂ Ry is the maximal ideal, so that I = ker(Rz → Rz)satisfies I2 = 0. By Nakayama’s lemma, to show that I is principal it suffices to prove thatI = I/myI is principal.

If we let W denote the completed etale local ring of Spec(Ok) at the point below z, thenRy and Rz pro-represent the deformation functors of (A0, A) and (A0, A, λ) to Artinian localW -algebras with residue field F. Let (A0,A,λ) be the universal deformation of (A0, A, λ)

to Rz, and let (A0, A) be the reduction to Rz of the universal object over Ry. Fix a Π ∈ Oksatisfying Ok = Z[Π], and set

J = Π⊗ 1− 1⊗Π ∈ Ok ⊗Z Rz.


Note that J generates the kernel of the natural map Ok ⊗Z Rz → Rz. The algebraic deRham homology (see Section 7.2) of A sits in a short exact sequence

0→ Fil1(A)→ HdR1 (A)→ Lie(A)→ 0,

of free Rz-modules, and similarly with A replaced by A0. Moreover, HdR1 (A0) is free of

rank one over Ok ⊗Z Rz, and

Fil1(A0) = JHdR1 (A0).

For the proof of this last equality, note that JLie(A0) = 0, which proves one inclusion;

equality then follows as both sides are rank one Rz-module direct summands of HdR1 (A0).

As the subsheaf F ⊂ Lie(A) is annihilated by J , the Lie algebra Lie(A) admits an

Rz-module basis ε1, . . . , εn such that F is generated by ε1, . . . , εn−1, and J acts on Lie(A)as

J =

0 · · · 0 j1...

. . ....

...0 · · · 0 jn

for some j1, . . . , jn ∈ Rz. The essential fact, proved in the course of proving [Ho3, Proposi-

tion 3.2.3], is that for any such basis the ideal (j1, . . . , jn) ⊂ Rz is principal.First suppose that N(r) ∈ F×, so that λ ∈ r−1L(A0,A) induces a map

λ : HdR1 (A0)→ HdR

1 (A).

As the kernel of Rz → Rz has square zero, the deformation theory of [Lan, Chapter 2] tellsus that this map has a canonical lift to a map


1 (A),

and the maximal quotient of Rz over which λ deforms is defined by the vanishing of thecomposition

Fil1(A)→ HdR1 (A)

λ−→ HdR1 (A)→ Lie(A).

Note that the deformation of λ to this maximal quotient automatically satisfies the extravanishing condition of (5.1.2), by Remark 5.1.7. On the other hand, we know by the uni-

versality of (A0,A,λ) that this maximal quotient is Rz/I = Rz. Thus Rz is the maximal

quotient of Rz in which

λ(JHdR1 (A0)) ∈ Lie(A)

vanishes. If we fix an Ok ⊗Z S-module generator σ ∈ HdR1 (A0) and write λ(σ) =

∑aiεi in

Lie(A) in terms of our basis εi, then

λ(Jσ) = Jλ(σ) = an∑

jiεi

vanishes if and only if anji = 0 for every i. Using the principality of the ideal (j1, . . . , jn),

this vanishing locus is defined by a single equation, and so the kernel of Rz → Rz is principal.Now suppose N(r) = 0 in F, so that δkRz = rRz. In this case δkλ determines a map

δkλ : HdR1 (A0)→ HdR

1 (A)

which admits a canonical deformation to


1 (A).

The maximal quotient of Rz to which λ deforms is defined by the relations

(1) δkλ(JHdR1 (A0)) = 0 in Lie(A),


(2) δkλ(HdR1 (A0)) = 0 in Lie(A)/F .

It is the second relation that guarantees that the deformation of λ satisfies the vanishingcondition of (5.1.2). But JF = 0, and so the second relation implies the first. Keeping σ as

above, we have now shown that Rz is the maximal quotient of Rz in which δkλ(σ) = 0 in

Lie(A)/F . As Lie(A)/F is free of rank one over Rz, the kernel of Rz → Rz is principal.It only remains to show that I 6= 0. If I = 0 then, as M is n dimensional and flat over

Ok, the same would be true of the local ring Rz. This would imply that Z(m, r)(C) containsan irreducible component of dimension n − 1. But the complex uniformization of Z(m, r)described in Section 6.1 shows that Z(m, r)(C) has dimension n− 2.

If S = Spec(F) for an algebraically closed field F, and ` 6= char(F) is a prime, then

(5.1.3) HomOk,`(T`(A0), T`(A))

carries a hermitian form defined in the same way as the form on (5.1.1). Here T` denotes`-adic Tate module. Although L(A0, A) need not be self-dual, an easy linear algebra exerciseshows that (5.1.3) always is; this is a consequence of the hypothesis that the polarizationsψ0 and ψ are principal.

If S = Spec(C) then we may form the Betti homology groups

(5.1.4) A0 = H1(A0(C),Z), A = H1(A(C),Z).

The Ok-module HomOk(A0,A) is equipped with a self-dual hermitian form defined exactly

as for (5.1.1), now of signature (n−1, 1). There is an alternate description of this hermitianform that will be useful later. Each of the homology groups in (5.1.4) is a self-dual hermitianOk-module. Indeed, the polarization on A0 induces a perfect Z-valued symplectic form ψ0

on A0, and there is a unique hermitian form hA0on A0 satisfying

ψ0(x, y) = Trk/QhA0(δ−1k x, y),

where δk =√dk is the square root lying in the upper half complex plane3. Similarly A is

equipped with a perfect symplectic form ψ, and a hermitian form hA satisfying

(5.1.5) ψ(x, y) = Trk/QhA(δ−1k x, y).

The hermitian Ok-modules A0 and A have signatures (1, 0) and (n − 1, 1), and, in thenotation of (2.1.2), there is an isomorphism of hermitian Ok-modules

L(A0,A) ∼= HomOk(A0,A).

The following is a slight refinement of Proposition 2.13(ii) of [KR2].

Proposition 5.1.10. For every algebraically closed field F and every (A0, A) ∈ M(F), there

is a unique incoherent self-dual hermitian (kR, Ok)-module L (A0, A) of signature (n, 0)satisfying

L (A0, A)` ∼= HomOk,`(T`(A0), T`(A))

for every prime ` 6= char(F). Furthermore, L (A0, A) depends only on the connected com-ponent of M containing (A0, A), and not on (A0, A) itself.

3More precisely, there is a choice of i =√−1 such that ψ0(ix, x) and ψ(ix, x) are positive definite, and

we choose δk to lie in the same connected component of C r R as i


Proof. First we prove uniqueness. If F has characteristic 0 then there is nothing to prove, asthe stated conditions determine L (A0, A) locally at every place of Q. Assume instead thatF has characteristic p > 0, and that L (A0, A) and L ′(A0, A) both satisfy the stated prop-erties. In particular L (A0, A)` ∼= L ′(A0, A)` for all places ` 6= p, including the archimedeanplace. The incoherence condition then forces

L (A0, A)p ⊗Zp Qp ∼= L ′(A0, A)p ⊗Zp Qp.

By a result of Jacobowitz [Jac] this kp-hermitian space contains at most one isomorphismclass of self-dual hermitian Ok-modules4, and so L (A0, A)p ∼= L ′(A0, A)p.

For existence, suppose first that F has characteristic 0. It suffices to treat the case F = C.We have no choice but to take L (A0, A)` ∼= HomOk,`

(T`(A0), T`(A)) at all finite primes,and must take L (A0, A)∞ to have signature (n, 0). The only thing to check is that theresulting L (A0, A) is incoherent. On the one hand, in the notation of (5.1.4), there is anisomorphism

(5.1.6) L(A0,A)⊗Z Z ∼= L (A0, A)f .

On the other hand, L(A0,A) has signature (n− 1, 1), while L (A0, A) has signature (n, 0).Thus the local invariants of L(A0,A)⊗Z A and L (A0, A) differ at exactly the archimedeanplace, and the coherence of the former implies the incoherence of the latter.

Now take F to be arbitrary and let R be the completed etale local ring of M at (A0, A),so that R is a strictly Henselian local domain with residue field F, and comes with a lift(A0, A) ∈ M(R) of (A0, A) to R. The flatness of M implies that the fraction field F = Frac(R)has characteristic zero, and so from the paragraph above we know the existence of the

hermitian (k∞, Ok)-module L (A0/F alg , A/F alg). One easily checks that

L (A0, A) = L (A0/F alg , A/F alg)

satisfies all the desired properties. Moreover, the pair (A0/F alg , A/F alg) ∈ M(F alg) is the

pullback of the universal object via the generic geometric point Spec(F alg) → M, and sodepends only on the connected component containing (A0, A). Hence the same is true ofL (A0, A).

From Proposition 5.1.10 we obtain a decomposition

(5.1.7) M =⊔L

ML

where L runs over all incoherent self-dual hermitian (kR, Ok)-modules of signature (n, 0),and ML is the union of those connected components of M for which L (A0, A) ∼= L atevery geometric point (A0, A). By setting ZL (m, r) = Z(m, r) ×M ML we obtain a similardecomposition

Z(m, r) =⊔L

ZL (m, r).

5.2. CM cycles. For an Ok-scheme S, an S-valued point

(A1, B) ∈ (M(0,1) ×OkM(n−1,0))(S)

determines an S-valued point A1 × B ∈ M(n−1,1)(S), where A1 × B is implicitly endowedwith the product polarization, the product action of Ok, and the Ok-stable OS-submodule

4See also Proposition 2.15 of [KR2]. We are using here the hypothesis that dk is odd.


Lie(B) ⊂ Lie(A1 × B) satisfying Kramer’s signature (n − 1, 1) condition. In other words,the construction (A1, B) 7→ A1 ×B defines a morphism

M(0,1) ×OkM(n−1,0) → M(n−1,1).

The algebraic stack

Y = M(1,0) ×OkM(0,1) ×Ok

M(n−1,0)

is smooth and proper of relative dimension 0 over Ok, and admits a finite and unramifiedmorphism Y → M defined by (A0, A1, B) 7→ (A0, A1 × B). The algebraic stack Y is a CMcycle, in the sense that for any triple (A0, A1, B) ∈ Y(S) the entries A0 and A1 are ellipticcurves with complex multiplication, while B is isogenous to a product of elliptic curves withcomplex multiplication.

For any S-valued point (A0, A1, B) ∈ Y(S) there is an orthogonal decomposition

(5.2.1) L(A0, A1 ×B) ∼= L(A0, A1)⊕ L(A0, B),

where L(A0, A1) = HomOk(A0, A1) and L(A0, B) = HomOk

(A0, B).

Lemma 5.2.1. Let S be an Ok-scheme, and let S → S a closed subscheme defined by anilpotent ideal sheaf. Suppose k and ` are positive integers. Every pair

(B1, B2) ∈(M(k,0) ×Ok

M(`,0)

)(S)

admits a unique deformation to an S-valued point

(B1, B2) ∈(M(k,0) ×Ok

M(`,0)

)(S),

and the restriction map

HomOk(B1, B2)→ HomOk

(B1, B2)

is an isomorphism.

Proof. The analogous statement for p-divisible groups, proved using Grothendieck-Messingtheory and assuming that p is locally nilpotent on S, is [Ho2, Proposition 2.4.1]. To provethe lemma, combine the argument of [loc. cit.] with the proof of [Ho3, Proposition 2.1.2],which is based instead on algebraic de Rham cohomology, and so is valid for abelian schemesover an arbitrary base.

Proposition 5.2.2. Let S = Spec(F) be the spectrum of an algebraically closed field, andsuppose (A0, A1, B) ∈ Y(F).

(1) There is a unique incoherent self-dual hermitian (kR, Ok)-module L0(A0, A1) ofsignature (1, 0) satisfying

L0(A0, A1)` ∼= HomOk,`(T`(A0), T`(A1))

for every prime ` 6= char(F).(2) The hermitian Ok-module L(A0, B) is self-dual of signature (n− 1, 0).

Moreover, the modules L0(A0, A1) and L(A0, B) depend only the connected component of Ycontaining (A0, A1, B), and not on the point (A0, A1, B) itself.

Proof. The claims concerning (A0, A1) are exactly as in Proposition 5.1.10, and we omittheir identical proofs. For the pair (A0, B), first suppose that F has characteristic 0, inwhich case we may assume that F = C. As in the discussion following (5.1.4), the Ok-modules A0 = H1(A0(C),Z) and B = H1(B(C),Z) come equipped with hermitian forms of


signatures (1, 0) and (n− 1, 0), respectively. It follows from the signature conditions on A0

and B that

A0(C) ∼= A0R/A0

B(C) ∼= BR/B,

where the complex structure on the right hand side of each isomorphism is determined bythe natural action of Ok ⊗Z R ∼= C. It follows that L(A0, B) ∼= L(A0,B), and elementarylinear algebra shows that the right hand side is self-dual of signature (n− 1, 0).

For the case of arbitrary F, let R be the completed etale local ring of Y at the geometricpoint (A0, A1, B). As Y is smooth of relative dimension 0 over Ok, the fraction field F =

Frac(R) has characteristic 0, and there is a canonical point (A0, B) ∈ Y(R) lifting (A0, B).

From the characteristic 0 case just treated, we know that L(A0/F alg , B/F alg) is self-dual ofsignature (n− 1, 0). It is easy to see from Lemma 5.2.1 and, for example, [FGA, Corollary

8.4.7] that the reduction map L(A0, B) → L(A0, B) is an isomorphism. Composing its

inverse with base change to F alg yields an isomorphism L(A0, B) ∼= L(A0/F alg , B/F alg),which shows that L(A0, B) is self-dual of signature (n− 1, 0).

The proof that L(A0, B) is constant on connected components is exactly as in the proofof Proposition 5.1.10.

From Proposition 5.2.2 we have a decomposition

(5.2.2) Y =⊔

(L0,Λ)

Y(L0,Λ),

where the indices run over all isomorphism classes of pairs (L0,Λ) consisting of

• an incoherent self-dual hermitian (kR, Ok)-module L0 of signature (1, 0),• a self-dual hermitian Ok-module Λ of signature (n− 1, 0).

The stack Y(L0,Λ) is the union of those connected components of Y along which L0(A0, A1) ∼=L0 and L(A0, B) ∼= Λ.

Remark 5.2.3. From (5.1.7) we obtain a decomposition

Y =⊔

(L0,Λ)

Y(L0,Λ).

Moreover, each pair (L0,Λ) as above determines an incoherent self-dual (kR, Ok)-moduleL0 ⊕ Λ of signature (n, 0), whose infinite and finite parts are, by definition,

(L0 ⊕ Λ)∞ = L0,∞ ⊕ (Λ⊗Z R)

(L0 ⊕ Λ)f = L0,f ⊕ (Λ⊗Z Z).

If we fix one pair (L0,Λ) and set L = L0⊕Λ, the morphism Y→ M restricts to a morphismY(L0,Λ) → ML .

5.3. Decomposition of the intersection. In this subsection and the next, fix one Y(L0,Λ)

as in (5.2.2), and set L = L0 ⊕Λ as in Remark 5.2.3. From the preceeding subsections wehave a cartesian diagram (this is the definition of the upper left corner)

ZL (m, r) ∩ Y(L0,Λ)//

Y(L0,Λ)

ZL (m, r) // ML .


Our goal is to decompose the intersection ZL (m, r)∩Y(L0,Λ) into smaller, more manageablesubstacks.

Given m1,m2 ∈ Q≥0 and r | dk, denote by X(L0,Λ)(m1,m2, r) the algebraic stack overOk whose functor of points assigns to a connected Ok-scheme S the groupoid of tuples(A0, A1, B, λ1, λ2) in which

• (A0, A1, B) ∈ Y(L0,Λ)(S),

• λ1 ∈ r−1L(A0, A1) satisfies 〈λ1, λ1〉 = m1,• λ2 ∈ r−1L(A0, B) satisfies 〈λ2, λ2〉 = m2,

and the map δkλ1 : A0 → A1 induces the trivial map

(5.3.1) δkλ1 : Lie(A0)→ Lie(A1)

for any generator δk ∈ dk. As in Remark 5.1.7, this last condition is automatically satisfiedif N(r) ∈ O×S .

Proposition 5.3.1. For every m ∈ Q>0 and every r | dk, there is an isomorphism ofOk-stacks

(5.3.2) ZL (m, r) ∩ Y(L0,Λ)∼=

⊔m1,m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r).

Proof. Suppose S is a connected Ok-scheme. An S-valued point on the right hand side of(5.3.2) consists of a triple

(A0, A, λ) ∈ ZL (m, r)(S)

and a triple

(A0, A1, B) ∈ Y(L0,Λ)(S)

together with an isomorphism A ∼= A1×B identifying Lie(B) with the subsheaf F ⊂ Lie(A).Under the orthogonal decomposition (5.2.1), λ ∈ r−1L(A0, A) decomposes as

λ = λ1 + λ2 ∈ r−1L(A0, A1)⊕ r−1L(A0, B)

in such a way that 〈λ, λ〉 = 〈λ1, λ1〉 + 〈λ2, λ2〉. If we set m1 = 〈λ1, λ1〉 and m2 = 〈λ2, λ2〉then the quintuple (A0, A1, B, λ1, λ2) defines an S-valued point of X(L0,Λ)(m1,m2, r). Thisdefines the desired isomorphism.

Now we completely determine the structure of the stacks appearing in the right hand sideof (5.3.2). We will see momentarily that each has dimension 0 or 1, depending on whetherm1 > 0 or m1 = 0. For any m ∈ Q≥0 and any r | dk, define the representation number

(5.3.3) RΛ(m, r) =∣∣λ ∈ r−1Λ : 〈λ, λ〉 = m

∣∣.For m ∈ Q>0 define a finite set of odd cardinality

(5.3.4) DiffL0(m) = primes p of Q : m is not represented by V0,p,

and note that every p ∈ DiffL0(m) is nonsplit in k.

Theorem 5.3.2. Fix m1,m2 ∈ Q≥0 with m1 > 0, and r | dk. Abbreviate

X = X(L0,Λ)(m1,m2, r).

(1) If |DiffL0(m1)| > 1, then X = ∅.


(2) If DiffL0(m1) = p, then X has dimension 0 and is supported in characteristic p.Furthermore, the etale local ring of every geometric point of X has length

νp(m1) = ordp(pm1) ·

1/2 if p is inert in k,

1 if p is ramified in k,

and the number of geometric points of X (counted with multiplicities) is

(5.3.5)∑

z∈X(Falgp )

1

|Aut(z)|=hkwk· RΛ(m2, r)

|Aut(Λ)|· ρ(m1N(s)

pε

)

where p is the unique prime of k above p, Falgp is an algebraic closure of its residue

field, ρ is defined by (1.5.1), s = r/(r + p) is the prime-to-p part of r, and

(5.3.6) ε =

1 if p is inert in k,

0 if p is ramified in k.

Proof. If X 6= ∅ then there is some point (A0, A1, B, λ1, λ2) ∈ X(F), where F is either C orFalgp for some prime p. Let A1 be the elliptic curve A1, but with the action of Ok replaced

by its complex conjugate. Thus λ1 : A0 → A1 is an Ok-conjugate-linear degree m1 quasi-isogeny between elliptic curves with complex multiplication, and Ok acts on the Lie algebrasof A0 and A1 through the same homomorphism Ok → F. The only way such a conjugatelinear quasi-isogeny can exist is if F has nonzero characteristic, p is nonsplit in k, and A0

and A1 are supersingular elliptic curves. In particular

HomZ`(T`(A0), T`(A1)) ∼= Hom(A0, A1)⊗Z Z`for every prime ` 6= p, and hence also

L0,`∼= L0(A0, A1)` ∼= HomOk,`

(T`(A0), T`(A1)) ∼= L(A0, A1)⊗Z Z`as hermitian Ok,`-modules. As 〈λ1, λ1〉 = m1 by definition of the moduli space X, wehave now shown that L0,` represents m1 for all finite primes ` 6= p. Therefore DiffL0

(m1)contains at most one prime, p. We have already remarked that this set has odd cardinality,and therefore DiffL0

(m1) = p.We interrupt the proof for a lemma.

Lemma 5.3.3. The etale local ring of X at every point

(A0, A1, B, x1, x2) ∈ X(Falgp )

is Artinian of length νp(m1).

Proof. We reduce the proof to calculations of Gross [Gr]. The tuple (A0, A1, B, λ1, λ2)

corresponds to a morphism z : Spec(Falgp ) → X, and by composing with the structure

morphism we obtain a geometric point Spec(Falgp ) → Spec(Ok). Let W be the completion

of the etale local ring of Spec(Ok) at this point. In less fancy language, W is the completionof the integer ring of the maximal unramified extension of kp. Let R be the completed etalelocal ring of X at (A0, A1, B, λ1, λ2). This ring pro-represents the deformation functor ofthe tuple (A0, A1, B, λ1, λ2) to Artinian local W -algebras with residue field F. Lemma 5.2.1implies that (A0, B, λ2) admits a unique lift to any such W -algebra. Thus the data of B andλ2 can be ignored in the deformation problem, and R pro-represents the deformation functorof (A0, A1, λ1). Equivalently, R pro-represents the deformation functor of (A0, A1, λ1).


By the Serre-Tate theorem we may replace A0 and A1 in the above deformation problemby their p-divisible groups, which are Ok,p-linearly isomorphic. Let G be the connected

p-divisible group of height 2 and dimension 1 over Falgp , and fix an action of Ok,p on G such

that the induced action on Lie(G) is through the structure morphism Ok,p → Falgp . Let

λ1 ∈ r−1 End(G)

be an Ok,p-conjugate-linear map such that δkλ1 : G → G induces the trivial map on Liealgebras. The quasi-endomorphism λ satisfies

ordp(Nrd(λ)) = ordp(m1)

where Nrd is the reduced norm on the quaternion order End(G), and R pro-represents the

functor that assigns to every Artinian local W -algebra with residue field Falgp the set of

deformations (G, λ) of (G,λ) with λ ∈ r−1 End(G) and

(5.3.7) δkλ : Lie(G)→ Lie(G)

equal to zero.Suppose first that N(r) ∈ Z×p . Then λ1 ∈ End(G), and Remark 5.1.7 implies that the

vanishing of (5.3.7) is automatically satisfied for any deformation. In this case, Gross’sresults immediately imply that R is Artinian of length νp(m1).

Now suppose N(r) /∈ Z×p , so that rOk,p = dkOk,p. If we set y = δkλ1 ∈ End(G), then thering R pro-represents the functor that assigns to every Artinian local W -algebra with residue

field Falgp the set of deformations (G, y) with y ∈ End(G) and y : Lie(G)→ Lie(G) equal to

zero. Let R′ be the ring pro-representing the same deformation problem, but without thecondition that y : Lie(G)→ Lie(G) vanish. By Gross’s results R′ ∼= W/pk+1, where

k = ordp(Nrd(y)) = νp(m1).

Let (Gk+1, yk+1) be the universal deformation of (G, y) to W/pk+1, and let (Gk, yk) be itsreduction to W/pk. To show that R ∼= W/pk it suffices to prove that

(5.3.8) yk+1 : Lie(Gk+1)→ Lie(Gk+1)

is nonzero, but that

(5.3.9) yk : Lie(Gk)→ Lie(Gk)

vanishes.For any `, let G` denote the canonical lift5 of G to R` = W/p`, and let D(G`) be the

Grothendieck-Messing crystal of G` evaluated at R`. Thus D(G`) is a free Ok⊗ZR`-moduleof rank one, and sits in an exact sequence free R`-modules

0→ Fil1D(G`)→ D(G`)→ Lie(G`)→ 0.

As in the proof of Proposition 5.1.9, Fil1D(G`) = JD(G`), where Ok = Z[Π] and

J = Π⊗ 1− 1⊗Π ∈ Ok ⊗Z R`

generates (as an R`-module), the kernel of the natural map Ok ⊗Z R` → R`. Note that theimage of

J = Π⊗ 1− 1⊗Π ∈ Ok ⊗Z R`

in R` is Π−Π, which generates the maximal ideal pR`.

5in the sense of [Gr], so G` is the unique defomation of G, with its action of Ok,p, to R`


Suppose we are given an Ok-conjugate-linear endomorphism y`−1 ∈ End(G`−1). ByGrothendieck-Messing theory, such an endomorphism induces an endomorphism y`−1 ofD(G`), and y`−1 lifts to End(G`) if and only if the composition

Fil1D(G`)→ D(G`)y`−1−−−→ D(G`)→ Lie(G`)

is trivial. It is now easy to see that each of the following statements is equivalent to thenext one:

(1) y`−1 lifts to End(G`),(2) the image of y`−1(JD(G`)) = Jy`(D(G`)) in Lie(G`) is trivial,(3) the image of y`−1(D(G`)) in Lie(G`) lies in p`−1Lie(G`),(4) the composition

D(G`)y`−1−−−→ D(G`)→ Lie(G`)→ Lie(G`−1)

vanishes,(5) the composition

D(G`−1)y`−1−−−→ D(G`−1)→ Lie(G`−1)

vanishes,(6) y`−1 : Lie(G`−1)→ Lie(G`−1) is trivial.

Thus y`−1 lifts to End(G`) if and only if it induces the zero endomorphism of Lie(G`−1),and the nonvanishing of (5.3.8) and vanishing of (5.3.9) follow immediately.

To complete the proof of Theorem 5.3.2, it only remains to prove (5.3.5). We do thisthrough a sequence of lemmas.

Lemma 5.3.4. Abbreviating Y = Y(L0,Λ), we have

(5.3.10)∑

z∈X(Falgp )

1

|Aut(z)|=∑L0

∑(A0,A1,B)∈Y(Falg

p )

L(A0,A1)∼=L0

RL0(m1, s)RΛ(m2, r)

|Aut(A0, A1, B)|,

where the outer sum on the right is over all hermitian Ok-modules L0 of rank one, and therepresentation number RL0

(m1, s) is defined in the same way as (5.3.3).

Proof. Directly from the definitions, we have∑z∈X(Falg

p )

1

|Aut(z)|=


p )

∑λ1∈r−1L(A0,A1)〈λ1,λ1〉=m1

Lie(δkλ1)=0

∑λ2∈r−1L(A0,B)〈λ2,λ2〉=m2

1

|Aut(A0, A1, B)|,

where the condition Lie(δkλ1) = 0 refers to the vanishing of (5.3.1).We claim that

λ ∈ r−1L(A0, A1) : Lie(δkλ) = 0 = s−1L(A0, A1)

for all A0 ∈ M(1,0)(Falgp ) and A1 ∈ M(0,1)(Falg

p ). If λ ∈ s−1L(A0, A1) then, as s is prime to p,λ induces a morphism of Lie algebras λ : Lie(A0) → Lie(A1). By the argument of Remark5.1.7, the image of this map is annihilated by δk, and so Lie(δkλ) = 0. Conversely, supposewe start with λ ∈ r−1L(A0, A1) satisfying Lie(δkλ) = 0. Let G be the connected p-divisiblegroup of height 2 and dimension 1, and set OB = End(G). Thus OB is the maximal orderin a quaternion division algebra over Qp. We may fix an embedding Ok,p → End(G) andisomorphisms A0[p∞] ∼= G ∼= A1[p∞] in such a way that the first is Ok,p-linear, and the


second is Ok,p-conjugate-linear. The hypothesis λ ∈ r−1L(A0, A1) implies that δkλ ∈ OB ,but we cannot have δkλ ∈ O×B (for then δkλ, and also Lie(δkλ), would be an isomorphism).Therefore δkλ lies in the unique maximal ideal of OB , and hence λ ∈ OB . This implies that

λ ∈ r−1L(A0, A1) ∩Hom(A0[p∞], A1[p∞]) = s−1L(A0, A1)

as desired.We have now shown that∑z∈X(Falg

p )

1

|Aut(z)|=


p )

∑λ1∈s−1L(A0,A1)〈λ1,λ1〉=m1

∑λ2∈r−1L(A0,B)〈λ2,λ2〉=m2

1

|Aut(A0, A1, B)|.

On the right hand side, each L(A0, A1) is a hermitian Ok-module of rank one, whileL(A0, B) ∼= Λ. The lemma follows immediately.

Let V0 be the incoherent hermitian space over Ak determined by L0, and recall fromRemark 2.1.2 that for every prime p nonsplit in k there is a coherent hermitian space V0(p),which is isomorphic to V0 everywhere locally away from p. We now repeat this construction

on the level of (kR, Ok)-modules. Define a new hermitian (kR, Ok)-module L0(p) by settingL0(p)` = L0,` for every place ` 6= p. For the p-component L0(p)p, take the same underlyingOk,p-module as L0,p, but replace the hermitian form 〈·, ·〉L0,p

on L0,p with the hermitianform

〈·, ·〉L0(p)p = cp〈·, ·〉L0,p,

where

cp =

any uniformizing parameter of Zp, if p is inert in k,

any element of Z×p that is not a norm from O×k,p, if p is ramified in k.

The resulting coherent (kR, Ok)-module L0(p) has V0(p) as its associated hermitian Ak-module. Note that if p is inert in k then L0(p) is not self-dual.

Lemma 5.3.5. Any triple (A0, A1, B) appearing in the final sum of (5.3.10) satisfies

L(A0, A1) ∈ gen(L0(p)).

Proof. It is easy to see that

L0(p)` ∼= L0,`∼= L0(A0, A1)` ∼= HomOk,`

(T`(A0), T`(A1)) ∼= L(A0, A1)⊗Z Z`for all primes ` 6= p, and that

L0(p)∞ ∼= L(A0, A1)⊗Z R,as both sides are positive definite. In particular the coherent Ak-hermitian spaces V0(p) andL(A0, A1) ⊗Z A are isomorphic at all places away from p, and by comparing invariants wesee that

(5.3.11) V0(p)p ∼= L(A0, A1)⊗Z Qpas kp-hermitian spaces. In order to strengthen (5.3.11) to an isomorphism

(5.3.12) L0(p)p ∼= L(A0, A1)⊗Z Zp,fix Ok,p-module generators x and y of the left hand side and right hand side, respectively,of (5.3.12), and define p-adic integers α = 〈x, x〉 and β = 〈y, y〉.

Let G be the unique connected p-divisible group of height 2 over Falgp , and fix an action

of Ok,p on G in such a way that the induced action on Lie(G) is through the structure map


Ok,p → Falgp . There is anOk,p-linear isomorphismG ∼= A0[p∞], and anOk,p-conjugate-linear

isomorphism G ∼= A1[p∞]. These choices identify L(A0, A1) ⊗Z Zp with the submodule ofOk,p-conjugate-linear endomorphisms

EndOk,p(G) ⊂ End(G),

and identify the quadratic form 〈·, ·〉 on L(A0, A1)⊗ZZp with a Z×p -multiple of the restrictionto EndOk,p

(G) of the reduced norm on the quaternionic order End(G). A routine calculation

with quaternion algebras, as in [KRY1, pp. 376–378], now implies that

ordp(β) =

1 if p is inert in k,

0 if p is ramified in k.

Comparing with the definition of L0(p) then shows that ordp(α) = ordp(β). The isomor-phism (5.3.11) implies that χk,p(α) = χk,p(β), and this information is enough to guaran-tee that α/β is a norm from O×k,p. This proves (5.3.12), and completes the proof of thelemma.

Lemma 5.3.6. For each L0 ∈ gen(L0(p)) there are hk isomorphism classes of triples

(A0, A1, B) ∈ Y(Falgp ) such that L(A0, A1) ∼= L0. Any such triple satisfies

|Aut(A0, A1, B)| = w2k · |Aut(Λ)|.

Proof. Let R be any complete local Noetherian ring with residue Falgp . Using Lemma 5.2.1

and Grothendieck’s formal existence theorem [FGA, Section 8.4.4], the triple (A0, A1, B)lifts uniquely to R, as do all of its automorphisms. Using this, we are easily reduced to thecorresponding counting problem in characteristic 0. The category Y(C) is easily describedin terms of linear algebra (see Section 6.1 below). Briefly, there are hk CM elliptic curvesA0 ∈ M(1,0)(C), and for each such A0 there is a unique CM elliptic curve A1 ∈ M(0,1)(C)satisfying L(A0, A) ∼= L0, and a unique B ∈ M(n−1,0)(C) satisfying L(A0, B) ∼= Λ. Each of

A0 and A1 has automorphism group O×k , and the automorphism group of B is isomorphicto Aut(Λ).

Lemma 5.3.7. Still assuming that DiffL0(m1) = p, we have

1

wk

∑L0∈gen(L0(p))

RL0(m1, s) = ρ

(m1N(s)

pε

).

Proof. As V0(p) is coherent, we may fix a hermitian space V0 over k such that

V0 ⊗Q A ∼= V0(p).

Note that DiffL0(m1) = p implies that V0 represents m1. Pick one vector λ0 ∈ V0 suchthat 〈λ0, λ0〉 = m1, and an Ok-lattice L0 ⊂ V0 such that L0 ∈ gen(L0(p)).

Let k1 denote the group of norm one elements in k×, and define O1k in the same way. As

h varies over k1/O1k, the lattices h · L0 ⊂ V0, with the hermitian forms restricted from V0,

vary over gen(L0(p)). Thus

1

wk

∑L0∈gen(L0(p))

RL0(m1, s) =

∑h∈k1/O1

k

1hs−1L0(λ0),


where 1 denotes characteristic function. If we fix any Ok-linear isomorphism Ok ∼= L0, the

hermitian form on L0 is identified with 〈x, y〉 = xypεu for some u ∈ O×k , and now

1

wk

∑L0∈gen(L0(p))

RL0(m1, s) =

∑h∈k1/O1

k

1Ok(h−1sλ0)

where λ0 ∈ k× satisfies uN(λ0) = m1/pε, and s ∈ k× satisfies sOk = s. The equality∑

h∈k1/O1k

1Ok(h−1sλ0) = ρ

(m1N(s)

pε

)is easily checked, as both sides admit a factorization over the prime numbers, and theprime-by-prime comparison is elementary.

Combining (5.3.10) and the four lemmas shows that∑z∈X(Falg

p )

1

|Aut(z)|=

∑L0∈gen(L0(p))


p )

L(A0,A1)∼=L0


|Aut(A0, A1, B)|

=hkw2k

∑L0∈gen(L0(p))


|Aut(Λ)|

=hkwk· RΛ(m2, r)

|Aut(Λ)|· ρ(m1N(s)

pε

),

and completes the proof of Theorem 5.3.2.

Theorem 5.3.2 implies that X(L0,Λ)(m1,m2, r) has dimension 0 whenever m1 > 0. Nowwe turn to the case of m1 = 0.

Proposition 5.3.8. Fix a positive m ∈ Q and r | dk.

(1) If RΛ(m, r) = 0 then X(L0,Λ)(0,m, r) = ∅.(2) If RΛ(m, r) 6= 0 then X(L0,Λ)(0,m, r) is nonempty, and is smooth of relative dimen-

sion 0 over Ok. In particular, it is a regular stack of dimension 1.

Proof. We know from Proposition 5.1.3 that Y(L0,Λ) → Spec(Ok) is smooth of relativedimension 0, and Lemma 5.2.1 implies that the map X(L0,Λ)(0,m, r)→ Y(L0,Λ) defined by

(A0, A1, B, 0, λ2) 7→ (A0, A1, B)

is formally etale. Hence the composition X(L0,Λ)(0,m, r) → Spec(Ok) is smooth of relativedimension 0.

It only remains to show that X(L0,Λ)(0,m, r) is nonempty if and only if RΛ(m, r) 6= 0. IfX(L0,Λ)(0,m, r) is nonempty then we may pick any geometric point

(A0, A1, B, λ1, λ2) ∈ X(L0,Λ)(0,m, r)(F).

Using L(A0, B) ∼= Λ, the homomorphism λ2 defines an element of Λ satisfying 〈λ2, λ2〉 = m,and in particular RΛ(m, r) 6= 0. Conversely, if RΛ(m, r) 6= 0 then pick some λ2 ∈ Λ satisfying〈λ2, λ2〉 = m. It follows from the uniformization (6.1.5) below that Y(L0,Λ)(C) 6= ∅, and forany choice of (A0, A1, B) ∈ Y(L0,Λ)(C) the vector λ2 defines an element of Λ ∼= L(A0, B).Setting λ1 = 0, the tuple (A0, A1, B, λ1, λ2) defines a complex point of X(L0,Λ)(0,m, r).


Remark 5.3.9. It follows from (5.3.2) and Theorem 5.3.2 that if RΛ(m, r) = 0, the intersec-tion Z(m, r) ∩ Y(L0,Λ) is isomorphic to the zero dimensional stack

(5.3.13)⊔

m1∈Q>0

m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r).

On the other hand, if RΛ(m, r) 6= 0 then Z(m, r) ∩ Y(L0,Λ) is the disjoint union of the zerodimensional stack (5.3.13) with the one dimensional stack X(L0,Λ)(0,m, r).

6. CM values of automorphic Green functions

As in Section 5.3, fix an n ≥ 2 and a pair (L0,Λ) consisting of an incoherent self-dual

hermitian (kR, Ok)-module L0 of signature (1, 0), and a self-dual hermitian Ok-module Λ of

signature (n− 1, 0). As in Remark 5.2.3, define an incoherent self-dual hermitian (kR, Ok)-module of signature (n, 0) by

L = L0 ⊕ Λ.

Let L (∞) be the unique coherent self-dual hermitian (kR, Ok)-module of signature (n−1, 1)satisfying L (∞)f ∼= Lf . In other words, L (∞) is obtained from L by changing thesignature at the archimedean place from (n, 0) to (n−1, 1). Similarly, let L0(∞) be obtainedfrom L0 by switching the signature at the archimedean place from (1, 0) to (0, 1).

As in Section 1.5, the finite Z-module d−1k Lf/Lf is equipped with a d−1

k Z/Z-valuedquadratic form Q(x) = 〈x, x〉, and we denote by ∆ its automorphism group as a finitequadratic space. The space SL of complex valued functions on d−1

k Lf/Lf is equipped withan action of ∆ and a commuting action ωL of SL2(Z) defined by the Weil representation.In particular, ∆ acts on the space H2−n(ωL ) of harmonic Maass forms with values in SL .In the exact same way the finite Z-modules d−1

k L0,f/L0,f and d−1k Λ/Λ are equipped with

quadratic forms (still denoted Q), and the spaces SL0and SΛ are equipped with actions

ωL0and ωΛ of SL2(Z). Moreover, the obvious isomorphism

SL∼= SL0

⊗C SΛ

is SL2(Z)-equivariant.The purpose of this section is twofold. First, we describe the complex uniformizations of

the orbifolds

ZL (m, r)(C) // ML (C) Y(L0,Λ)(C)oo

constructed in Section 5.1. Comparing these uniformizations with the complex Shimuravarieties considered earlier will allow us to use the construction (3.3.1) to define Greenfunctions for the divisors ZL (m, r). More generally, we will associate to any ∆-invariantharmonic Maass form f ∈ H2−n(ωL ) a divisor ZL (f) and a Green function ΦL (f). Thesecond purpose of this section is to find an explicit formula for the value of ΦL (f) at thecomplex points of Y(L0,Λ)

6.1. Complex uniformization. Here we recall the uniformization of the smooth complexorbifold ML (C) and its Kudla-Rapoport divisors. The complex uniformization is explainedin [KR2, Section 3] and [Ho2, Section 3.8], and so we only sketch the main ideas.

To each point (A0, A) ∈ ML (C) there is an associated pair (A0,A) of self-dual hermitianOk-modules as in (5.1.4). As in (2.1.2) there is an associated self-dual hermitian Ok-moduleL(A0,A) of signature (n− 1, 1), and the isomorphism (5.1.6) implies that

L(A0,A) ∈ gen(L (∞)).


The pair (A0,A) depends on the connected component of ML (C) containing (A0, A), butnot on (A0, A) itself, and the formation of (A0,A) from (A0, A) establishes a bijection fromthe set of connected components of ML to the set of isomorphism classes of pairs (A0,A) inwhich

• A0 is a self-dual hermitian Ok-module of signature (1, 0),• A is a self-dual hermitian Ok-module of signature (n− 1, 1),• L(A0,A) ∈ gen(L (∞)).

Remark 6.1.1. For each L ∈ gen(L (∞)) there are exactly hk pairs (A0,A) as above satisfy-ing L(A0,A) ∼= L. Indeed, there are hk choices of A0, and for each choice there is a uniqueA for which L(A0,A) ∼= L.

We now give an explicit parametrization of the connected component of ML (C) indexedby one pair (A0,A). Let D(A0,A) be the space of negative kR-lines in L(A0,A)R. The group

Γ(A0,A) = Aut(A0)×Aut(A)

sits in a short exact sequence

(6.1.1) 1→ µ(k)→ Γ(A0,A) → Aut(L(A0,A))→ 1

in which the arrow µ(k)→ Γ(A0,A) is the diagonal inclusion, and Γ(A0,A) → Aut(L(A0,A))

sends (γ0, γ) to the automorphism λ 7→ γ λ γ−10 . In particular, Γ(A0,A) acts on D(A0,A).

There is a morphism of (n− 1)-dimensional orbifolds

Γ(A0,A)\D(A0,A) → ML (C)

defined by sending the negative line z ∈ D(A0,A) to the pair (A0, Az), where A0(C) = A0R/A0

and Az(C) = AR/A as real Lie groups with Ok-actions. The complex structure on A0(C) isdefined by the natural action of kR ∼= C on A0R, but the complex structure on Az(C) dependson z. A choice of nonzero vector a0 ∈ A0 determines a homomorphism L(A0,A)R → ARby λ 7→ λ(a0). The image of z under this homomorphism is a negative line z ⊂ AR, whichdoes not depend on the choice of a0. Of course AR inherits a complex structure from itsOk-action and the isomorphism kR ∼= C, but this does not define the complex structure onAz(C). Instead, define an R-linear endomorphism Iz of AR by

Iz(a) =

i · a if a ∈ z⊥

−i · a if a ∈ z

and use this new complex structure Iz to make Az(C) into a complex Lie group. Thesymplectic form ψ on A defined by (5.1.5) defines a polarization on Az(C), and the subspace

z⊥ ⊂ AR ∼= Lie(Az)

satisfies Kramer’s signature (n − 1, 1) condition. From the discussion above we find thecomplex uniformization

(6.1.2) ML (C) ∼=⊔

(A0,A)

Γ(A0,A)\D(A0,A).

Remark 6.1.2. The uniformization (6.1.2) shows that ML (C) is almost the same as

(6.1.3)⊔

L∈gen(L (∞))

Aut(L)\DL,

where DL is the space of negative kR-lines in LR. The difference is that each connectedcomponent of (6.1.3) appears in ML (C) with multiplicity hk, by Remark 6.1.1, and the


points in ML (C) have wk times as many automorphisms as the corresponding points of(6.1.3), by the exactness of (6.1.1).

Remark 6.1.3. If either n > 2, or n = 2 and L (∞) is isotropic, then the strong approxi-mation theorem implies that gen(L (∞)) consists of 21−o(dk)hk isometry classes of self-dualhermitian Ok-lattices. Consequently, ML (C) has 21−o(dk)h2

k components.

Now we turn to the complex uniformization of the Kudla-Rapoport divisors. For any m ∈Q>0 and any r | dk, the algebraic stack of Definition 5.1.5 admits a complex uniformization

(6.1.4) ZL (m, r)(C) ∼=⊔

(A0,A)

(Γ(A0,A)\

⊔λ∈r−1L(A0,A)〈λ,λ〉=m

D(A0,A)(λ)),

in which D(A0,A)(λ) ⊂ D(A0,A) is the space of negative lines orthogonal to λ. The essentialpoint is that D(A0,A)(λ) is precisely the locus of points z ∈ D(A0,A) for which the R-linearmap λ : A0R → AR is C-linear relative to the complex structure Iz.

Finally, we describe the complex uniformization of the CM cycle Y(L0,Λ) of Section 5.2.Fix a triple (A0,A1,B) in which

• A0 and A1 are self-dual hermitian Ok-modules of signatures (1, 0) and (0, 1), respec-tively, satisfying L(A0,A1) ∈ gen(L0(∞)),

• B is a self-dual hermitian Ok-module of signature (n−1, 0) satisfying L(A0,B) ∼= Λ.

We attach to this triple the point (A0, A1, B) ∈ Y(L0,Λ)(C), where

A0(C) = A0R/A0

A1(C) = A1R/A1

B(C) = BR/B

as real Lie groups with Ok-actions. The complex structure on A0(C) is given by the naturalaction of kR ∼= C on A0R, and similarly for the complex structure on B(C). The complexstructure on A1(C), however, is given by the complex conjugate of the natural action of kR ∼=C on A1R. Of course the elliptic curves A0 and A1 are endowed with their unique principalpolarizations, while B is endowed with the polarization determined by the symplectic formψB on B ∼= H1(B(C),Z) defined, as in (5.1.5), by

ψB(x, y) = trk/Q hB(δ−1k x, y).

The construction (A0,A1,B) 7→ (A0, A1, B) establishes a bijection from the set of isomor-phism classes of all such triples to the set of isomorphism classes of the category Y(L0,Λ)(C),and defines an isomorphism of 0-dimensional complex orbifolds

(6.1.5) Y(L0,Λ)(C) ∼=⊔

(A0,A1,B)

Γ(A0,A1,B)\y(A0,A1,B),

where y(A0,A1,B) is a single point on which

Γ(A0,A1,B) = Aut(A0,A1,B)

acts trivially. The morphism Y(L0,Λ)(C) → ML (C) is easy to describe in terms of (6.1.5)and (6.1.2). For each triple (A0,A1,B) we set A = A1 ⊕B, and send the point y(A0,A1,B)

to the point of D(A0,A) defined by the negative kR-line L(A0,A1)R ⊂ L(A0,A)R.


Remark 6.1.4. The uniformization (6.1.5) implies that Y(L0,Λ)(C) has 21−o(dk)h2k points,

each with w2k · |Aut(Λ)| automorphisms. Thus the rational number

degC Y(L0,Λ) =∑

y∈Y(L0,Λ)(C)

1

|Aut(y)|

is given by the explicit formula

degC Y(L0,Λ) =h2k

w2k

· 21−o(dk)

|Aut(Λ)|.

Moreover, we have the following proposition.

Proposition 6.1.5. Assume that either n > 2, or n = 2 and L (∞) is isotropic.

(1) The rule L0 7→ L0 ⊕ Λ establishes a bijection

gen(L0(∞))→ gen(L (∞)).

(2) The set Y(L0,Λ)(C) has exactly one point in every connected component of ML (C).

Proof. (1) Since both sets have 21−o(dk)hk elements, it suffices to show that the map isinjective. For any hermitian Ok-modules L0, L

′0 ∈ gen(L0(∞)) there are fractional ideals b

and b′ such that b ∼= L0 and b′ ∼= L′0, where the hermitian forms on b and b′ are defined by−xy. If L0 ⊕ Λ ∼= L′0 ⊕ Λ, then taking top exterior powers (in the category of Ok-modules)shows that L0

∼= L′0 as Ok-modules. But this implies that b and b′ lie in the same idealclass, and hence are isomorphic as hermitian Ok-modules. Therefore L0

∼= L′0 as hermitianOk-modules. Claim (2) is an immediate consequence of (1).

6.2. Automorphic Green functions for Kudla-Rapoport divisors. For every m ∈Q/Z and every r | dk, define a ∆-invariant function ϕm,r ∈ SL as the characteristic functionof the subset

λ ∈ r−1Lf/Lf : Q(λ) = m ⊂ d−1k Lf/Lf .

Using [Se73, Chapter IV.1.7], it is easy to check that ϕm,r 6= 0 if and only if m ∈ N(r)−1Z/Z.

By Witt’s theorem the finitely many nonzero ϕm,r’s form a basis of S∆L . We call d−1

k Lf/Lf

isotropic if d−1k Lp/Lp represents 0 non-trivially for every prime p dividing dk. This condi-

tion is equivalent to the existence of an isotropic element of order |dk| in d−1k Lf/Lf .

Lemma 6.2.1. For any m ∈ Q>0 and any r | dk, there is an fm,r ∈ H2−n(ωL )∆ withholomorphic part of the form

f+m,r(τ) = ϕm,r · q−m +

∑k∈Q≥0

c+m,r(k) · qk,

for some c+m,r(k) ∈ SL . Furthermore

(1) if n > 2, then fm,r is unique;

(2) if n = 2 and d−1k Lf/Lf is not isotropic, then fm,r is again unique;

(3) if n = 2 and d−1k Lf/Lf is isotropic, then any two such fm,r differ by a constant,

and fm,r is uniquely determined if we impose the further condition that c+m,r(0) ∈ SL

vanishes at the trivial coset of d−1k Lf/Lf . That is to say, c+m,r(0, 0) = 0.

Remark 6.2.2. When n = 2 and d−1k Lf/Lf is isotropic, we always choose fm,r so that

c+m,r(0, 0) = 0.


Proof. The existence statement follows from [BF, Proposition 3.11]. To prove the uniquenessstatement when n > 2, we note that a harmonic Maass form f ∈ Hk(ωL ) with vanishingprincipal part is automatically holomorphic [BF, Proposition 3.5]. Since the weight is neg-ative, it vanishes identically. Now suppose that n = 2. Using the same argument as forn > 2, we see that any two fm,r differ by an element of M0(ωL )∆, that is, by an element ofSL which is invariant under the action of the group SL2(Z)×∆.

If d−1k Lf/Lf is not isotropic then it is easily seen that M0(ωL )∆ = 0. If d−1

k Lf/Lf

is isotropic then it follows from [Sch2, Theorem 5.4] that the space of invariants M0(ωL )∆

has dimension 1. Moreover, the map from M0(ωL )∆ to C given by evaluation at the trivialcoset of d−1

k Lf/Lf defines an isomorphism.

The proof of the following lemma is similar to the uniqueness part of Lemma 6.2.1, andis left to the reader.

Lemma 6.2.3. Any f ∈ H2−n(ωL )∆ may be decomposed as a C-linear combination

f(τ) = const +∑

m∈Q>0

r|dk

αm,r · fm,r(τ)

where “const” is a constant form in M2−n(ωL )∆. This constant form is necessarily 0,except when n = 2 and d−1

k Lf/Lf is isotropic.

For any f as in Lemma 6.2.3, define a divisor on ML with complex coefficients

(6.2.1) ZL (f) =∑

m∈Q>0

r|dk

αm,r · ZL (m, r).

Remark 6.2.4. Although the decomposition of Lemma 6.2.3 is not unique, the divisor (6.2.1)does not depend on the choice of decomposition. This amounts to verifying that ZL (m, r) =0 whenever fm,r = 0, which is clear: if fm,r = 0 then ϕm,r = 0, which implies thatm 6∈ N(r)−1Z. Thus ZL (m, r) = 0 by Remark 5.1.6.

We now construct a Green function for ZL (f). It suffices to do this on each connectedcomponent Γ(A0,A)\D(A0,A) of (6.1.2). The assumption that

L(A0,A) ∈ gen(L (∞))

implies that L(A0,A) ∼= Lf as hermitian Ok-modules. Exactly as with Lf , the Z-module

d−1k L(A0,A)/L(A0,A) is equipped with a d−1

k Z/Z-valued quadratic form whose automor-phism group we again denote by ∆, and there is an isomorphism of quadratic spaces

(6.2.2) d−1k L(A0,A)/L(A0,A) ∼= d−1

k Lf/Lf ,

which we now fix. This isomorphism identifies SL with the space SL(A0,A) of complexvalued functions on the left hand side of (6.2.2). This identification depends on the choiceof (6.2.2), but the restriction

(6.2.3) S∆L(A0,A)

∼= S∆L

to ∆-invariants is independent of the choice. Using the isomorphism (6.2.3), we view

f ∈ H2−n(ωL )∆

as a ∆-invariant SL(A0,A)-valued harmonic Maass form. The construction (3.3.1) definesa function ΦL(A0,A)(f) on Γ(A0,A)\D(A0,A) with logarithmic singularities along the divisorZL (f)(C). By repeating the above construction on every connected component of (6.1.2),


we obtain a Green function ΦL (f) for the divisor ZL (f) on ML . In particular, if we takef = fm,r as in Lemma 6.2.1, we obtain a Green function ΦL (fm,r) for the divisor ZL (m, r).

Remark 6.2.5. Throughout this section we are assuming that L is presented to us in theform L = L0 ⊕ Λ, but this decomposition is needed only to define the CM cycle Y(L0,Λ).The construction of the divisor ZL (f) on ML and its Green function ΦL (f) are valid for

any incoherent self-dual hermitian (kR, Ok)-module L of signature (n, 0).

In the remaining subsections we compute the value of ΦL (f) at the points of Y(L0,Λ)(C).

6.3. Coherent and incoherent Eisenstein series. Let P ⊂ SL2(Z) be the subgroupof upper triangular matrices. For each ϕ ∈ SL0 , define an incoherent Eisenstein series ofweight 1

(6.3.1) EL0(τ, s, ϕ) =∑

γ∈P\ SL2(Z)

ωL0(γ)ϕ(0) · (cτ + d)−1 · Im(γτ)s2 .

Here γ =(a bc d

), τ = u+ iv ∈ H, and s is a complex variable with Re(s) 0. The Eisenstein

series has meromorphic continuation to all s, and is holomorphic at s = 0. As (6.3.1) islinear in ϕ, it may be viewed as a function EL0

(τ, s) taking values in the dual space S∨L0.

This particular Eisenstein series was studied in [Scho] and [BY, Section 2]. Indeed, if wepick any L0 ∈ gen(L0(∞)) then (6.3.1) is precisely the Eisenstein series denoted EL0

(τ, s, 1)in [BY], and depends only on the genus of L0, not on L0 itself.

By [BY, Proposition 2.5], the completed Eisenstein series

E∗L0(τ, s, ϕ) = Λ(χk, s+ 1) · EL0

(τ, s, ϕ)

satisfies the functional equation E∗L0(τ,−s, ϕ) = −E∗L0

(τ, s, ϕ), where

Λ(χk, s) = |dk|s2π−

s+12 Γ(s+ 1

2

)L(χk, s).

In particular EL0(τ, 0) = 0. The central derivative E′L0

(τ, 0) at s = 0 is a harmonic Maassform of weight 1 with representation ω∨L0

, whose holomorphic part we denote by (as in [BY,(2.26)])

(6.3.2) EL0(τ) =

∑m∈Q≥0

a+L0

(m) · qm.

Up to a change of notation, the following proposition is due to Schofer [Scho]; see also[BY, Theorem 2.6]. Be warned that both references contain minor misstatements. Theformula of part (4) is misstated in [Scho], but the error is corrected in [BY]. The formulaof (2) is correct in [Scho], but is misstated in [BY].

Proposition 6.3.1. Recall the finite set DiffL0(m) of odd cardinality from (5.3.4), and thefunction ρ of (1.5.1). The coefficients a+

L0(m) ∈ S∨L0

are given by the following formulas.

(1) If m < 0 then a+L0

(m) = 0.

(2) The constant term is

a+L0

(0, ϕ) = ϕ(0) ·(γ + log

∣∣∣∣4πdk∣∣∣∣− 2

L′(χk, 0)

L(χk, 0)

)for every ϕ ∈ SL0

.(3) If m > 0 and |DiffL0(m)| > 1, then a+

L0(m) = 0.


(4) If m > 0 and DiffL0(m) = p for a single prime p, then

a+L0

(m,ϕ) = − wk2hk· ρ(m|dk|pε

)· ordp(pm) · log(p)

∑µ∈d−1

k L0,f/L0,f

Q(µ)=m

2s(µ)ϕ(µ).

On the right hand side, s(µ) is the number of primes q | dk such that µq = 0, ε isdefined by (5.3.6), and Q(µ) = m is understood as an equality in Q/Z.

Define a coherent Eisenstein series of weight −1 associated to L0(∞) by

(6.3.3) EL0(∞)(τ, s, ϕ) =∑

γ∈P\ SL2(Z)

ωL0(γ)ϕ(0) · (cτ + d) · Im(γτ)

s2 +1.

This is the Eisenstein series denoted EL0(τ, s,−1) in [BY, Section 2], for any choice of L0 ∈gen(L0(∞)). The following relationship between the coherent and incoherent Eisensteinseries was first observed by Kudla [Ku3, (2.17)], and is a special case of [BY, Lemma 2.3].

Proposition 6.3.2. For any ϕ ∈ SL0the Eisenstein series (6.3.1) and (6.3.1) are related

by the equality−2 · ∂(E′L0

(τ, 0, ϕ)dτ) = EL0(∞)(τ, 0, ϕ) · v−2du ∧ dvof smooth 2-forms on H.

Suppose L0 ∈ gen(L0(∞)). Exactly as in (6.2.3), there is an SL2(Z)-equivariant isomor-

phism S∆0

L0

∼= S∆0

L0, which allows us to define a non-holomorphic theta series θL0

: H →(S∨L0

)∆0 by

θL0(τ, ϕ) = v

∑λ∈d−1

k L0

ϕ(λ)e2πi〈λ,λ〉τ .

for any ϕ ∈ S∆0

L0. The following proposition follows from [BY, Proposition 2.2].

Proposition 6.3.3 (Siegel-Weil formula). The coherent Eisenstein series (6.3.3) is relatedto the above theta series by

2o(dk)

hk

∑L0∈gen(L0(∞))

θL0(τ) = EL0(∞)(τ, 0).

6.4. CM values and the Rankin-Selberg L-function. For each ϕ ∈ SΛ define

RΛ(m,ϕ) =∑

λ∈d−1k Λ

〈λ,λ〉=m

ϕ(λ).

These representation numbers are the Fourier coefficients of a holomorphic S∨Λ-valued mod-ular form

θΛ(τ, ϕ) =∑m∈Q

RΛ(m,ϕ) · qm ∈Mn−1(ω∨Λ).

Given any F ∈ Sn(ωL ) with Fourier expansion

F (τ) =∑

m∈Q≥0

b(m) · qm,

define the Rankin-Selberg convolution L-function

(6.4.1) L(F, θΛ, s) = Γ(s

2+ n− 1

) ∑m∈Q>0

b(m), RΛ(m)

(4πm)

s2 +n−1

.


On the right hand side the pairing is the tautological pairing between SL and its dual. Theinclusion d−1

k Λ/Λ→ d−1k Lf/Lf determines a canonical surjection SL → SΛ, and hence an

injection on dual spaces. In particular, this allows us to view RΛ(m) as an element of S∨L .The usual unfolding method shows that

(6.4.2) L(F, θΛ, s) =

∫SL2(Z)\H

F (τ), EL0

(τ, s)⊗ θΛ(τ)vn−2 du dv,

where τ = u + iv and EL0(τ, s) is the incoherent Eisenstein series of (6.3.1). On the righthand side we are using the canonical isomorphism S∨L

∼= S∨L0⊗S∨Λ to view EL0

(τ, s)⊗θΛ(τ)as an S∨L -valued function. Of course L(F, θΛ, 0) = 0, as the Eisenstein series vanishes ats = 0.

Theorem 6.4.1. For every f ∈ H2−n(ωL )∆ the CM value

ΦL (Y(L0,Λ), f) =∑

y∈Y(L0,Λ)(C)

ΦL (y, f)

|Aut(y)|

satisfies

1

degC Y(L0,Λ)· ΦL (Y(L0,Λ), f) = −L′

(ξ(f), θΛ, 0

)+ CT

[f+, EL0 ⊗ θΛ

].

Here the differential operator

ξ : H2−n(ωL )→ Sn(ωL )

is defined by (2.2.3), and CT[f+, EL0⊗ θΛ] is the constant term of the q-expansion of

f+, EL0 ⊗ θΛ.

Proof. This is really a special case of [BY, Theorem 4.7], but beware that the statementof [loc. cit.] contains a sign error. We sketch the main ideas for the convenience of thereader. Recall from Section 6.1 that the complex points of Y(L0,Λ) are indexed by triples(A0,A1,B). If we fix such a point y(A0,A1,B) ∈ Y(L0,Λ)(C) and abbreviate L0 = L(A0,A1)and L = Λ⊕L0, then the theta function ΘL(τ, z) appearing in (3.3.1) admits a factorization

ΘL(τ, y(A0,A1,B)) = θΛ(τ)⊗ θL0(τ)

at z = y(A0,A1,B). Hence

ΦL (y(A0,A1,B), f) =

∫ reg

SL2(Z)\Hf, θΛ ⊗ θL0

dµ(τ).

Summing over all points complex points of Y(L0,Λ) yields

ΦL (Y(L0,Λ), f) =∑

(A0,A1,B)L(A0,A1)∈L0(∞)

L(A0,B)∼=Λ

1

|Aut(A0,A1,B)|

∫ reg

SL2(Z)\H

f, θΛ ⊗ θL(A0,A1)

dµ(τ)

=hk

w2k|Aut(Λ)|

∑L0∈gen(L0(∞))

∫ reg

SL2(Z)\Hf, θΛ ⊗ θL0

dµ(τ)

=degC Y(L0,Λ)

2

∫ reg

SL2(Z)\Hf(τ), θΛ(τ)⊗ EL0(∞)(τ, 0) dµ(τ)


by the Siegel-Weil formula (Proposition 6.3.3) and Remark 6.1.4. Applying Proposition6.3.2 and Stokes’ theorem, a simple calculation, as in the proof of [BY, Theorem 4.7], showsthat

1

degC Y(L0,Λ)· ΦL (Y(L0,Λ), f)

= −∫ reg

SL2(Z)\Hf, θΛ ⊗ ∂E′L0

(τ, 0) dτ

= −∫ reg

SL2(Z)\Hdf, θΛ ⊗ E′L0

(τ, 0) dτ −∫

SL2(Z)\Hξ(f), θΛ ⊗ E′L0

(τ, 0)vn dµ(τ)

= CT[f+(τ), θΛ(τ)EL0(τ)]− L′(ξ(f), θΛ, 0),

as claimed.

Remark 6.4.2. Let ϕr ∈ SL be the characteristic function of

r−1Lf/Lf ⊂ d−1k Lf/Lf .

By abuse of notation we denote again by ϕr the similarly defined elements of SL0and SΛ.

If A is an element of S∨L , S∨L0, or S∨Λ abbreviate A(r) = A(ϕr). For example

RΛ(m, r) = RΛ(m,ϕr) =∑

λ∈r−1Λ〈λ,λ〉=m

1.

The following is a restatement of Theorem 6.4.1 in the case f = fm,r.

Corollary 6.4.3. The harmonic form fm,r of Lemma 6.2.1 satisfies

1

degC Y(L0,Λ)· ΦL (Y(L0,Λ), fm,r) = −L′(ξ(fm,r), θΛ, 0) + c+m,r(0, 0) · a+

L0(0, r) ·RΛ(0, r)

+∑

m1,m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r).

7. Metrized line bundles and derivatives of L-functions

Throughout Section 7 we fix, as in Section 6, a pair (L0,Λ) consisting of

• an incoherent self-dual (kR, Ok)-module L0 of signature (1, 0),• a self-dual hermitian Ok-module Λ of signature (n− 1, 0).

As in Remark 5.2.3, define an incoherent self-dual (kR, Ok)-module L = L0⊕Λ of signature(n, 0).

In this section we state and prove a preliminary form of our main result. This preliminaryform, Theorem 7.3.1 below, is an intersection formula for the Kudla-Rapoport divisors(endowed with the automorphic Green functions constructed eariler) with the CM cycleY(L0,Λ). The intersection takes place on the open Shimura variety ML , and is expressed inthe language of metrized line bundles. In Section 8 the result will be reformulated in termsof arithmetic Chow groups on the canonical toroidal compactification of ML .


7.1. Generalities on metrized line bundles. In this subsection, M is any equidimensionalregular algebraic stack, flat and of finite type over Ok. A metrized line bundle L = (L, || · ||)on M consists of a line bundle L and a hermitian metric || · || on the complex points L(C). The

isomorphism classes of metrized line bundles form a group Pic(M) under tensor product. An

arithmetic divisor D = (D,Φ) is a pair consisting of a divisor D on M and a Green function6

on M(C) for the complex fiber D(C). The arithmetic divisors form a group Div(M) under

addition. If we start with an arithmetic divisor D, the constant function 1 on M defines arational section s of the line bundle L = O(D) associated to D, and there is a unique metric|| · || on L satisfying − log ||s||2z = Φ(z). This establishes a surjection

Div(Y)→ Pic(Y).

Suppose that Y is smooth over Ok of relative dimension 0, and comes equipped with afinite and unramified morphism i : Y → M. As in [KRY2, Chapter 2.1] there is a linearfunctional

deg : Pic(Y)→ R

defined by

deg L =∑Di

∑p⊂Ok

∑y∈Di(Falg

p )

mi log(N(p))

|Aut(y)|−∑y∈Y(C)

log ||σ||2y|Aut(y)|

for any rational section σ of L with divisor div(σ) =∑imiDi. The composition

Pic(M)i∗−→ Pic(Y)

deg−−→ R,

called arithmetic intersection against Y, is denoted

[ · : Y] : Pic(M)→ R.

Keeping i : Y→ M as above, suppose that we are given a Cartesian diagram

Z ∩ Y

// Y

i

Z

j// M

in which Z is Cohen-Macaulay of dimension dim(M) − 1, and the morphism j is finite,unramified, and representable. In the notation of [Vi, Section 3], Z determines a divisorj∗[Z] on M. This can be made more explicit as follows. The hypotheses on j imply, as in[Vi, Lemma 1.19] that any geometric point of M admits an etale neighborhood U → M withthe property that Z/U → U restricts to a closed immersion of schemes on each connectedcomponent of Z/U . In particular each connected component of Z/U defines a divisor on U ,and (j∗[Z])/U is their sum. When no confusion is possible we use the same letter Z to denotethe stack, the associated divisor, and the associated line bundle.

Suppose Φ is a Green function for Z, and set Z = (Z,Φ). Assuming that Z ∩ Y has

dimension 0, we have the explicit formula [Z : Y] = I(Z : Y) + Φ(Y) where the first term is

6Throughout Section 7, all divisors have integer coefficients, and all Green functions are real-valued.


the finite intersection multiplicity7

I(Z : Y) =∑p⊂Ok

log(N(p))∑

y∈(Z∩Y)(Falgp )

lengthOZ∩Y,y(OZ∩Y,y)

|Aut(y)|,

and the second term is

Φ(Y) =∑y∈Y(C)

Φ(y)

|Aut(y)|.

7.2. The cotautological bundle and the Chowla-Selberg formula. Let S be a schemeand π : A→ S an abelian scheme. As in [Lan, Section 2.1.6], define a coherent OS-module,the algebraic de Rham cohomology of A, as the hypercohomology Hi

dR(A) = Riπ∗(Ω•A/S) of

the de Rham complex0→ OA → Ω1

A/S → Ω2A/S → · · · .

The algebraic de Rham homology HdR1 (A) = HomOS (H1

dR(A),OS) sits in an exact sequence

0→ Fil1(A)→ HdR1 (A)→ Lie(A)→ 0,

and Fil1(A) is canonically isomorphic to the OS-dual of Lie(A∨).

Definition 7.2.1. Let (Auniv0 , Auniv) denote the universal object over M, and recall that

Auniv is endowed with anOk-stableOM-submodule Funiv ⊂ Lie(Auniv) such that the quotientLie(Auniv)/Funiv is locally free of rank one. The cotautological bundle on M is the line bundle

T = HomOM(Fil1(Auniv

0 ),Lie(Auniv)/Funiv).

Denote by TL the restriction of T to ML .

Recall from Section 6.1 that each connected component of ML (C) admits a uniformiza-tion D(A0,A) → ML (C), where D(A0,A) is the space of negative kR-lines in L(A0,A). Thehermitian symmetric domain D(A0,A) carries a tautological bundle L, whose fiber at a pointz is the line z, made into a complex vector space using the fixed isomorphism kR ∼= C. Thefollowing proposition and remark explain the connection between L and the cotautologicalbundle.

Proposition 7.2.2. At any point z ∈ D(A0,A) there is a canonical kR-linear isomorphism

fz : TL ,z → z

from the fiber at z of (the pullback of) TL to negative line z.

Proof. Identify k ⊗Q C ∼= C× C in such a way that x⊗ 1 7→ (x, x), and define idempotentse = (1, 0) and e = (0, 1). If X is any complex vector space with a commuting action ofk, then k acts on eX through the fixed embedding k → C, and acts on eX through theconjugate embedding.

Let (A0, Az) ∈ ML (C) be the image of z under D(A0,A) → ML (C). The map

A0C ∼= HdR1 (A0)

e−→ eHdR1 (A0) ∼= Fil1(A0)

restricts to a kR-linear isomorphism A0R ∼= Fil1(A0), while the quotient map

AC ∼= HdR1 (Az)→ Lie(Az)

7As both Z and Y are Cohen-Macaulay, the finite intersection multiplicity agrees with the more naturalSerre intersection multiplicity defined as in [SABK, p. 11] or [Ho3, Section 3.1]. This follows from [Se00,

p. 111].


restricts to a kR-linear isomorphism AR ∼= Lie(Az). Thus we obtain isomorphisms

L(A0,A)R ∼= HomkR(A0R,AR) ∼= HomkR(Fil1(A0),Lie(A)z),

and tracing through the constructions of Section 6.1 shows that their composition identifiesz⊥ with HomkR(Fil1(A0),Fz). The surjection

L(A0,A)R → HomkR(Fil1(A0),Lie(Az)/Fz)

therefore has kernel z⊥, and identifies z ∼= HomkR(Fil1(A0),Lie(Az)/Fz) as desired.

Remark 7.2.3. Proposition 7.2.2 does not say that the pullback of TL to D(A0,A) is iso-morphic to the tautological bundle L. The point is that the signature conditions imposedon Lie(Auniv

0 ) and Lie(Auniv)/Funiv imply that the action of kR on the fiber at z of TL

is through the complex conjugate of the fixed isomorphism kR ∼= C. Thus the isomor-phism fz of Proposition 7.2.2 is complex-conjugate-linear. It is more natural to consider thecomplex-linear isomorphism

f∨z : TL ,z∼= HomC(z,C)

defined by f∨z (s) = 〈·, fz(s)〉, which identifies the pullback of TL with the dual of thetautological bundle.

We use Proposition 7.2.2 to metrize the cotautological bundle TL : the norm of a sections is defined to be

(7.2.1) ||s||2z = −4πeγ · 〈fz(s), fz(s)〉,where γ = −Γ′(1) is Euler’s constant. The cotautological bundle endowed with the abovemetric is denoted

TL ∈ Pic(ML ).

The remainder of this subsection is devoted to studying the image of TL under the arithmeticintersection

[ · : Y(L0,Λ)] : Pic(ML )→ Rof Section 7.1. In particular, we will explain how the Chowla-Selberg formula implies thefollowing theorem.

Theorem 7.2.4. For any ϕ ∈ SL0 , the metrized cotautological bundle satisfies

ϕ(0) · [TL : Y(L0,Λ)] = −degC Y(L0,Λ) · a+L0

(0, ϕ),

where a+L0

(0) ∈ S∨L0is defined by (6.3.2), and degC Y(L0,Λ) is defined in Remark 6.1.4.

In particular, taking ϕ to be the characteristic function of r−1L0,f/L0,f , as in Remark6.4.2, shows that

[T⊗RΛ(m,r)L : Y(L0,Λ)] = −degC Y(L0,Λ) · a+

L0(0, r) ·RΛ(m, r)

for all m ∈ Q≥0 and r | dk.

The proof requires some preparation. First we state the Chowla-Selberg formula in a formsuited to our purposes. Suppose E is an elliptic curve over C with complex multiplicationby Ok. Fix a model of E over a finite extension K/Q contained in C and large enough thatE has everywhere good reduction, and let π : E → Spec(OK) be the Neron model of E overOK . Let ω be a nonzero rational section of the line bundle π∗Ω

1E/OK on Spec(OK) with

divisordiv(ω) =

∑q

m(q) · q,


where the sum is over the closed points q ∈ Spec(OK). The Faltings height of E is definedas

hFalt(E) =1

[K : Q]

(∑q

log(N(q)) ·m(q)− 1

2

∑τ :K→C

log

∣∣∣∣∣∫Eτ (C)

ωτ ∧ ωτ∣∣∣∣∣).

It is independent of the choice of K, the model of E over K, and the section ω. TheChowla-Selberg formula [Co] implies that

(7.2.2) −2hFalt(E) = log(2π) +1

2log |dk|+

L′(χk, 0)

L(χk, 0).

Recall that Y(L0,Λ) carries over it a universal triple of abelian schemes (Auniv0 , Auniv

1 , Buniv).Define a line bundle

coLie(Auniv0 ) = π∗Ω

1Auniv

0 /Y(L0,Λ)

on Y(L0,Λ), where π : Auniv0 → Y(L0,Λ) is the structure morphism. A vector ω ∈ coLie(Auniv

0,y )

in the fiber at a complex point y ∈ Y(L0,Λ)(C) is a global holomorphic 1-form on Auniv0,y (C),

and we denote by

coLie(Auniv0 ) ∈ Pic(Y(L0,Λ))

the line bundle coLie(Auniv0 ) endowed with the metric

(7.2.3) ||ω||2y =

∣∣∣∣∣∫Auniv

0,y (C)

ω ∧ ω

∣∣∣∣∣ .With this definition,

(7.2.4) deg coLie(Auniv0 ) =

∑(A0,A1,B)∈Y(L0,Λ)(C)

2hFalt(A0)

|Aut(A0, A1, B)|.

Of course Lie(Auniv0 ) is isomorphic to the dual of coLie(Auniv

0 ). Denote by

(7.2.5) Lie(Auniv0 ) ∈ Pic(Y(L0,Λ))

the line bundle Lie(Auniv0 ) with the metric dual to (7.2.3). More explicitly, if we endow

A0 = H1(Auniv0,y (C),Z) with its hermitian form hA0

as in Section 5.1, then Lie(Auniv0,y ) ∼= A0R

as real vector spaces, and ||v||2y = |dk|−12hA0(v, v) for any v ∈ Lie(Auniv

0,y ).

Lemma 7.2.5. The metrized line bundle Lie(Auniv0 ) satisfies

1

degC Y(L0,Λ)· deg Lie(Auniv

0 ) = log(2π) +1

2log |dk|+

L′(χk, 0)

L(χk, 0).

Of course we may define Lie(Auniv1 ) in the same manner as (7.2.5), and the stated equality

holds with Auniv0 replaced by Auniv

1 .

Proof. Combine (7.2.4) and the Chowla-Selberg formula (7.2.2).

For any positive c ∈ R, define the twisted trivial bundle

1(c) ∈ Pic(Y(L0,Λ))

as the structure sheaf OY(L0,Λ)endowed with the metric ||f ||2y = c · |f(y)|2. It is clear from

the definitions that

(7.2.6) deg 1(c) = − log(c) · degC Y(L0,Λ).


Lemma 7.2.6. There is an isomorphism

TL |Y(L0,Λ)∼= Lie(Auniv

0 )⊗ Lie(Auniv1 )⊗ 1

(16π3eγ

)of metrized line bundles on Y(L0,Λ).

Proof. Recall that ML carries a universal pair of abelian schemes (Auniv0 , Auniv), and that

Auniv comes with a universal OML -submodule Funiv ⊂ Lie(Auniv). By definition of themorphism Y(L0,Λ) → ML , the universal objects over ML and Y(L0,Λ) are related8 by

(Auniv0 , Auniv)/Y(L0,Λ)

∼= (Auniv0 , Auniv

1 ×Buniv),

and the isomorphism

Lie(Auniv)|Y(L0,Λ)∼= Lie(Auniv

1 ×Buniv)

identifies Funiv|Y(L0,Λ)∼= Lie(Buniv). In particular there is a canonical isomorphism

TL |Y(L0,Λ)∼= Hom(Fil1(Auniv

0 ),Lie(Auniv1 )).

For any elliptic curve A0 → Spec(R) over a ring, the short exact sequence

0→ Fil1(A0)→ HdR1 (A0)→ Lie(A0)→ 0

of R-modules is dual to

0→ H0(A0,Ω1A0/R

)→ H1dR(A0)→ H1(A0,OA0)→ 0,

and there is a canonical identification H1(A0,OA0) ∼= Lie(A∨0 ). In particular there is canon-

ical perfect pairing Lie(A∨0 )⊗R Fil1(A0) → OS , and identifying Lie(A0) ∼= Lie(A∨0 ) via theunique principal polarization, we obtain a perfect pairing

(7.2.7) Lie(A0)⊗R Fil1(A0)→ R.

Applying this with A0 = Auniv0 yields the second isomorphism in

TL |Y(L0,Λ)∼= Hom(Fil1(Auniv

0 ),Lie(Auniv1 )) ∼= Lie(Auniv

0 )⊗ Lie(Auniv1 ).

All that remains is to keep track of the metrics under this isomorphism. This is routine,once one knows an explicit formula for the pairing (7.2.7) when A0 ∈ M(1,0)(C) is the complexelliptic curve with homology A0 = H1(A0(C),Z), as in the discussion surrounding (5.1.4).Taking e and e as in the proof of Proposition 7.2.2, the compositions

A0R → A0C ∼= HdR1 (A0)

e−→ eHdR1 (A0) ∼= Lie(A0)

and

A0R → A0C ∼= HdR1 (A0)

e−→ eHdR1 (A0) ∼= Fil1(A0)

are kR-linear isomorphisms. Thus the pairing (7.2.7) corresponds to a pairing

A0R × A0R → C,

which is hermitian with respect to the action of kR ∼= C. Using the proof of [Ho1, Proposition4.4], one can show that this pairing is 2π|dk|−1/2 · hA0 . The rest of the proof is elementarylinear algebra, and is left to the reader.

8There is a mild abuse of notation: we are using Auniv0 to denote both the universal elliptic curve over

ML , and the universal elliptic curve over Y(L0,Λ).


Proof of Theorem 7.2.4. Combining Lemmas 7.2.5 and 7.2.6 with (7.2.6) shows that

[TL : Y(L0,Λ)] = deg Lie(Auniv0 ) + deg Lie(Auniv

1 ) + deg 1(16π3eγ

)= degC Y(L0,Λ)

(2L′(χk, 0)

L(χk, 0)+ log

∣∣∣∣dk4π

∣∣∣∣− γ) ,and comparing with Proposition 6.3.1 completes the proof.

7.3. The main result, preliminary form. Let f ∈ H2−n(ωL ) be a ∆-invariant harmonicMaass form. The divisor ZL (f) on ML of (6.2.1), with its Green function ΦL (f) of Section6.2, determines an arithmetic divisor

(7.3.1) ZL (f) ∈ Div(ML ),

and we denote in the same way the corresponding metrized line bundle. Throughout therest of Section 7 we fix m ∈ Q>0 and r | dk, and take f = fm,r to be the harmonic form ofLemma 6.2.1, with holomorphic part

f+m,r(τ) = ϕm,rq

−m +∑k∈Q≥0

c+m,r(k) · qk.

The underlying divisor of (7.3.1) is then simply the Kudla-Rapoport divisor ZL (m, r).

Theorem 7.3.1. Keeping the notation of Section 6.4, the harmonic form fm,r satisfies

[ZL (fm,r) : Y(L0,Λ)] + c+m,r(0, 0) · [TL : Y(L0,Λ)] = −degC Y(L0,Λ) · L′(ξ(fm,r), θΛ, 0

).

The proof of the theorem will occupy the rest of Section 7, although we have enoughinformation to prove a special case right now. Fix m1,m2 ∈ Q≥0. In Section 5.3 wedefined an Ok-stack X(L0,Λ)(m1,m2, r) equipped with a finite, unramified, and representablemorphism

X(L0,Λ)(m1,m2, r)→ Y(L0,Λ).

If m1 > 0, then X(L0,Λ)(m1,m2, r) has dimension zero, and defines a divisor on Y(L0,Λ),necessarily supported in nonzero characteristic. By endowing this divisor with the trivialGreen function, we obtain an arithmetic divisor

(7.3.2) X(L0,Λ)(m1,m2, r) ∈ Div(Y(L0,Λ)).

Proposition 7.3.2. Suppose m = m1 +m2 with m1 ∈ Q>0 and m2 ∈ Q≥0. The arithmeticdivisor (7.3.2) satisfies

deg X(L0,Λ)(m1,m2, r) = −degC Y(L0,Λ) · a+L0

(m1, r) ·RΛ(m2, r).

Proof. This follows by comparing Theorem 5.3.2 with Proposition 6.3.1. Abbreviate

X = X(L0,Λ)(m1,m2, r).

If |DiffL0(m1)| > 1 then both sides of the desired equality are 0, so assume DiffL0

(m1) = pfor a prime p, necessarily nonsplit in k, and let p be the prime of k above p. Theorem 5.3.2implies

deg X(L0,Λ)(m1,m2, r) = log(N(p))∑

x∈X(Falgp )

length(OX,x)

|Aut(x)|

=hk log(p)

wk|Aut(Λ)|· ordp(pm1) ·RΛ(m2, r) · ρ

(m1N(s)

pε

),


where s is the prime-to-p part of r, and ε is defined by (5.3.6). On the other hand, Proposition6.3.1 tells us that

a+L0

(m1, r) = −wk log(p)

2hk· ordp(pm1) · ρ

(m1|dk|pε

) ∑µ∈d−1

k L0,f/L0,f

Q(µ)=m1

2s(µ)ϕr(µ),

where ϕr is the characteristic function of r−1L0,f/L0,f ⊂ d−1k L0,f/L0,f .

The proposition follows from the above equalities and Remark 6.1.4, once we prove

ρ

(m1|dk|pε

) ∑µ∈d−1

k L0,f/L0,f

Q(µ)=m1

2s(µ)ϕr(µ) = 2o(dk)ρ

(m1N(s)

pε

).

Both sides factor as a product of local terms, and the equality of local terms for primes notdividing dk is obvious. It therefore suffices to prove, for every prime ` | dk, the relation

(7.3.3) ρ`(m1|dk|)∑

µ∈d−1k L0,`/L0,`

Q(µ)=m1

2s`(µ)ϕr,`(µ) = 2ρ`(m1N(s)),

where ρ`(k) is the number of ideals in Ok,` of norm kZ`,

s`(µ) =

1 if µ = 0

0 otherwise,

and ϕr,` is the characteristic function of r−1L0,`/L0,` ⊂ d−1k L0,`/L0,`.

Case 1: If ord`(m1) ≥ 0, then only the term µ = 0 contributes to the left hand side of(7.3.3), both sides of the equality are equal to 2, and we are done.

Case 2: If ord`(m1) < −1, then ρ`(m1|dk|) = ρ`(m1N(s)) = 0, and we are done.Case 3: If ord`(m1) = −1 and ` - N(r), then ρ`(m1N(s)) = 0. On the left hand side of

(7.3.3), the assumption ord`(m1) = −1 implies that any µ appearing in the summust be nonzero, and hence ϕr,`(µ) = 0. Thus in this case both sides of (7.3.3)vanish.

Case 4: If ord`(m1) = −1 and ` | N(s), then ρ`(m1|dk|) = ρ`(m1N(s)) = 1. On the otherhand, as ` 6∈ DiffL0

(m1), the rank one kp-hermitian space V0,` represents m1. It

follows from the self-duality of L0,` that d−1k L0,` represents m1, and from this it is

easy to see that the rank one Ok/p-quadratic space d−1k L0,`/L0,` has two distinct

nonzero solutions to Q(µ) = m1. Thus∑µ∈d−1

k L0,`/L0,`

Q(µ)=m1

2s`(µ) = 2

and again (7.3.3) holds.Case 5: If ord`(m1) = −1 and ` = p, then ρ`(m1N(s)) = 0. On the left hand side of (7.3.3),

the sum over µ is empty: any µ ∈ d−1k L0,p/L0,p representing m1 ∈ Qp/Zp could be

lifted to µ ∈ d−1k L0,p representing m1 ∈ Qp, contradicting p ∈ DiffL0

(m1). Thusboth sides of (7.3.3) vanish.

This exhausts all cases, and completes the proof.


We can now prove Theorem 7.3.1 under the simplifying hypothesis RΛ(m, r) = 0. Recallfrom Remark 5.3.9 that under this hypothesis

ZL (m, r) ∩ Y(L0,Λ)∼=

⊔m1∈Q>0

m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r),

and each stack appearing on the right has dimension zero. Proposition 7.3.2 shows that

1

degC Y(L0,Λ)· I(ZL (m, r) : Y(L0,Λ)) =

1

degC Y(L0,Λ)

∑m1∈Q>0

m2∈Q≥0

m1+m2=m

deg X(L0,Λ)(m1,m2, r)

= −∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r),

while Corollary 6.4.3 shows that

1

degC Y(L0,Λ)· ΦL (Y(L0,Λ), fm,r) = −L′

(ξ(fm,r), θΛ, 0

)+ c+m,r(0, 0) · a+

L0(0, r) ·RΛ(0, r)

+∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r).

Adding these together gives

1

degC Y(L0,Λ)· [ZL (fm,r) : Y(L0,Λ)] = c+m,r(0, 0) · a+

L0(0, r) ·RΛ(0, r)− L′

(ξ(fm,r), θΛ, 0

),

and an application of Theorem 7.2.4 completes the proof. Proving Theorem 7.3.1 in generalrequires treating improper intersections, and requires a bit more work.

7.4. The adjunction formula. The proof of Theorem 7.3.1 in full generality revolvesaround the study of a canonical section

(7.4.1) σm,r ∈ Γ(Z♥L (m, r)|Y(L0,Λ)

)of the line bundle

(7.4.2) Z♥L (m, r) = ZL (m, r)⊗ T⊗−RΛ(m,r)L

restricted to Y(L0,Λ). This subsection is devoted to the construction of (7.4.1), and thecalculation of its divisor, which the reader may find in Proposition 7.4.9 below.

Because of the minor nuisance that the natural maps Y → M and Z(m, r) → M are notclosed immersions, the section (7.4.1) will be constructed by patching together sections onan etale open cover. Accordingly, we define a sufficiently small etale open subscheme of ML

to be a scheme U together with an etale morphism U → ML such that

(1) on each connected component Z ⊂ ZL (m, r)/U the natural map Z → U is a closedimmersion,

(2) on each connected component Y ⊂ Y(L0,Λ)/U the natural map Y → U is a closedimmersion, and the universal object (A0, A1, B) over Y satisfies L(A0, B) ∼= Λ.


As in the discussion of Section 7.1, the stack ML admits a finite cover by sufficiently smalletale open subschemes.

Fix a sufficiently small etale open subscheme U → ML , and connected components

Z ⊂ ZL (m, r)/U(7.4.3)

Y ⊂ Y(L0,Λ)/U .

The smoothness of Y(L0,Λ) over Ok, together with our hypotheses on U , imply that Y isa reduced and irreducible one-dimensional closed subscheme of U . The closed subschemeZ ⊂ U is perhaps neither reduced nor irreducible, but an easy deformation theory argumentshows that the generic fiber of ZL (m, r) is smooth, and hence Z/k is a smooth variety ofdimension n− 1. The intersection Z ∩ Y = Z ×U Y is a closed subscheme of Y , and henceis either all of Y or is of dimension 0.

Definition 7.4.1. Given connected components (7.4.3), we say that

(1) Z is Y -proper if Z ∩ Y has dimension 0,(2) Z is Y -improper if Z ∩ Y = Y .

Proposition 7.4.2. The number of Y -improper components Z ⊂ ZL (m, r)/U is RΛ(m, r).

Proof. Let η ∈ Y be the generic point, so that k(η) is a finite extension of k, and let η → ηbe the geometric generic point above η. Denote by (A0,η, A1,η, Bη) and (A0,η, A1,η, Bη) thepullbacks to η and η of the universal object over Y(L0,Λ). The fiber

ZL (m, r)η = ZL (m, r)/U ×U η

is a disjoint union of copies of η, one for every

λ ∈ r−1L(A0,η, A1,η ×Bη)

satisfying 〈λ, λ〉 = m. Under the decomposition 5.2.1, any such λ takes the form λ = λ1+λ2,but we must have λ1 = 0 as there are no nonzero Ok-linear maps A0,η → A1,η. Indeed,such a nonzero map would induce an Ok-linear isomorphism of Lie algebras, which cannotexist as A0,η satisfies the signature (1, 0) condition, while A1,η satisfies the signature (0, 1)condition. Thus all such λ lie in

r−1L(A0,η, Bη) ∼= r−1Λ,

and it follows that ZL (m, r)η is a disjoint union of RΛ(m, r) copies of η. Moreover, ourdefinition of a sufficiently small etale open guarantees that r−1L(A0,η, Bη) ∼= r−1Λ, andso all such f are already defined over η. In other words, ZL (m, r)η is a disjoint union ofRΛ(m, r) copies of η. It follows that there are RΛ(m, r) connected components of ZL (m, r)/Uwhose image in U contains η, and the claim follows easily.

As in the proof of Proposition 5.1.9, write Ok = Z[Π] and define elements of Ok ⊗Z OUby

J = Π⊗ 1− 1⊗Π

J = Π⊗ 1− 1⊗Π.

An elementary calculation shows that the sequence

(7.4.4) · · · J−→ Ok ⊗Z OUJ−→ Ok ⊗Z OU

J−→ Ok ⊗Z OUJ−→ · · · .

is exact.


Lemma 7.4.3. Every geometric point y → Y admits an affine etale neighborhood Spec(R)→U over which the R-module Lie(A) is free, and admits a basis ε1, . . . , εn such that

(1) ε1, . . . , εn−1 is a basis for the universal subsheaf F ⊂ Lie(A),(2) the operator J on Lie(A) has the form

(7.4.5) J =

0 · · · 0 j1...

. . ....

...0 · · · 0 jn

∈Mn(R)

for some j1, . . . , jn ∈ R satisfying (j1, . . . , jn) = (jn) = dkR,(3) there is unique Ok-linear endomorphism

j : Lie(A)→ Lie(A)

such that J = δk j as endomorphisms of Lie(A),(4) the endomorphism j preserves the subsheaf F ⊂ Lie(A), and j induces an automor-

phism of the quotient Lie(A)/F .

Here A is the pullback to R of the universal object via U → M→ M(n−1,1).

Proof. Certainly there is an affine etale neighborhood over which

0→ Fil1(A)→ HdR1 (A)→ Lie(A)→ 0

is an exact sequence of free R-modules. If ε1 . . . , εn is any basis of HdR1 (A) such that

ε1 . . . , εn−1 generates F , then the matrix of J is (7.4.5) for some j1, . . . , jn ∈ R, simplybecause JF = 0. Futhermore, J acts on the quotient Lie(A)/F through the complexconjugate of the structure map Ok → R, and so jn = Π − Π. It is easy to check that(Π−Π)Ok = dk, and so (jn) = dkR.

To prove that (j1, . . . , jn) = dkR, after possibly shrinking the etale neighborhood Spec(R),it suffices to prove this equality after replacing R by the completion of the etale local ring aty. We then know from the proof of [Ho3, Proposition 3.2.3] (as in the proof of Proposition5.1.9), that the ideal (j1, . . . , jn) is principal. Everything we have said so far holds for anygeometric point of U . Now we exploit the hypothesis that y is a geometric point of Y . LetI ⊂ R be the ideal defining the closed subscheme Y ×U Spec(R) ⊂ Spec(R), and let A′

be the reduction of A to R/I. By definition of the morphism Y → M of Section 5.2, theabelian scheme A′ comes with a decomposition A′ ∼= A1×B, and the subsheaf F ′ ⊂ Lie(A′)is F ′ = Lie(B). In particular, F ′ admits the Ok-stable, and hence J-stable, OY -directsummand Lie(A0). Thus there is some basis of Lie(A′) with respect to which

J =

0 · · · 0 0...

. . ....

...0 · · · 0 00 · · · 0 jn

∈Mn(R/I).

As the ideal of R/I generated by the entries of J is independent of the choice of basis, wesee that

(j1, . . . , jn) = (jn) = (δk)

in R/I. Now pick a generator γ ∈ R of the principal ideal (j1, . . . , jn). We have shown that(δk) ⊂ (γ), with equality after reducing modulo I. Furthermore, R/I is an integral domainof characteristic 0, as Y is reduced, irreducible, and flat over Ok. It follows that if we writeδk = uγ with u ∈ R, then u is a unit in R/I, and hence is also a unit in the local ring R.This shows that (j1, . . . , jn) = (δk) in R.


All remaining claims of the lemma are obvious. The only thing to note is that M, hencealso R, is flat over Ok, and so δk ∈ R is not a zero divisor. Thus the entries of the matrixJ are uniquely divisible by δk.

Keeping Z and Y as in (7.4.3), denote by IZ ⊂ OU the ideal sheaf defining the closedsubscheme Z → U , and by O(Z) = I−1

Z the line bundle on U determined by the divisorZ. Let IY ⊂ OU be the ideal sheaf defining the closed subscheme Y → U . The first orderinfinitesimal neighborhood of Y is the closed subscheme Y → U defined by the ideal sheafI2Y ⊂ OU . The picture is

Z

Y

22

,,

// Y ∩ Z

<<

""

U // ML

Y

@@

where Y ∩ Z = Y ×U Z.Let (A0, A, λ) be the pullback to Y of the universal object over ZL (m, r). Of course the

pair (A0, A) has a canonical extension (A0, A) to Y , obtained by pulling back the universal

pair over M via Y → U → M, but there is no such canonical extension of λ to Y . Indeed, Y ∩Zis the maximal closed subscheme of Y over which λ extends to an element of r−1L(A0, A)satisfying the vanishing condition of (5.1.2). Recall that, by virtue of the moduli problem

defining M(n−1,1), the OY -module Lie(A) comes equipped with a corank one submodule F .We will now construct a canonical OY -module map

obst(λ) : Fil1(A0)→ Lie(A)/F ,

the obstruction to deforming λ, whose zero locus is Y ∩Z. The scheme U may be covered byopen subschemes Ui with the property that on each Ui either N(r) ∈ O×Ui or rOUi = dkOUi .In the construction of obst(λ) we are free to assume that U itself satisfies one of these twoproperties.

The construction of obst(λ) is quite simple if N(r) ∈ O×U , so we treat this case first.Under this hypothesis, λ determines an Ok-linear map λ : HdR

1 (A0) → HdR1 (A) of OY -

modules, which, by the deformation theory of [Lan, Proposition 2.1.6.4] extends canonicallyto an Ok-linear map


1 (A)

of OY -modules. Define obst(λ) as the composition

(7.4.6) Fil1(A0)→ HdR1 (A0)

λ−→ HdR1 (A)→ Lie(A)/F .

Lemma 7.4.4. If N(r) ∈ O×U , the zero locus of obst(λ) is Y ∩ Z.

Proof. Using the notation of (7.4.6), denote by obst∗(λ) the composition

Fil1(A0)→ HdR1 (A0)

λ−→ HdR1 (A)→ Lie(A),

so that obst(λ) is the composition

Fil1(A0)obst∗(λ)−−−−−−→ Lie(A)→ Lie(A)/F .


By deformation theory, the zero locus of obst∗(λ) is the maximal closed subscheme of Y

over which λ extends to an element of r−1L(A0, A). For such an extension the vanishing of

(5.1.2) is automatic by Remark 5.1.7, and so the zero locus of obst∗(λ) is Y ∩ Z. Thus weare reduced proving that obst∗(λ) and obst(λ) have the same zero locus.

If dk ∈ O×U the argument is simple, and exploits the splitting

Ok ⊗Z OY ∼= OY ×OY .

The orthogonal idempotents on the right hand side induce a splitting N = eN ⊕ eN of anyOk⊗ZOY -module N , in which eN is the maximal submodule on which Ok acts through thestructure map Ok → OY , and eN is the maximal submodule on which Ok acts through the

complex conjugate. Kramer’s signature condition on F implies that F = eLie(A), and so Fadmits a canonical rank one OY -module direct summand eLie(A) on which Ok acts through

the complex conjugate of the structure morphism. As Ok also acts on Fil1(A0) through thecomplex conjugate of the structure morphism, the image of the Ok-linear map obst∗(λ) is

contained in eLie(A) ∼= Lie(A)/F . Thus obst∗(λ) vanishes if and only if obst(λ) vanishes,as desired.

Returning to the general case, clearly the zero locus of obst(λ) contains the zero locus

of obst∗(λ). To prove the other inclusion, let V ⊂ Y be the vanishing locus of obst(λ). Wewill prove that obst∗(λ)|V = 0. Fix a geometric point y ∈ V (F) of V and let R be the etale

local ring of V at y. Denote by (A0,A) the pullback of (A0, A) through Spec(R)→ V → Y ,and similarly denote by


1 (A)

the pullback of λ : HdR1 (A0)→ HdR

1 (A). Using Lemma 7.4.3, there is an R-basis ε1, . . . , εnof Lie(A) such that ε1, . . . , εn−1 generates the corank one R-submodule FA ⊂ Lie(A), andJ acts on Lie(A) as

J =

0 · · · 0 j1...

. . ....

...0 · · · 0 jn

∈Mn(R),

where j1, . . . , jn ∈ R satisfy (j1, . . . , jn) = (jn).Fix an Ok ⊗Z R-module generator σ ∈ HdR

1 (A0), and write

λ(σ) = λ1ε1 + · · ·+ λnεn ∈ Lie(A)

with λi ∈ R. Consider obst∗(λ)|R, which is the composition


λ−→ HdR1 (A0)→ Lie(A)

and obst(λ)|R, which is the composition of obst∗(λ)|R with the quotient map

Lie(A)→ Lie(A)/FA.

In particular, the image of obst(λ)|R is generated by

λ(Jσ) = λnjnεn ∈ Lie(A)/FA.

On the other hand, we know from the definition of V that obst(λ)|R = 0, and so λnjn =0. As in the proof of Proposition 5.1.9, Fil1(A0) = JHdR

1 (A0). Therefore the image ofobst∗(λ)|R is generated by

λ(Jσ) = J · λ(σ) = λn(j1ε1 + · · ·+ jnεn) ∈ Lie(A),


which also vanishes, as (j1, . . . , jn) = (jn) and λnjn = 0. This shows that obst∗(λ)|R = 0for every etale local ring R of V , we deduce that obst∗(λ)|V = 0.

We now give the construction of obst(λ) assuming rOU = dkOU . Lemma 7.4.3 implies

that the OY -module Lie(A) has a canonical Ok-linear endomorphism j satisfying J = δk j.As the closed subscheme Y → Y is defined by a square-zero ideal sheaf, the map δkλ : A0 →A induces a map


1 (A).

Once again using Fil1(A0) = JHdR1 (A0), define obst(λ) by

(7.4.7) obst(λ)(Js) = j · δkλ(s) ∈ Lie(A)/F

for every s ∈ HdR1 (A0).

Remark 7.4.5. To see that (7.4.7) is well-defined, use the local freeness of HdR1 (A0) over

Ok⊗ZOY and the exactness of (7.4.4). If Js1 = Js2 then s1−s2 ∈ JHdR1 (A0). The signature

condition on F implies that J annihilates Lie(A)/F , and therefore δkλ(s1) = δkλ(s2) in

Lie(A)/F .

Remark 7.4.6. If δk ∈ O×U , so that N(r) ∈ O×U and rOU = dkOU , then the constructions(7.4.6) and (7.4.7) agree.

Lemma 7.4.7. If rOU = dkOU , the zero locus of obst(λ) is Y ∩ Z.

Proof. Recall that Y ∩Z is the maximal closed subscheme of Y over which λ extends to anelement of r−1L(A0, A) satisfying the vanishing of (5.1.2). By deformation theory, this isthe same as the maximal closed subscheme over which the compositions

Fil1(A0)→ HdR1 (A)

δkλ−−→ HdR1 (A)→ Lie(A)

and

HdR1 (A0)

δkλ−−→ HdR1 (A0)→ Lie(A)→ Lie(A)/F

vanish. Using JF = 0 and Fil1(A0) = JHdR1 (A0), one easily checks that the vanishing

of the second composition implies the vanishing of the first. Using the invertibility of j onLie(A)/F , the vanishing of the second composition is equivalent to the vanishing of obst(λ),completing the proof of the lemma.

The following result is reminiscent of the classical adjunction isomorphism as in [Liu,Lemma 9.1.36], however our result is particular to the moduli space M. Indeed, it is astatement about the cotautological bundle T, which we have defined using the moduli inter-pretation of M.

Theorem 7.4.8 (Adjunction). Assuming that Z is Y -improper, there is a canonical iso-morphism

(7.4.8) O(Z)|Y ∼= TL |Yof line bundles on Y .

Proof. View the obstruction

obst(λ) : Fil1(A0)→ Lie(A)/Fas a section

obst(λ) ∈ Γ(TL |Y )


with zero locus Y ∩ Z. Under the inclusion OU ⊂ O(Z) of OU -modules, the constantfunction 1 on U defines a section s ∈ Γ(O(Z)) with zero locus Z. Hence the restriction

s|Y ∈ Γ(O(Z)|Y ) also has zero locus Y ∩ Z.After passing to a Zariski open cover of U , we are free to assume that U = Spec(R) is

affine, and that the line bundles O(Z) and TL |U are trivial. Fix isomorphisms of R-modulesΓ(O(Z)) ∼= R and Γ(TL |U ) ∼= R, and let I ⊂ R be the ideal defining the closed subschemeY ⊂ U . Note that R/I is an integral domain of characteristic 0. Let f, g ∈ R/I2 be theelements corresponding to the sections obst(λ) and s|Y . As these sections have the samezero locus, (f) = (g) and we may write f = vg and g = uf for some u, v ∈ R/I2. Inparticular g · (1− uv) = 0.

Suppose we have some x ∈ R/I2 satisfying gx = 0. The claim is that x ∈ I/I2. Supposenot. The element g ∈ R/I2 comes with a lift to R, defined by s, and we fix any lift of x to R,necessarily with x 6∈ I. In particular x is a unit in the localization RI . Therefore g ∈ I2RIand the natural surjection RI → RI/(g) induces an isomorphism on tangent spaces. This isa contradiction, as we know from the smoothness of the generic fibers of U and Z that thecompleted local rings of RI and RI/(g) are power series rings (over some finite extension ofk) in n− 1 and n− 2 variables, respectively.

If we apply the above to x = 1− uv we see that 1 = uv in R/I. Therefore the map

Γ(TL |Y ) ∼= R/I2 u−→ R/I2 ∼= Γ(O(Z)|Y ),

which takes obst(λ) 7→ s|Y , becomes an isomorphism after tensoring with R/I ∼= OY .Although g = uf does not determine u uniquely, any other such u has the same image inR/I. In view of the discussion above, the desired isomorphism (7.4.8) may be defined asfollows: there is some homomorphism

TL |Y → O(Z)|Ysatisfying obst(λ) 7→ s|Y . Such a homomorphism is not unique, but any two have the samerestriction to Y , and this restriction is an isomorphism.

At last we construct the promised section (7.4.1). Fix one connected component Y ⊂Y(L0,Λ)/U , and regard ZL (m, r) as a line bundle on ML . Its pullback to a line bundle on Usatisfies

ZL (m, r)|U ∼=⊗Z

O(Z),

where the tensor product is over all connected components Z ⊂ ZL (m, r)/U . Combiningthe adjunction isomorphism (7.4.8) with Proposition 7.4.2 yields an isomorphism

TRΛ(m,r)L |Y ∼=

⊗Y -improper Z

O(Z)|Y ,

and hence an isomorphism

ZL (m, r)|Y ∼= TRΛ(m,r)L |Y ⊗

⊗Y -proper Z

O(Z)|Y ,

which we rewrite as

(7.4.9) Z♥L (m, r)|Y ∼=⊗

Y -proper Z

O(Z)|Y .


Each line bundle O(Z) ⊃ OU has a canonical section s ∈ Γ(O(Z)), corresponding to theconstant function 1 in OU , satisfying

div(s|Y ) = div(s) ∩ Y = Z ∩ Y

as divisors on Y . Therefore (7.4.9) determines a section

σm,r|Y ∈ Γ(Z♥L (m, r)|Y )

corresponding to the section ⊗s|Y on the right hand side of (7.4.9), and this section satisfies

(7.4.10) div(σm,r|Y ) =∑

Y -proper Z

(Z ∩ Y ).

Note that each s|Y appearing in the tensor product is nonvanishing at the generic point ofY , precisely because the tensor product is over only the Y -proper Z’s. In particular σm,r|Yis nonzero.

By repeating the above construction on each connected component Y of Y(L0,Λ)/U we

obtain a section of the pullback of Z♥L (m, r) to Y(L0,Λ)/U . As U varies over a cover of ML bysufficiently small etale opens, these sections (being truly canonical) agree on the overlaps,and the desired section (7.4.1) is defined by patching them together.

Proposition 7.4.9. As divisors on Y(L0,Λ), we have

div(σm,r) =∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r).

Proof. As in the construction of σm,r, fix a sufficiently small etale open subscheme U →ML and a connected component Y ⊂ Y(L0,Λ)/U . It follows from (5.3.2) that there is anisomorphism of Y -schemes⊔

Z

(Z ∩ Y ) ∼=⊔

m1,m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r)/Y

where the disjoint union on the left is over all connected components Z ⊂ ZL (m, r)/U . Eachside of this isomorphism has a well-defined 0-dimensional part: the disjoint union of all its0-dimensional connected components. Taking the 0-dimensional parts, using Theorem 5.3.2and Proposition 5.3.8 for the right hand side, and then viewing the 0-dimension parts asdivisors on Y , we find ∑

Y -proper Z

(Z ∩ Y ) =∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r)/Y .

Combining this with (7.4.10) shows that

div(σm,r)/Y =∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

X(L0,Λ)(m1,m2, r)/Y ,

and the claim follows immediately.


7.5. Adjunction in the complex fiber. Fix one point y ∈ Y(L0,Λ)(C). We will givea purely analytic construction of the fiber of σm,r at y using the complex uniformization(6.1.2). Recall from Section 6.1 that to the point y there is associated a triple (A0,A1,B)of hermitian Ok-modules such that L(A0,B) ∼= Λ and

L(A0,A1) ∈ gen(L0(∞)).

Set A = A1 ⊕B so that L(A0,A) ∈ gen(L (∞)) and

L(A0,A) ∼= L(A0,A1)⊕ Λ.

Recall that the connected component of ML (C) containing y admits an orbifold presentation

Γ\D → ML (C)

in which D is the space of negative lines in L(A0,A)R, and that under this presentation thepoint y corresponds to the negative line

L(A0,A1)R ⊂ L(A0,A)R.

Denote by Z(m, r) the pullback to D of the divisor ZL (m, r). By (6.1.4) the correspondingline bundle is

(7.5.1) Z(m, r) ∼=⊗

λ∈r−1L(A0,A)〈λ,λ〉=m

O(λ).

On the right hand side O(λ) is the line bundle on D defined by the divisor D(λ) of negativelines orthogonal to λ. We must explain the meaning of the infinite tensor product on theright. Denote by s(λ) the constant function 1 on D, viewed as a section of O(λ). Forany open set U ⊂ D with compact closure there are only finitely many λ ∈ r−1L(A0,A)satisfying 〈λ, λ〉 = m for which D(λ)∩U 6= ∅. For λ not in this finite set the section s(λ) isnonvanishing on U , and defines a trivialization of O(λ). Thus, after restricting to any suchU all but finitely many of the factors of (7.5.1) are trivialized, and the meaning of (7.5.1) isclear. The line bundle (7.5.1) has a canonical section

sm,r =⊗

λ∈r−1L(A0,A)〈λ,λ〉=m

s(λ)

corresponding to the constant function 1 in OD ⊂ Z(m, r).Let T denote the pullback of the cotautological bundle TL to D. The irreducible com-

ponents of the divisor Z(m, r) passing through y are indexed by the set

(7.5.2) λ ∈ r−1Λ : 〈λ, λ〉 = m ⊂ L(A0,A).

Recall from Remark 7.2.3 that at every point z ∈ D there is a canonical isomorphism

Tz ∼= HomC(z,C),

which was called f∨z , but which we now suppress from the notation. For each z ∈ D andeach λ in (7.5.2), denote by λz the orthogonal projection of λ to z. There is a uniqueholomorphic section obstan(λ) ∈ Γ(D, T ) whose fiber at every point z satisfies

obstanz (λ) = 〈·, λz〉.

Of course the zero locus of obstan(λ) is the divisor D(λ), and hence there is a uniqueisomorphism of line bundles O(λ) ∼= T satisfying s(λ) 7→ obstan(λ). This isomorphism isthe analytic analogue of the adjunction isomorphism of Theorem 7.4.8.


Define a holomorphic section

obstanm,r =

⊗λ∈r−1Λ〈λ,λ〉=m

obstan(λ)

of

TRΛ(m,r) =⊗


T.

The pullback of (7.4.2) to D is

(7.5.3) Z♥(m, r) ∼= Z(m, r)⊗ T⊗−RΛ(m,r),

which has the holomorphic section sm,r ⊗ (obstanm,r)

−1.

Lemma 7.5.1. The fiber at y of sm,r⊗ (obstanm,r)

−1 agrees with the fiber at y of the sectionσm,r.

Proof. The main thing to explain is the relation between the analytically constructed sectionobstan(λ) and the algebraically constructed section obst(λ) of the previous subsection. Wewill express both constructions in terms of parallel transport with respect to the Gauss-

Manin connection. Let R = OD,y be the completion of the ring of germs of holomorphicfunctions at y. Equivalently, R is the completed etale local ring of ML /C at y, a powerseries ring over C in n − 1 variables. Let (A0, A) ∈ ML (C) be the pair represented by thepoint y, and let (A0,A) be the universal deformation of (A0, A) over R. If m ⊂ R is the

maximal ideal, set R = R/m2. Thus Spec(R) is the first order infinitesimal neighborhood

of y. Denote by (A0, A) the reduction of (A0,A) to R. Let

T = HomR(Fil1(A0),Lie(A)/FA)

be the pullback to R of the cotautological bundle, and let T be its reduction to R.Each λ in the set (7.5.2) determines a C-linear map λ : HdR

1 (A0) → HdR1 (A), which,

using the Gauss-Manin connection, has a canonical deformation


1 (A)

defined by parallel transport. See [Lan] and [Vo] for the Gauss-Manin connection, and[BeOg] for the algebraic theory of parallel transport. The composition


λ−→ HdR1 (A)→ Lie(A)/FA,

viewed as an element of T , is precisely the pullback of obstan(λ) to T . On the other hand,

the reduction of λ to R is precisely the map


1 (A)

appearing in (7.4.6). Thus the reduction map T → T sends obstan(λ) 7→ obst(λ). Withthis in mind, the rest of the proof follows by tracing through the definitions of the twosections in question.

Define a metrized line bundle on ML by

(7.5.4) Z♥L (fm,r) = ZL (fm,r)⊗ T⊗−RΛ(m,r)L .


Proposition 7.5.2. For any point y ∈ Y(L0,Λ)(C), the section σm,r constructed in Section7.4 satisfies

− log ||σm,r||2y = ΦL (y, fm,r)

with respect to the metric on Z♥L (fm,r)|y determined by (7.5.4).

Proof. The metrized line bundle (7.5.4) pulls back to a metrized line bundle on D, whoseunderlying line bundle is (7.5.3). It is easy to compute the norm of the section sm,r ⊗(obstan

m,r)−1 with respect to this metric. For any z ∈ D not contained in the support of

Z(m, r), the section sm,r satisfies

− log ||sm,r||2z = ΦL (z, fm,r),

by definition of the metric on ZL (fm,r), while

log ||obstanm,r||2z =

∑λ∈r−1Λ〈λ,λ〉=m

log ||obstan(λ)||2z =∑


(log |〈λz, λz〉|+ log(4π) + γ

),

by definition (7.2.1) of the metric on TL . It follows that

− log ||σm,r||2y = limz→y

(− log ||sm,r||2z + log ||obstan

m,r||2z)

= limz→y

(ΦL (z, fm,r) +

∑λ∈r−1Λ〈λ,λ〉=m

(log |〈λz, λz〉|+ log(4π) + γ

)).

By Corollary 3.3.2, this is the value at y of the (discontinuous) function ΦL (z, fm,r).

7.6. Completion of the proof. Again we consider the metrized line bundle

Z♥L (fm,r) = ZL (fm,r)⊗ T⊗−RΛ(m,r)L

on ML . Its restriction to Y(L0,Λ) has a canonical nonzero section (7.4.1), which, as in Section7.1, determines an arithmetic divisor

div(σm,r) = (div(σm,r),− log ||σm,r||2) ∈ Div(Y(L0,Λ))

satisfying

[Z♥L (fm,r) : YL0,Λ] = deg div(σm,r).

Of course this relation holds for any nonzero section of the restriction of Z♥L (fm,r) to Y(L0,Λ).The advantage of the section σm,r of (7.4.1) is that its construction is sufficiently explicitto allow for the calculation of its arithmetic degree. Indeed, Proposition 7.4.9 implies thatthe arithmetic divisor

divfin(σm,r) = (div(σm,r), 0)

satisfies

(7.6.1) deg divfin(σm,r) =∑

m1∈Q>0

m2∈Q≥0

m1+m2=m

deg X(L0,Λ)(m1,m2, r).

On the other hand, it is immediate from Proposition 7.5.2 that the arithmetic divisor

div∞(σm,r) = (0,− log ||σm,r||2)

satisfies

(7.6.2) deg div∞(σm,r) = ΦL (Y(L0,Λ), fm,r).


At last we have all the necessary ingredients to prove Theorem 7.3.1.

Proof of Theorem 7.3.1. Combining (7.6.1) with Proposition 7.3.2 shows that

deg divfin(σm,r) = −degC Y(L0,Λ)

∑m1∈Q>0

m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r).

The m1 = 0 term absent from the right hand side instead appears in the equality

[T⊗RΛ(m,r)L : Y(L0,Λ)] = −degC Y(L0,Λ) · a+

L0(0, r) ·RΛ(m, r)

of Theorem 7.2.4, and combining all of this with (7.6.2) gives the final equality in

[ZL (fm,r) : Y(L0,Λ)] = [T⊗RΛ(m,r)L : Y(L0,Λ)] + [Z♥L (fm,r) : Y(L0,Λ)]

= [T⊗RΛ(m,r)L : Y(L0,Λ)] + deg divfin(σm,r) + deg div∞(σm,r)

= ΦL (Y(L0,Λ), fm,r)− degC Y(L0,Λ)

∑m1,m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r).

Corollary 6.4.3 shows that

1

degC Y(L0,Λ)· ΦL (Y(L0,Λ), fm,r) = −L′

(ξ(fm,r), θΛ, 0

)+ c+m,r(0, 0) · a+

L0(0, r) ·RΛ(0, r)

+∑

m1,m2∈Q≥0

m1+m2=m

a+L0

(m1, r) ·RΛ(m2, r),

and hence

1

degC Y(L0,Λ)· [ZL (fm,r) : Y(L0,Λ)] = −L′

(ξ(fm,r), θΛ, 0

)+ c+m,r(0, 0) · a+

L0(0, r) ·RΛ(0, r).

Another application of Theorem 7.2.4 then shows that

[ZL (fm,r) : Y(L0,Λ)] = −degC Y(L0,Λ) · L′(ξ(fm,r), θΛ, 0

)− c+m,r(0, 0) · [TL : Y(L0,Λ)],

and completes the proof.

8. Arithmetic cycles classes and derivatives of L-functions

Throughout Section 8 we fix a pair (L0,Λ) as in Sections 6 and 7, and set L = L0⊕Λ asin Remark 5.2.3. Let ∆ be the automorphism group of the finite quadratic space d−1

k Lf/Lf ,and fix a ∆-invariant harmonic Maass form f ∈ H2−n(ωL ) with holomorphic part

(8.0.3) f+(τ) =∑m∈Qm0

c+(m) · qm.

The goal of Section 8 is to reformulate Theorem 7.3.1 in terms of the arithmetic Chowgroup of the canonical toroidal compactification of ML , and to explain the connection witharithmetic theta lifts in the sense of Kudla.


8.1. Compactification and the arithmetic Chow group. The moduli space M(n−1,1)

defined in Section 5.1 admits a canonical toroidal compactification M∗(n−1,1). Over Ok[1/dk]

the construction is found in [Lan]; the extension to Ok is contained in [Ho3]. The Ok-stack

M∗ = M(1,0) × M∗(n−1,1)

is regular, proper and flat over Ok of relative dimension n− 1, and smooth over Ok[1/dk].It contains M as a dense open substack, and the boundary M∗ r M, when endowed with itsreduced substack structure, is proper and smooth over Ok of relative dimension n − 2.Exactly as in (5.1.7), there is a decomposition

M∗ =⊔L

M∗L

in which M∗L is, by definition, the Zariski closure of ML in M∗.

Let CH1

R(M∗L ) be the arithmetic Chow group with real coefficients and log-log growthalong the boundary in the sense of Burgos-Kramer-Kuhn [BKK, BBK]. For concreteness,we use the definition of [Ho3, Section 3.1] (which is slightly different from [BKK, BBK]; see[Ho3, Remark 3.1.4]). The canonical map Y(L0,Λ) → ML , induces a linear functional

CH1

R(M∗L )→ R,

the arithmetic degree along Y(L0,Λ), defined in [Ho3, Section 3.1] and denoted Z 7→ [Z, Y(L0,Λ)].Extend this functional C-linearly to

CH1

C(M∗L ) = CH1

R(M∗L )⊗R C.

Let BL = M∗L r ML be the boundary of M∗L . We will now define a divisor BL (f) on M∗L ,supported on BL . As the boundary is smooth, it suffices to assign a multiplicity to eachconnected component of the generic fiber BL /k. Start with a component C of the geometricfiber BL /kalg . This component lies on some connected component of M∗L /kalg , which, as in

Section 6.1, corresponds to a pair (A0,A) with L(A0,A) ∈ gen(L (∞)). As in Section 4,the component C then corresponds to the Γ(A0,A)-orbit of an isotropic k-line I ⊂ L(A0,A)Q.Let a = I ∩ L(A0,A). By Lemma 4.3.1 there is a decomposition

L(A0,A) = E(C)⊕ a⊕ b

in which b is an isotropic Ok-submodule of rank one, and E(C) is orthogonal to a⊕b. Underany such decomposition, E(C) is a self-dual hermitian Ok-module of signature (n − 2, 0).The multiplicity of C with respect to f is defined, in accordance with (4.3.3) and Remark4.3.4, as follows. If n > 2 then

(8.1.1) multC(f) =∑

m∈Q>0

m

n− 2

∑λ∈d−1

k E(C)〈λ,λ〉=m

∑ν∈d−1

k a/a

c+(−m,λ+ ν),

where c+(−m) is viewed as a function on

d−1k E(C)/E(C)⊕ d−1

k a/a ⊂ d−1k L(A0,A)/L(A0,A) ∼= d−1

k Lf/Lf .

If n = 2 then E(C) = 0, and in view of the second formula of Remark 4.3.4 we instead define

(8.1.2) multC(f) = −2∑

m∈Z≥0

∑ν∈d−1

k a/a

c+(−m, ν)σ1(m).


Exactly as in [Ho3, Section 3.7], the isomorphism class of the hermitian module E(C) isconstant on the Gal(kalg/k)-orbit of C, and so summing over all geometric components C

yields a divisor

BL (f) =∑C

multC(f) · C

on M∗L /k. Denote in the same way the divisor on M∗L obtained by taking the Zariski closure.

Let Z∗L (f) be the Zariski closure in M∗L of the Kudla-Rapoport divisor ZL (f), and definethe total Kudla-Rapoport divisor

ZtotalL (f) = Z∗L (f) + BL (f).

The analysis, carried out in Section 4.4, of the Green function ΦL (f) near the boundaryshows that the pair

ZtotalL (f) =

(Ztotal

L (f),ΦL (f))

defines a class in CH1

C(M∗L ).The universal abelian scheme Auniv → M(n−1,1) extends to a semi-abelian scheme A∗ →

M∗(n−1,1), and the universal subsheaf Funiv ⊂ Lie(Auniv) extends canonically to a subsheaf

F∗ ⊂ Lie(A∗) by [Ho3, Theorem 2.5.2]. The cotautological bundle T of Section 7.2 thereforeextends to a line bundle

T∗ = Hom(Fil1(Auniv0 ),Lie(A∗)/F∗)

on M∗, and the restriction of T∗ to M∗L is denoted T∗L . Fix a rational section s of T∗L , andrecall the metric (7.2.1) on TL . It follows from the proof of Theorem 4.3.3 (see especiallystep 2) that

div(s) = (div(s),− log ||s||2)

is an arithmetic divisor on M∗L in the sense of [Ho3, Section 3.1], and defines a class

T∗L ∈ CH1

R(M∗L ),

which does not depend on the choice of s.

8.2. The main result, first form. Theorem 7.3.1 relates the central derivative of aRankin-Selberg L-function to the arithmetic intersection against Y(L0,Λ) of a certain metrizedline bundle on the open Shimura variety ML . We now reformulate this result in terms ofthe arithmetic Chow group of the compactified Shimura variety M∗L .

Definition 8.2.1. The arithmetic theta lift of f is the class

ΘL (f) = ZtotalL (f) + c+(0, 0) · T∗L ∈ CH

1

C(M∗L ).

Theorem 8.2.2. The arithmetic theta lift satisfies

[ΘL (f) : Y(L0,Λ)] = −degC Y(L0,Λ) · L′(ξ(f), θΛ, 0),

where

ξ : H2−n(ωL )→ Sn(ωL ).

the complex-conjugate-linear homomorphism of (2.2.3).


Proof. If f = fm,r then this is immediate from Theorem 7.3.1. If f = c+(0) is a constantfunction (this can only happen when n = 2) then ZL (f) = 0 and Theorems 6.4.1 and 7.2.4imply

[ΘL (f) : Y(L0,Λ)] = ΦL (Y(L0,Λ), f) + c+(0, 0) · [TL : Y(L0,Λ)]

= −degC Y(L0,Λ) · L′(ξ(f), θΛ, 0)

+ degC Y(L0,Λ) · c+(0), EL0 ⊗ θΛ+ c+(0, 0) · [TL : Y(L0,Λ)]

= −degC Y(L0,Λ) · L′(ξ(f), θΛ, 0).

The general case now follows from Lemma 6.2.3.

8.3. A modularity conjecture. In this subsection we assume either that n > 2, or thatn = 2 and d−1

k Lf/L is not isotropic in the sense of Section 6.2. In the case we are

excluding, in which n = 2 and d−1k Lf/L is isotropic, recall that there was some ambiguity

in the definition of fm,r, removed by fiat in Remark 6.2.2, and an unwanted constant termin the decomposition of Lemma 6.2.3.

Conjecture 8.3.1. If ξ(f) = 0 then ΘL (f) = 0.

Of course the conjecture implies that the arithmetic theta lift

ΘL : H2−n(ωL )∆ → CH1

C(M∗L )

factors through ξ to define a complex-conjugate-linear homomorphism (denoted the sameway)

ΘL : Sn(ωL )∆ → CH1

C(M∗L ).

We now give an alternate interpretation of the arithmetic theta lift, in the spirit of Kudla[Ku4]. For any positive m ∈ Q and any divisor r | dk, define the total arithmetic Kudla-Rapoport divisor

ZtotalL (m, r) = Ztotal

L (fm,r) ∈ CH1(ML ).

Extend the definition to m = 0 by setting

ZtotalL (0, r) = T∗L .

Next, define a function

ZtotalL (m) : SL → CH

1

C(M∗L )

by the following rule: given any ∆-invariant ϕ ∈ SL , decompose

ϕ =∑k∈Q/Zr|dk

αk,r · ϕk,r

as a linear combination of the functions ϕk,r ∈ SL of Section 6.2, and define

ZtotalL (m,ϕ) =

∑r|dk

αm,r · ZtotalL (m, r).

Extend the definition to arbitrary ϕ ∈ SL by first projecting to the subspace of ∆-invariants.If we view these functions as elements

ZtotalL (m) ∈ S∨L ⊗C CH

1

C(M∗L ),


then the equality

(8.3.1) ΘL (f) =∑m≥0

c+(−m), Ztotal

L (m)∈ CH

1

C(M∗L )

is trivially verified when f = fm,r, and so holds for all f by Lemma 6.2.3.

Conjecture 8.3.2. The formal generating series

(8.3.2) ΘL (τ) =∑

m∈Q≥0

ZtotalL (m) · qm,

valued in S∨L ⊗C CH1

C(M∗L ), is a modular form of weight n and representation ω∨L . Moreprecisely, (8.3.2) is the q-expansion of a function

ΘL ∈Mn(ω∨L )⊗C CH1

C(M∗L ).

Note that (8.3.1) is simply the constant term of the formal q-expansion f+, ΘL .

Proposition 8.3.3. Conjectures 8.3.2 and 8.3.1 are equivalent. If these conjectures are

true, then for any F ∈ Sn(ωL )∆ the arithmetic theta lift ΘL (F ) is equal to the Peterssoninner product

〈F, ΘL 〉 =

∫SL2(Z)\H

F (τ), ΘL (τ) v2−ndu dv.

Proof. A theorem of Borcherds [Bo2, Theorem 3.1] implies that Conjecture 8.3.2 is equiva-lent to the vanishing of (8.3.1) for every f ∈M !

2−n(ωL )∆. By the exactness of (2.2.4), thisis equivalent to Conjecture 8.3.1. The final claim follows from [BF, Proposition 3.5].

The following theorem provides some evidence for Conjecture 8.3.2. Similarly, the righthand side of Theorem 8.2.2 obviously vanishes if ξ(f) = 0, providing evidence for Conjecture8.3.1.

Theorem 8.3.4. The S∨L valued generating series

[ΘL (τ) : Y(L0,Λ)] =∑

m∈Q≥0

[ZtotalL (m) : Y(L0,Λ)] · qm

is a holomorphic modular form of weight n and representation ω∨L .

Proof. Using the modularity criterion of Borcherds [Bo2, Theorem 3.1], it suffices to showthat ∑

m∈Q≥0

c+(−m), [Ztotal

L (m) : Y(L0,Λ)]

= 0

for every weakly harmonic form f ∈ M !2−n(SL ) with holomorphic part (8.0.3). But this is

clear: ∑m∈Q≥0

c+(−m), [Ztotal

L (m) : Y(L0,Λ)]

= [ΘL (f) : YL ]

by (8.3.1), and the vanishing of the right hand side follows from Theorem 8.2.2, after notingthat ξ(f) = 0 by the exactness of (2.2.4).

Remark 8.3.5. To obtain Theorem B of the introduction, evaluate the S∨L valued modularform of Theorem 8.3.4 at the characteristic function ϕ0 ∈ SL of 0 to obtain a scalar valuedmodular form.


9. Scalar valued modular forms

In this section, we reformulate our main result, Theorem 8.2.2, in terms of scalar valuedmodular forms. The data (L0,Λ) and L = L0 ⊕ Λ remain fixed, as in earlier sections.Abbreviate D = |dk|, and recall our running assumption that D is odd. In particular,D ≡ 3 (mod 4).

9.1. Preliminaries. Recall that χk is the idele class character associated to the quadraticfield k. If we view χk as a Dirichlet character modulo D, then any factorization D = Q1Q2

induces a factorization

χk = χQ1χQ2

where χQi : (Z/QiZ)× → C× is a quadratic Dirichlet character. For each positive divisorQ | D, fix a matrix

RQ =

(α βDQγ Qδ

)∈ Γ0(D/Q)

with α, β, γ, δ ∈ Z, and define the Aktin-Lehner operator

WQ =

(Qα βDγ Qδ

)= RQ

(Q

1

).

Fix a normalized cuspidal new eigenform

g =∑m

a(m)qm ∈ Sn(Γ0(D), χnk).

For every divisor Q | D, define another modular form gQ(τ) =∑aQ(m)qm by the following

conditions:

aQ(m) = χnQ(m)a(m) if (m,Q) = 1,

aQ(m) = χnD/Q(m)a(m) if (m,D/Q) = 1,

aQ(m1m2) = aQ(m1)aQ(m2) if (m1,m2) = 1.

According to [As, Theorem 2], the relation

g|nWQ = χnQ(β)χnD/Q(α)εQ(g) · gQ(9.1.1)

holds for the fourth root of unity εQ(g) defined by

εQ(g) =∏q|Q

χnQ(Q/q) · λq,

where

λq = a(q) ·

−q1−n2 if n ≡ 0 (mod 2),

δqq1−n

2 if n ≡ 1 (mod 2),

and δq is defined by

(9.1.2) δq =

1 if q ≡ 1 (mod 4),

i if q ≡ 3 (mod 4).

Remark 9.1.1. If n is even, then the Fourier coefficients of g are totally real. It follows thatgQ = g, and g|nWQ = εQ(g)g for every divisor Q | D. Furthermore,

εQ(g) = −p1−n2 a(p) = ±1.


Define a vector valued modular form ~g ∈ Sn(ωL )∆ by

~g(τ) =∑

γ∈Γ0(D)\ SL2(Z)

(g|nγ)(τ) · ω(γ−1)ϕ0,(9.1.3)

where ϕ0 ∈ SL is the characteristic function of 0 ∈ d−1k Lf/Lf . The construction g 7→ ~g

defines a surjectionSn(Γ0(D), χnk)→ Sn(ωL )∆

by [Sch2, Corollary 5.5], but we will never need this fact. There is an adjoint construction

Mn−1(ω∨Λ)→Mn−1(Γ0(D), χn−1k )

from S∨Λ valued modular forms to scalar valued modular forms, defined by evaluation atϕ0 ∈ SΛ. This map takes the vector valued theta series θΛ ∈Mn−1(ω∨Λ) to the scalar valuedtheta series

θscΛ (τ) =

∑m∈Z≥0

RscΛ (m) · qm,

where RscΛ (m) is the number of ways to represent m by Λ.

Let GUΛ be the general unitary group associated with the hermitian lattice Λ. For anyZ-algebra R its R-valued points are given by

GUΛ(R) = h ∈ GLOk(Λ⊗Z R) : 〈hx, hy〉 = ν(h)〈x, y〉 for all x, y ∈ Λ⊗Z R,

where ν(h) ∈ R× denotes the similitude factor of h. Note the relation

N(det(h)) = ν(h)n−1.(9.1.4)

For h ∈ GUΛ(R) the similitude factor ν(h) belongs to R>0. Since Λ is positive definite, theShimura variety

XΛ = GUΛ(Q)\GUΛ(Q)/GUΛ(Z)

has dimension zero. The natural map Resk/QGm → GUΛ to the center induces an action of

the class group CL(k) = k×\k×/O×k on XΛ:

CL(k)×XΛ −→ XΛ, ([α], [h]) 7→ [αh].(9.1.5)

For any h ∈ GUΛ(Q) there exists a unique ν0(h) ∈ Q>0 such that ν(h) ∈ ν0(h)Z×. Wedefine a self-dual hermitian Ok-module of signature (n− 1, 0) by endowing the Ok-module

Λh = (Λ⊗Z Q) ∩ hΛ

with the hermitian form〈x, y〉h = ν0(h)−1〈x, y〉.

Note that if h ∈ GUΛ(Z), then ν0(h) = 1 and Λh = Λ. If h ∈ GUΛ(Q), then Λh ∼= Λ ashermitian modules. Hence, the assignment h 7→ Λh defines a map from XΛ to the set ofisometry classes of self-dual hermitian Ok-module of signature (n− 1, 0).

Similarly, for any h ∈ GUΛ(Q) we define a self-dual incoherent (kR, Ok)-module of sig-nature (1, 0) by endowing

L0,h = det(h)L0

with the hermitian form〈x, y〉h = ν0(h)1−n〈x, y〉.

The assignment h 7→ L0,h defines a map from XΛ to the set of isometry classes of self-dual

(kR, Ok)-module of signature (1, 0).

Lemma 9.1.2. For any h ∈ GUΛ(Q) we have L ∼= L0,h ⊕ Λh.


Proof. Let p be a prime. Recall from Section 2.1 that a hermitian space over kp is determinedby its dimension and invariant. The relations

det(Λh ⊗Z Qp) = ν0(h)1−n det(Λ⊗Z Qp),det(L0,h ⊗Z Qp) = ν0(h)1−n · det(L0 ⊗Z Qp),

combined with (9.1.4), show that L ⊗Z Qp and (L0,h ⊕ Λh) ⊗Z Qp have the same localinvariant, and hence are isomorphic. A result of Jacobowitz [Jac] shows that every self-dual lattice in L ⊗Z Qp is isomorphic to L ⊗Z Zp as a hermitian Ok,p-module, and soL ⊗Z Zp ∼= (L0,h ⊕ Λh)⊗Z Zp. This implies the assertion.

Let η be an algebraic automorphic form for GUΛ which is trivial at∞ and right GUΛ(Z)-invariant, i.e., a function

η : XΛ −→ C.

Throughout we assume that under the action (9.1.5) of CL(k) on XΛ the function η trans-forms with a character χη : CL(k)→ C×, that is,

η(αh) = χη(α)η(h).(9.1.6)

We associate a theta function with η by putting

θscη,Λ =

∑h∈XΛ

η(h)

|Aut(Λh)|· θsc

Λh∈Mn−1(Γ0(D), χn−1

k ).

This form is cuspidal when the the character χη is non-trivial. We denote its Fourierexpansion by

θscη,Λ(τ) =

∑m≥0

Rscη,Λ(m) · qm.

Similarly, we may define

(9.1.7) θη,Λ(τ) =∑h∈XΛ

η(h)

|Aut(Λh)|· θΛh(τ),

but this is only a formal sum: as h varies the forms θΛh take values in the varying spacesS∨Λh .

Lemma 9.1.2 allows us to make sense of the convolution L-function L(~g, θΛh , s) of Section6.4, and to define

(9.1.8) L(~g, θη,Λ, s) =∑h∈XΛ

η(h)

|Aut(Λh)|· L(~g, θΛh , s).

In the next subsection we will compare this to the convolution L-function

(9.1.9) L(g, θscη,Λ, s) = Γ

(s2

+ n− 1) ∞∑m=1

a(m)Rscη,Λ(m)

(4πm)s2 +n−1

of the scalar valued forms g and θscη,Λ.


9.2. Rankin-Selberg L-functions for scalar and vector valued forms. In this sub-section we prove a precise relation between (9.1.8) and (9.1.9). First, we give an explicitformula for the Fourier coefficients of ~g in terms of those of g.

Recall that V is the hermitian space over Ak attached to L . Let ψ =∏ψp be the

usual unramified additive character of Q\A, satisfying ψ∞(x) = e(x). Recall that the Weilrepresentation ω = ωV ,ψ of SL2(A) on the space of Schwartz-Bruhat functions S(V ) isdetermined by the formulas

ω(n(b))ϕ(x) = ψ(b〈x, x〉)ϕ(x)

ω(m(a))ϕ(x) = χnk(a)|a|n+1ϕ(ax)

ω(w)ϕ(x) = γ(V )

∫V

ϕ(y)ψ(− trk/Q〈x, y〉) dy

for all b ∈ A and a ∈ A×, where

n(b) =

(1 b0 1

), m(a) =

(a 00 a−1

), w =

(0 −11 0

),

dy is the self-dual Haar measure with respect to ψ(trk/Q〈x, y〉), and γ(V ) =∏p≤∞ γp(V )

is a certain 8th root of unity (the product of the local Weil indices). For any Q | D, define

γQ(V ) =∏p|Q

γp(V ).

Lemma 9.2.1. Every finite prime p satisfies

γp(V ) =

δ−np · χnk,p(p) · invp(V ) if p | D,invp(V ) if p - D,

where δp is defined by (9.1.2), and invp(V ) is the local invariant of Vp defined in Section2.1.

Proof. Let V 0p be Vp with the Qp-quadratic form Q(x) = 〈x, x〉. This is slightly different

from the space RVp of [Ku1]. By [Ku1, Theorem 3.1] and [Ku1, Lemma 3.4], one has (inthe notation of [loc. cit.]) that

γp(V ) = βVp(w) = γQp(ψp V 0p )−1 = γQp(det V 0, ψp)

−1γQp(ψ)−2nhQp(V 0p )−1.

Here hQp(V 0p ) is the Hasse invariant of V 0

p , and det V 0p = (det Vp)2Dn is the determinant

of V 0p . A direct calculation gives

hQp(V 0p ) = (D,D)

n(n−1)2

p invp(V ).

On the other hand, [Ra, Appendix A.5] implies that

γQp(Dn, ψp)γQp(ψ)2n = γQp(D,ψp)n(D,D)

n(n−1)2

p γQp(−1, ψp)−n

= γQp(−D,−ψp)n(−D,−1)np (D,D)n(n−1)

2p .

Applying [Ra, Proposition A.11] and [Ra, Proposition A.9] shows that

γQp(−D,−ψp) =

(p,D)pδp if p | D,1 if p - D.


Putting everything together and using χk,p(x) = (−D,x)p yields the formula in the lemma.For example, when p|D,

γp(V )−1 = γQp(Dn, ψp)γQp(ψ)2nhQp(V 0p )

= δnp invp(V )(D, p)np (−D,−1)np

= δnpχnk,p(p)invp(V )(−Dp,−1)np

= δnpχnk,p(p)invp(V ),

as (−Dp,−1)p = (−D/p,−1)p = 1.

Remark 9.2.2. If n is even and p | D, the equality of Lemma 9.2.1 simplifies to

γp(V ) =

(−1

p

)n/2invp(V ).

Remark 9.2.3. In the special case where D is prime, the root of unity γD(V ) can be madeeven more explicit. Indeed, the self-duality of L implies that invp(V ) = 1 for all p 6= D,and so the incoherence condition on V forces invD(V ) = −1. Similar reasoning shows thatχk,p(p) = 1, and so

γD(V ) = i−ninvD(V ) =

(−1)

n2 +1 if n is even,

in if n is odd.

For any Q | D and µ ∈ d−1k Lf/Lf , define

Qµ =∏p|Dµp 6=0

p,

where µp is the image of µ in d−1k Lf,p/Lf,p. Let ϕµ ∈ SL be the characteristic function of

µ.

Proposition 9.2.4. Let g =∑m a(m)qm and ~g be as in the previous subsection. Define

a(m,µ) =∑

Qµ|Q|D

Q1−nεQ(g)γQ(V )aQ(mQ)

and ~a(m) =∑µ a(m,µ)ϕµ ∈ SL . Then ~g has the Fourier expansion

~g(τ) =∑

m∈Q>0

~a(m)qm.

Proof. The statement can be deduced from the definition (9.1.3), using the decomposition

SL2(Z) =⊔Q|D

⊔j∈Z/QZ

Γ0(D)RQn(j)

and identity (9.1.1). We refer the reader to [Sch1, Theorem 6.5] for a more general result.

Proposition 9.2.5. The convolution L-function (6.4.1) satisfies

L(~g, θΛ, s) =∑Q|D

Qs2 εQ(g)γQ(V )L(gQ, θ

scΛq, s),


where q ∈ k× is such that q2O×k = QO×k . Moreover, for any η : XΛ → C satisfying (9.1.6)the L-functions (9.1.8) and (9.1.9) are related by

L(~g, θη,Λ, s) =∑Q|D

Qs2 εQ(g)γQ(V )χη(q−1)L(gQ, θ

scη,Λ, s).

Proof. Proposition 9.2.4 implies

L(~g, θΛ, s)

Γ( s2 + n− 1)=

∑µ∈d−1

k Λ/Λ

∑m∈Q>0

∑Qµ|Q|D

Q1−nεQ(g)γQ(V ) · aQ(mQ)RΛ(m,ϕµ)

(4πm)s2 +n−1

=∑Q|D

Q1−nεQ(g)γQ(V )∑

m∈ 1QZ>0

aQ(mQ)

(4πm)s2 +n−1

∑µ∈d−1

k Λ/ΛQµ|Q

RΛ(m,ϕµ)

=∑Q|D

Qs2 εQ(g)γQ(V )

∑m∈Z>0

aQ(m)

(4πm)s2 +n−1

∑µ∈d−1

k Λ/ΛQµ|Q

RΛ(m/Q,ϕµ).

The first claim now follows from the relation∑µ∈d−1

k Λ/ΛQµ|Q

RΛ(m/Q, µ) = RΛq−1 (m, 0) = RΛq(m, 0).

For the second claim, if we replace Λ by Λh and L0 by L0,h for h ∈ XΛ, then L and γQ(V )remain unchanged. The above calculations therefore imply that

L(~g, θη,Λ, s) =∑Q|D

εQ(g)γQ(V )Qs2

∑h∈XΛ

η(h)

|Aut(Λh)|L(gQ, θ

scΛqh

, s)

=∑Q|D

εQ(g)γQ(V )Qs2

∑h∈XΛ

η(q−1h)

|Aut(Λh)|L(gQ, θ

scΛh, s)

=∑Q|D

Qs2 εQ(g)γQ(V )χη(q−1)L(gQ, θ

scη,Λ, s).

Here we have used (9.1.6) and the fact that |Aut(Λh)| is equal to |Aut(Λqh)|. This provesthe proposition.

Corollary 9.2.6. If n is even, then

L(~g, θη,Λ, s) = L(g, θscη,Λ, s) ·

∏p|D

(1 + χη(p−1)εp(g)γp(V )p

s2

).

Proof. This is immediate from Proposition 9.2.5 and Remark 9.1.1.

9.3. The main result, second form. Now we are ready to state the main result in termsof scalar valued modular forms. For every h ∈ XΛ, the isomorphism L ∼= L0,h ⊕ Λh ofLemma 9.1.2 allows us to define a codimension n− 1 cycle

Y(L0,h,Λh) −→ M∗L .


Recall that the degree degC Y(L0,h,Λh) is given by the explicit formula in Remark 6.1.4. Eachalgebraic automorphic form η : XΛ → C satisfying (9.1.6) now determines a cycle

ηY(L0,Λ) =∑h∈XΛ

η(h) · Y(L0,h,Λh)

on M∗L with complex coefficients, and a corresponding linear functional CH1

C(M∗L ) → Cdenoted

Z 7→ [Z : ηY(L0,Λ)].

When the character χη is non-trivial, then ηY(L0,Λ)(C) has degree zero. Under the as-sumption of Proposition 6.1.5, this cycle actually has degree zero on each component ofM∗L (C).

Starting with the form g ∈ Sn(Γ0(D), χnk) of Section 9.1, pick any f mapping to ~g underthe differential operator

ξ : H2−n(ωL ) −→ Sn(ωL )

of (2.2.3), and let

ΘL (f) ∈ CH1

C(M∗L )

be the arithmetic cycle class of Definition 8.2.1. Combining Theorem 8.2.2 with Proposition9.2.5 and Corollary 9.2.6, we obtain the following theorem.

Theorem 9.3.1. In the notation above,

[ΘL (f) : Y(L0,Λ)] = −degC Y(L0,Λ) ·∑Q|D

Qs2 εQ(g)γQ(V )L′(gQ, θ

scΛq, 0),

where q ∈ k× is such that q2O×k = QO×k . Moreover, if n is even, and η : XΛ → C satisfies(9.1.6), then

[ΘL (f) : ηY(L0,Λ)] = −21−o(dk) h2k

w2k

· dds

[L(g, θsc

η,Λ, s) ·∏p|D

(1 + χη(p−1)εp(g)γp(V )p

s2

)]∣∣s=0

,

where p ∈ k× such that p2O×k = pO×k . Note that in the first formula the sum is over allpositive divisors Q | D, while in the second the product is over the prime divisors p | D. Inboth formulas, the choice of L0 intervenes on the right hand side only through the factorsγQ(V ).

9.4. The Rankin-Selberg convolution with an Eisenstein series. In this subsection,we assume throughout that D is prime and that g ∈ Sn(Γ0(D), χnk) is a newform withFourier expansion g =

∑m a(m)qm. Then the twist gχk

=∑χk(m)a(m)qm by the qua-

dratic character χk is a newform in Sn(Γ0(D2), χnk).We consider a special example of an algebraic automorphic form for GUΛ, namely the

function η given by

η(h) =

(∑

Λ1∈gen(Λ)1

|Aut(Λ1)|

)−1

if h ∈ GUΛ(Q) UΛ(Q) GUΛ(Z),

0 otherwise.

Notice that it satisfies (9.1.6) with χη = 1 (since D is prime, every ideal class of k can be

represented by aa−1). We write

Ygen(Λ) = ηYL0,Λ =

( ∑Λ1∈gen(Λ)

1

|Aut(Λ1)|

)−1 ∑Λ1∈gen(Λ)

Y(L0,Λ1).


Consider the normalized genus theta function

E(τ) = κ(n)

(√D

2π

)n−1

L(χn−1k , n− 1)Γ(n− 1)θη,Λ,

where κ(n) = 1/2 or 1 depending on whether n = 2 or n > 2. Its associated scalar formEsc(τ) ∈Mn−1(Γ0(D), χn−1

k ) is a normalized Eisenstein series by the Siegel-Weil formula.In this subsection we relate the derivative L′(~g,E, 0) to derivatives (and values) of the

usual Hecke L-functions of g and gχk. The results will be expressed in terms of the completed

L-functions

Λ(χk, s) = Ds2π−

s+12 Γ

(s+ 1

2

)L(χk, s),

Λ(g, s) =

(√D

2π

)sΓ(s)L(g, s),

Λ(gχk, s) =

(D

2π

)sΓ(s)L(gχk

, s).

Although the method is the same, we obtain slightly different results depending onwhether n is even or odd. We first consider the case that n is even. The next proposi-tion follows from an explicit calculation using [KY] and [Ya, Section 3].

Proposition 9.4.1. Assume that n is even, and write

Esc(τ) = b(0) +

∞∑m=1

b(m)mn−2qm.

Each b(m) has a factorization b(m) = bD(m)bD(m) in which

bD(m) = 1 + (−D)2−n

2 χk,D(m)D(2−n) ordD(m),

and bD(m) =∏p6=D bp(p

ordpm) with

bp(pk) =

1−(χk,p(p)p

2−n)k+1

1− χk,p(p)p2−n

for all p 6= D.

Proposition 9.4.2. If n is even, then

(−4)n−2

2 Dn−1Λ(χk, s+ 1)L(~g,E, s)

= (D−s4 − (−1)

n2 εD(g)D

s4 )

×[Λ(g, 1 +

s

2)Λ(gχk

, 1− s

2) + (−1)

n2 εD(g)Λ(gχk

, 1 +s

2)Λ(g, 1− s

2)].

In particular, when n = 2, one has

DΛ(χk, s+ 1)L(~g,E, s) = 2(D−s4 + εD(g)D

s4 )Λ(g, 1 +

s

2)Λ(gχk

, 1 +s

2).

Proof. Since n is even, the cusp form g has trivial nebentypus character, and we have thefunctional equations

Λ(g, s) = inεD(g)Λ(g, n− s),Λ(gχk

, s) = −inΛ(gχk, n− s).


Here the latter identity follows from the fact that

gχk|WD2 = −gχk

,(9.4.1)

which can be proved similarly as [Mi, Lemma 4.3.10].We abbreviate

L =

∞∑m=1

a(m)b(m)

ms2 +1

=1

(1 + (−1)n2 +1εD(g)D

s2 )· (4π)

s2 +n−1

Γ( s2 + n− 1)· L(~g,E, s).

Here the second equality follows from Corollary 9.2.6 and the relation a(m) = a(m) notedin Remark 9.1.1. Proposition 9.4.1 implies

L =∑

(m,D)=1

a(m)bD(m)

ms2 +1

∞∑k=0

a(D)kbD(mDk)

Dk( s2 +1)

=1

1− a(D)D−s2−1

∑(m,D)=1

a(m)bD(m)

ms2 +1

+(−D)

2−n2

1− a(D)D−( s2 +n−1)

∑(m,D)=1

χk,D(m)a(m)bD(m)

ms2 +1

=1

1− a(D)D−s2−1

∏p 6=D

Lp(1) +(−D)

2−n2

1− a(D)D−( s2 +n−1)

∏p6=D

Lp(χk),

where for an idele class character χ of Q we denote

Lp(χ) =

∞∑k=0

χp(p)ka(pk)bp(p

k)

pk( s2 +1).

Write

Lp(g, s) =1

(1− α(p)p−s)(1− β(p)p−s),

where α(p) and β(p) are the roots of X2 − a(p)X + pn−1. Proposition 9.4.1 now implies

Lp(χ) =Lp(gχ,

s2 + 1)Lp(gχk

, s2 + n− 1)

Lp(χk, s+ 1).

Putting everything together, and using LD(gχk, s) = 1 and LD(g, s) = 1

1−a(D)D−s , gives

L =L(g, s2 + 1)L(gχk

, s2 + n− 1) + (−D)2−n

2 L(gχk, s2 + 1)L(g, s2 + n− 1)

L(χk, s+ 1).

Using the functional equations for Λ(g, s) and Λ(gχk, s), we obtain

(−1)n−2

2 2n−2Dn−1Λ(χk, s+ 1)L(~g,E, s)

= (D−s4 − (−1)

n2 εD(g)D

s4 )

×[(−1)

2−n2 Λ(g,

s

2+ 1)Λ(gχk

,s

2+ n− 1) + Λ(gχk

,s

2+ 1)Λ(g,

s

2+ n− 1)

]= (D−

s4 − (−1)

n2 εD(g)D

s4 )

×[Λ(g, 1 +

s

2)Λ(gχk

, 1− s

2) + (−1)

n2 εD(g)Λ(gχk

, 1 +s

2)Λ(g, 1− s

2)]


as claimed.

Corollary 9.4.3. Assume that n is even, and put

C(n) = (−1)n2

2πn2−1Γ(n2 )

Dn−1Γ(n− 1)Λ(χk, 1)Λ(χk, n− 1).

(1) If n > 2 and (−1)n2 εD(g) = 1, then

[ΘL (f) : Ygen(Λ)] = −C(n) degC Ygen(Λ) · log(D) · Λ(g, 1)Λ(gχk, 1).

(2) If n > 2 and (−1)n2 εD(g) = −1, then

[ΘL (f) : Ygen(Λ)] = 2C(n) degC Ygen(Λ) ·d

ds

[Λ(g, s+ 1)Λ(gχk

, 1− s)]|s=0.

(3) If n = 2, then

[ΘL (f) : Ygen(Λ)] = D−1 ·

log(D) · Λ(g, 1)Λ(gχk

, 1) if εD(g) = −1,

−2Λ′(g, 1)Λ(gχk, 1) if εD(g) = 1.

Note that the case n = 2 is special, as s = 1 is the center of the functional equation ofL(g, s).

Proof. It is clear from Theorem 8.2.2 and the definitions that

[ΘL (f) : Ygen(Λ)] =C(n)

κ(n)· h

2k

w2k

· (−4)n−2

2 Dn−1Λ(χk, 1)L′(~g,E, 0).

If εD(g) = (−1)n2 , then Proposition 9.4.2 implies

(−4)n−2

2 Dn−1Λ(χk, 1)L′(~g,E, 0) = − log(D) · Λ(g, 1)Λ(gχk, 1).

If instead εD(g) = (−1)n2 +1, then

(−4)n−2

2 Dn−1Λ(χk, 1)L′(~g,E, 0) = 2d

ds

[Λ(g, s+ 1)Λ(gχk

, 1− s)]|s=0.

All parts of the claim follow from this, together with the class number formula

C(2)

κ(2)· h

2k

w2k

= − 1

D.

Now we turn to the case where n is odd.

Proposition 9.4.4. Assume that n is odd. The function b(m) defined by

Esc(τ) = b(0) + (1− (−D)1−n

2 )−1∞∑m=1

b(m)mn−2qm

is multiplicative, and satisfies

b(pk) =

1−p(2−n)(k+1)

1−p2−n if p 6= D,

(−D)n−1

2

[−1 + (1 + (−D)

1−n2 ) 1−D(2−n)(k+1)

1−D2−n

]if p = D.

The following results can be proved analogously to Proposition 9.4.2 and Corollary 9.4.3.


Proposition 9.4.5. If n is odd, then

2n−2Dn−1

2 Λ(χk, s+ 1)(1− (−D)1−n

2 )L(~g,E, s)

= (Ds4 − inεD(g)D−

s4 )

×[D−

s4 Λ(g, 1− s

2)Λ(g, n− 1− s

2) + inεD(g)D

s4 Λ(g, 1 +

s

2)Λ(g, n− 1 +

s

2)].

Corollary 9.4.6. Suppose n is odd, and set

C(n) = − hkwk

πn−1

Dn−1Γ(n− 1)ζ(n− 1)

1

(1− (−D)1−n

2 ).

(1) If inεD(g) = 1, then

[ΘL (f) : Ygen(Λ)] = C(n) log(D) · Λ(g, 1)Λ(g, n− 1).

(2) If inεD(g) = −1, then

[ΘL (f) : Ygen(Λ)] = −2C(n) · dds

[D−

s2 Λ(g, 1 + s

)Λ(g, n− 1 + s

)]∣∣s=0

.

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Fachbereich Mathematik, Technische Universitat Darmstadt, Schlossgartenstrasse 7, D–64289Darmstadt, Germany

E-mail address: [email protected]

Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA

02467, USA


Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, Madison, WI53706, USA


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