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MODULI OF ABELIAN SCHEMES AND SERRE’S TENSOR CONSTRUCTION

by

Zavosh Amir-Khosravi

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2013 by Zavosh Amir-Khosravi

Abstract

Moduli of Abelian Schemes and Serre’s Tensor Construction

Zavosh Amir-KhosraviDoctor of Philosophy

Graduate Department of MathematicsUniversity of Toronto

2013

In this thesis we study moduli stacksMnΦ, indexed by an integer n > 0 and a CM-type (K, Φ),

which parametrize abelian schemes equipped with action by OK and an OK-linear princi-

pal polarization, such that the representation of OK on the relative Lie algebra of the abelian

scheme consists of n copies of each character in Φ. We do this by systematically applying

Serre’s tensor construction, and for that we first establish a general correspondence between

polarizations on abelian schemes M⊗R A arising from this construction and polarizations on

the abelian scheme A, along with positive definite hermitian forms on the module M. Next

we describe a tensor product of categories and apply it to the category Hermn(OK) of finite

non-degenerate positive-definite OK-hermitian modules of rank n and the category fibred in

groupoidsM1Φ of principally polarized CM abelian schemes. Assuming n is prime to the class

number of K, we show that Serre’s tensor construction provides an identification of this tensor

product with a substack of the moduli spaceMnΦ, and that in some cases, such as when the

base is finite type over C or an algebraically closed field of characteristic zero, this substack is

the entire space. We then use this characterization to describe the Galois action on MnΦ(Q),

by using the description of the action on M1Φ(Q) supplied by the main theorem of complex

multiplication

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Dedication

Dedicated to the memory of Arzhang Amir-Khosravi, who once to keep a petulant child busy,asked him to find all right triangles with whole number sides. It worked very well.

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Acknowledgements

I would like to thank my advisor, Stephen S. Kudla, for sharing his invaluable mathemati-cal vision and expertise, and for his generosity, wisdom, and infinite patience.

I would also like to thank Professor James Arthur from whom I have had the privilege tolearn much mathematics throughout the years.

I am forever indebted to Ed Barbeau for his early encouragement when I began to studymath, and for tremendous acts of kindness when I asked for his help.

Many thanks to Professor Kumar Murty for the helpful discussions and fruitful sugges-tions.

I would also like to express my gratitude towards Marie Bachtis, Ida Bulat, Abe Igelfeld,and Mike Lorimer for the friendly environment they collectively created at the University ofToronto.

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Contents

0 Introduction 10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Serre’s Construction of Abelian Schemes 61.1 Serre’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 The Dual Abelian Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.3 The Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 Ampleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.4 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3 First Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Tensor Product of Monoidal Categories 312.1 Action by a monoidal category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Tensor product of categories over a monoidal category . . . . . . . . . . . . . . 332.3 Tensor product over a 2-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 The case of a category fibred in groupoids . . . . . . . . . . . . . . . . . . . . . . 38

3 Application to moduli problems 433.1 The moduli stackMn

Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.2 Constructing objects ofMn

Φ(S) . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Homomorphisms between constructed objects inMn

Φ . . . . . . . . . . . . . . . 453.3 Second Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Stackification of Hermn(R)⊗H1 M1

Φ . . . . . . . . . . . . . . . . . . . . . . . . 553.4.1 The essential image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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3.4.2 Complex points ofMnΦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.3 Schemes of finite type over C . . . . . . . . . . . . . . . . . . . . . . . . . 603.5 Criteria for being in the image of Hermn(R)⊗H1 M1

Φ . . . . . . . . . . . . . . . 61

Interlude: Groups with Compact Factors 65

4 Galois action and complex multiplication 694.1 Hermitian fractional ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Galois action on triples over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.2 Description of the action onMn

Φ(Q) . . . . . . . . . . . . . . . . . . . . . 744.3 Double cosets of the unitary group . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 Computation of the reflex norm . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 85

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Chapter 0

Introduction

We start with the precise definition of the moduli spaces in consideration before we proceedwith some brief remarks on the larger context in which they appear. We will end the introduc-tion with an overview of our results.

0.1 Background

Let (K, Φ) be a fixed CM field with a CM type, and let n be a fixed positive integer. We associateto this data a moduli problem of abelian schemes.

Let L denote the reflex field of (K, Φ). Recall that L is obtained by adjoining to Q all num-bers of the form

∑φ∈Φ

φ(a), a ∈ K.

For each locally notherian scheme S over SpecOL, we define a categoryMnΦ(S) as follows.

The objects ofMnΦ(S) are triples (A, ι, λ) where

• A is an abelian scheme over S.

• ι : OK → EndS(A) is a ring homomorphism.

• λ : A → A∨ is an OK-linear principal polarization, so that λ ι(a) = ι(a)∨ λ for alla ∈ OK.

In addition we require that the action of OK on the relative Lie algebra of A over S satisfy thefollowing signature condition:

char(ι(a)|LieS(A), T) = ∏φ∈Φ

(T − φ(a))n, a ∈ OK,

where the right hand side is a polynomial with coefficients in OL, interpreted as a globalsection of OS[T] via the map OL[T] → OS[T] induced by the structure map of S/OL. Inparticular, A will have relative dimension ng over S, where 2g = [K : Q].

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CHAPTER 0. INTRODUCTION 2

A morphism (A, ι, λ) → (B, , ν) ofMnΦ(S) is an OK-linear homomorphism φ : A → B of

abelian schemes, with φ∗ν = λ.

The moduli spacesMnΦ are a variation of the moduli spacesMk(n− r, r) defined by Kudla

and Rapoport in [KR09], which are defined for a quadratic number field k ⊂ C and a pair ofintegers (n− r, r) with 0 < r < n. The latter are Deligne-Mumford stacks which are arithmeticmodels of Shimura varieties attached to unitary groups of signature (n − r, r). They host afamily of Kudla-Rapoport cycles whose intersection theory contains arithmetic informationof interest. In fact, Kudla has conjectured precise relations between the Arakelov interesectiontheory of cycles on arithmetic models of Shimura varieties and modular forms with coefficientsin arithmetic Chow groups.

The moduli problemsMnΦ, when K is a quadratic imaginary number field, coincide with

Kudla-Rapoport spaces for r = 0. This condition rigidifies the moduli space so that the cor-responding stack has relative dimension zero, and does not admit Green functions as in thegeneral case. Nevertheless these spaces have arithmetic properties and play a part within alarger context. When n = 1 they parametrize elliptic curves with complex multiplication byOK. For general n, their complex points are (finite) double coset spaces of the unitary groupsU(n). The latter have compact factors and so fall technically foul of Deligne’s axioms forShimura data. Nevertheless, they behave like zero-dimensional Shimura varieties and havesimilar arithmetic properties.

For CM-types Φ obtained by extending a quadratic imaginary number field, the spacesMn

Φ have been considered and fruitfully employed by Ben Howard in [How12] with arith-metic results. There they provide a cycle of absolute dimension 1 on integral models forM(n− 1, 1). The Kudla-Rapoport cycles then intersect this particular cycle with intersectionmultiplicities which are related to Fourier coefficients of an Eisenstein series.

0.2 Overview

In this work we study the moduli spaces MnΦ for a general CM-type Φ by applying Serre’s

tensor construction. The latter can be used to construct an abelian scheme, denoted M⊗R A,out of a given abelian scheme A with action by a ring R and a finite projective R-module M.This will provide a method for relating different moduli spaces of abelian schemes. To carrythis out in particular for the moduli spacesMn

Φ, we must equip M⊗R A with a principal po-larization. If we assume that A comes with an OK-linear principal polarization, the extra dataneeded to accomplish this is the choice of a hermitian form h on M, which is non-degenerateand positive-definite. This is the main result of the first chapter, summarized as follows:

Theorem A. Let (A, ι, λ) be an abelian scheme A over S with an R-action ι : R → EndS(A) andan R-linear polarization λ. Let the M ⊂ Kn be a full-rank lattice and h : M → M∨ an R-linear map.Then h⊗ λ : M⊗R A → M∨ ⊗R A∨ is a polarization if and only if (M, h) is hermitian and positivedefinite. Furthermore, if EndS(A) = R and λ is principal, then h⊗ λ is principal if and only if h is


non-degenerate.

Armed with this theorem we may systematically construct objects of MnΦ using objects

in M1Φ and finite, non-degenerate positive definite OK-hermitian modules (M, h) of rank n.

The latter form the objects of a groupoid Hermn(R). In order to describe systematically allthe objects in Mn

Φ which arise in this way we utilize a tensor product of categories over amonoidal category. Chapter two is devoted to this topic, providing an explicit constructionwith generators and relations.

The monoidal category acting on M1Φ and Hermn(R) is the groupoid H1 = Herm1(R),

with tensor product of modules giving the monoidal structure. In fact, Herm1(R) has the extrastructure of a 2-group: aside from its morphisms, its objects are also invertible. This allows acomparison between the morphisms inM1

Φ ⊗H1 Hermn(R) and those inMnΦ. In chapter two

we characterize the morphisms of a general tensor product of categories over a 2-group, andin chapter three we compare that result with another characterization, of a similar form, of themorphisms between between those objects in Mn

Φ that arise from Serre’s construction. Theresult of this comparison is as follows.

Theorem B. If n is prime to the class number of K, then the functor

Σ : Hermn(R)⊗H1 M1Φ →Mn

Φ,

induced by Serre’s tensor construction is fully faithful. It is an equivalence over Spec K, for any alge-braically closed field K of characteristic zero.

In fact, under the condition on n, we show that the stackification of Hermn(R)⊗H1 M1Φ is

all ofMnΦ if we restrict the base S to schemes of finite type over C. In other words over such a

scheme each object ofMnΦ arises etale locally from Serre’s tensor construction.

An easy consequence of this equivalence is a description, provided in chapter four, of theaction of the Galois group Gal(Q/K∗) onMn

Φ(Q), where K∗ is the reflex field of (K, Φ).

Proposition C. Let (A, λ, ι) ∈ MnΦ(Q), and σ ∈ Gal(Q/K∗). Pick s ∈ A×K∗ such that s

corresponds to σ|K∗ab under the reciprocity map. Then there exists an isomorphism

ψs,σ : (A, λ, ι)σ ∼−→ (aΦ,s, hΦ,s)⊗OK (A, ι, λ),

which is canonical in the triple (A, ι, λ).

Here the pair (aΦ,s, hΦ,s) is an object in Herm1(OK) associated to s and Φ in a natural way.The proposition follows from the description of the Galois action on M1

Φ(Q) given by themain theorem of complex multiplication, along with theorem B.

If G is a connected reductive group over Q with GadR semi-simple and compact, then the

double coset spaces of G are finite sets. Nonetheless as zero-dimensional varities the definitionof a canonical model may be applied to them. In these cases the definition of a canonical modeltrivially guarantees a unique such model. In the special case of unitary groups U(n), some ofthe double coset spaces have a description in terms of the complex points of Mn

Φ(Q). We


will check that the natural Galois action described by proposition C indeed matches the oneprescribed by the definition of a canonical model. This provides some evidence that the doublecoset spaces of U(n) may be considered as zero-dimensional Shimura varieties.

0.3 Notation

The following notations and definitions are used throughout this document. They are definedin the main text prior to use, but we have collected them here as well for reference.

• K is a CM-field of degree 2g over Q, with F its maximal totally real subextension. Thepair (K, Φ) denotes a CM-type, and L is always the reflex field.

• R is a ring with unity, not necessarily commutative, and finite free as a Z-module. Itsopposite ring is denoted Rop. It is often equipped with an involution R ∼−→ Rop : r 7→ r.In applications R is taken to be the ring of integers OK, with complex conjugation as theinvolution.

• The involution above is called positive if the hermitian form (x, y) 7→ TrRQ/Q(yx) is posi-tive definite.

• M is a finite projective right R-module. The dual module M∨ = HomR(M, R) is the setof right R-linear homomorphisms to R and is naturally a right Rop-module: (r f )(m) =

r f (m) for r ∈ R, f ∈ M∨. When R is equipped with an involution, we consider M∨ aright R-module via R ' Rop.

• A pair (M, h) consists of a module M as above and a right R-linear map h : M → M∨.The map h is hermitian if h(x)(y) = h(y)(x).

• A Jordan algebra over a field k is a commutative, but not necessarily associative algebrasuch that (u u) (u v) = u ((u u) v), for all u, v ∈ A, where denotes the algebramultiplication. A Jordan algebra is formally real if u u + v v = 0 implies u = v = 0. IfA is formally real over R, an element u ∈ A is called positive if all the eigenvalues of thelinear transformation Lu(v) = u v are positive.

• An (R-valued) hermitian form h : M → M∨ will be called positive-definite if it is positiveas an element of the formally real Jordan algebra EndR(M)⊗Z R. A positive-definite his called non-degenerate if it is an isomorphism.

• An R-lattice is a finitely generated projective right R-module M, such that MQ is free as aright RQ-module. An R-lattice is called hermitian if it is equipped with a hermitian formh : M→ M∨.


• Hermn(R) is the groupoid whose objects (M, h) are non-degenerate positive definite her-mitian lattices of rank n. Its morphisms φ : d(M, h) → (N, k) are all isomorphismsφ : M→ N such that φ∗(k) = h.

• A denotes an abelian scheme over a base scheme S. The dual abelian scheme will be de-noted A∨, and if φ : A→ B is a homomorphism of abelian schemes, the dual morphismwill be denoted φ∨ : B∨ → A∨.

• A pair (A, ι) denotes an abelian scheme A over S, and a ring homomorphism ι : R →EndS(A). The map ι is also called an R-action on A.

• A polarization λ : A → A∨ of abelian schemes is a symmetric isogeny such that for allgeometric points s ∈ S, λs is a polarization of abelian varieties.

• A triple (A, ι, λ) consists of an abelian scheme A with an R-action ι, as well as a polariza-tion λ : A→ A∨.

• The polarization λ in (A, ι, λ) is said to compatible with the Rosati involution if the followingdiagram commutes:

A

λ

ι(r) // A

λ

A∨ι(r)∨ // A∨

This is equivalentl to saying λ : A → A∨ is R-linear, where the R-action ι∨ : R →EndS(A∨) on A∨ is defined by by ι∨(r) = ι(r)∨.

Chapter 1

Serre’s Construction of AbelianSchemes

If (A, ι) is an abelian scheme with an R-action, and if M is a finite projective right R-module,Serre’s tensor construction provides a new abelian scheme M⊗R A. To apply this technique tomoduli problems of polarized abelian schemes, we must equip the resulting abelian schemesM ⊗R A with polarizations. In this chapter we show that this can be done if A itself comeswith a polarization, and if M is furnished with a positive-definite R-valued hermitian form.In later chapters we use these results to construct maps between moduli spaces of polarizedabelian schemes.

1.1 Serre’s Construction

1.1.1 Definition and First Properties

Let (A, ι) be an abelian R-module scheme. That is to say, A is an abelian scheme, such thateach abelian group A(T) is a left R-module on which r ∈ R acts via composition by its imageunder ι : R→ EndS(A)

Definition 1 (Serre’s Construction). Let A be a left R-module scheme, and M a right R-module M.The functor which associates to each S-scheme T the abelian group M⊗R A(T) is denoted by M⊗R Aand known as Serre’s (tensor) construction.

Our interest in this construction begins with a representability result.

Proposition 2. Let M be a finite projective right R-module, and A/S an R-module scheme. Thefunctor T 7→ M⊗R A(T) on S-schemes is representable by a group scheme, denoted M⊗R A.

Proof. Since M is projective and finite, there exists a surjective map of right R-modules p :Rn M. The map p admits a section s : M → Rn so that p s = idM. Letting K denote thekernel of p, s induces an isomorphism K ⊕M ∼= Rn by identifying (k, m) with k + s(m). The

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CHAPTER 1. SERRE’S CONSTRUCTION OF ABELIAN SCHEMES 7

map t : Rn → Rn given by t(x) = x− s p(x) is evidently a projection onto K. Thus we obtainthe following finite presentation of the module M:

Rn t // Rn p // M // 0

Since M ∼= s(M) is the kernel of t, we get an exact sequence:

0 // M s // Rn t // Rn p // M // 0 (1.1)

If we apply the functor −⊗R A(T) to the above sequence, it remains exact, again by pro-jectivity of M. The functor T 7→ Rn ⊗R A(T) is represented by the abelian variety An, and soafter identifying the T-points we obtain the exact sequence:

0 // M⊗R A(T)sA(T) // An(T)

tA(T) // An(T) //pA(T)// M⊗R A(T) // 0

The group homomorphisms tA(T) for various S-schemes T together define a morphismof abelian varieties tA : An → An. The above exact sequence shows that the functor T 7→M⊗R A(T) coincides with the one represented by the kernel of tA, a closed S-group subschemeof An.

The following results about Serre’s construction are well-known and will be stated withoutproof. For a more complete statement, see for example [Con04, Theorem 7.2].

Proposition 3. Let A→ S be an R-module scheme, and M a finite projective (right) R-module. Thenif A is any of the following, so is M ⊗R A: smooth, proper, of connected geometric fibres, separated,flat, locally of finite presentation, locally of finite type, and quasi-compact. Furthermore, the formationof Serre’s construction commutes with arbitrary base change.

An R-module scheme need not be an abelian scheme, but if it is, we will call it an abelianR-module scheme for short.

Corollary 4. If the A→ S is an abelian R-module scheme, then M⊗R A is an abelian scheme.

Proof. This follows from the first three properties listed in the proposition.

Corollary 5. If A is an abelian scheme and Tl(A) is the l-adic Tate module of A, then M⊗R Tl(A) ∼=Tl(M⊗R A).

Proof. Since Serre’s construction commutes with base change, there are canonical group schemeisomorphisms

M⊗R A[ln] ∼= (M⊗R A)[ln].

Taking projective limits, we obtain the result.


Let P be another ring. If the finite projective right R-module M is also a left P-module ina compatible way, i.e. if M is a P− R-module, then the abelian scheme M⊗R A is naturally aP-module scheme. The P-action ιM : P→ EndS(M⊗R A) is given on T-points by:

ιM(p)T : M⊗R A(T)→ M⊗R A(T), m⊗ t 7→ pm⊗ t, p ∈ P.

Now if N is a finite right P-module scheme, it is possible to apply Serre construction againto form the abelian scheme N ⊗P (M ⊗R A). Evidently, this is the same abelian scheme as(N ⊗P M) ⊗R A. Indeed, since the associativity isomorphism of modules is canonical, it in-duces a functorial isomorphism between the points of these two schemes.

We now investigate the morphisms between abelian schemes arising from Serre’s construc-tion. For a proof of the following, see for example [Con04].

Proposition 6. Let M, N be finite projective right R-modules and A, B two R-module schemes. Thereis a canonical ring isomorphism

N ⊗R HomS(A, B)⊗R M∨ ∼−→ HomS(M⊗R A, N ⊗R A′)

defined by n⊗ φ⊗ f 7→: (m⊗ t 7→ n⊗ f (m)φ(t)) on T-points t ∈ A(T).

When the module M is a bi-module, as noted before, the group scheme M⊗R A is itself anR-module scheme via the left R-module structure of M. We will be interested in the R-linearmaps between such abelian schemes.

Corollary 7. Let A, B be R-module schemes and M, N finite projective R bi-modules. There is anatural isomorphism

N ⊗R HomR(A, B)⊗R M∨ ∼−→ HomR(M⊗R A, N ⊗R B)

Proof. As M and N are projective we can pick isomorphisms M⊕M′ ' Rm and N ⊕ N′ ' Rn

for some modules M′ and N′. The associated projection Rm M and the injection N →Rn give an embedding of HomS(M ⊗R A, N ⊗R B) inside HomS(Am, Bn). An element of thelatter can be identified with an m × n matrix with elements in HomS(A, B). Such a matrixcorresponds to an R-linear map if and only if every matrix entry is R-linear. Such matricescorrespond to N ⊗R HomR(A, B)⊗R M∨.

Corollary 8. Let R be a commutative ring, M, N finite projective R-modules, and let A, B be R-moduleschemes. There is a canonical isomorphism

HomR(M, N)⊗R HomR(A, B) ∼−→ HomR(M⊗R A, N ⊗R B),

induced by f ⊗ φ 7→ (m⊗ t 7→ f (m)⊗ φ(t)) on T-points t ∈ A(T)


Proof. When R is commutative, all R-modules can be considered as bimodules. Since the leftand right R-actions on HomR(A, B) agree, it is also a bimodule. The tensor product of bimod-ules being associative, the corollary follows from the isomorphism HomR(M, N) ∼= M∨ ⊗R Nwhich holds when M is projective.

1.1.2 The Dual Abelian Scheme

In this section we characterize the dual of abelian schemes arising from Serre’s construction,and the dual morphisms between them. In doing this, we establish relations between theduality theories of abelian schemes, and that of projective modules. First, some notation.

For a right R-module M, let us denote by M∨ = Hom−R(M, R) the set of right R-linearhomomorphism from M to R. M∨ is a right Rop-module with an action of r on f ∈ M∨ givenby

( f ∗ r)(m) = r( f (m)).

We let f ∨ : N∨ → M∨ denote the dual, given by pre-composition, of a right Rop-linear mapf : M→ N.

The right Rop-module R∨ = Hom−R(R, R) is canonically isomorphic to Rop itself. Indeed,let ψ : R∨ → Rop be defined by ψ( f ) = f (1). This map is canonical and Rop-linear:

ψ( f ∗ r) = ( f ∗ r)(1) = r f (1) = rψ( f ) = ψ( f ) ∗ r.

Evidently, the map ψ is an isomorphism of modules. Onwards, we will implicitly identify Rop

and R∨ via ψ.

Now let (A, ι) be an abelian R-module scheme over S, and let A∨ be the abelian schemedual to A. For each r ∈ R, the map ι(r) : A→ A induces a dual ι(r)∨ : A∨ → A∨. We obtain aring homomorphism

ι∨ : Rop → EndS(A),

by definingι∨(r) = ι(r)∨.

Thus (A∨, ι∨) becomes an abelian Rop-module scheme, dual to (A, ι).

If f : M → N is a map of right R-modules, and A is an R-module scheme, for each S-scheme T we let fA(T) denote the map

f ⊗ idA(T) : M⊗R A(T)→ N ⊗R A(T).

As T varies, the collection of maps fA(T) induce a morphism of functors M⊗R A → N ⊗R

A. We denote this functor, and the morphism of schemes it represents, by fA.

We will investigate how the two duality functors mentioned above interact through Serre’sconstruction. We will do this by first considering the case of free modules.


Recall that products and coproducts are categorical duals, so they are interchanged by con-travariant equivalence functors, up to canonical isomorphism. Recall also that finite productsand coproducts coincide in the category of right R-modules, as well as the category of abelianschemes.

Now let si : R → Rn and pi : Rn → R denote the canonical injection and projection mapsthat identify Rn as a direct sum and a direct product, respectively. Let also s′i : Rop → (Rop)n

and p′i : (Rop)n → Rop denote the analogous maps in the category of right Rop-modules. Theduality functor

Hom−R(−, R) : PModR −→ PModRop ,

is a contravariant equivalence of categories mapping R to Rop. Since it must interchange directsums and direct products, up to canonical identifications we have

s′i = p∨i , p′i = s∨i . (1.2)

Similarly, the duality functor from R-module schemes to Rop-module schemes must inter-change direct sums and products. Let σi : A → An and σ′i : A∨ → (A∨)n denote the canonicalinjection maps, and πi : An → A, π′i : (A∨)n → A∨ the projections. Then up to canonicalidentifications

σ′i = π∨i , π′i = σ∨i . (1.3)

For an S-scheme T, consider the map

(pi)A(T) : Rn ⊗R A(T)→ R⊗R A(T), (r1, ..., rn)⊗ t 7→ ri ⊗ t.

Under the natural identifications Rn ⊗R A ∼= An and R⊗R A ∼= A, the above map coincideswith the projection

πi : An → A, (t1, ..., tn) 7→ ti.

Similarly, the maps si : R→ Rn which identify Rn as a direct sum of modules, induce maps

(si)A : R⊗R A→ Rn ⊗R A, r⊗ t 7→ si(r)⊗ t,

which after canonical identifications coincide with the canonical injections σi : A → An. Inother words we have

πi = (pi)A , σi = (si)A. (1.4)


Applying the same discussion to Rop and A∨ gives

π′i = (p′i)A∨ , σ′i = (s′i)A∨ .

Comparing the above relations with those previously obtained, we have

πi = (pi)A = ((si)A∨)∨ , σi = (si)A = ((pi)A∨)

∨. (1.5)

As a first consequence of the above relations, we identify the dual of a map induced by amatrix.

Lemma 9. Let Φ : Am → Bn be a right R-linear morphism of R-module schemes, given by the matrix(aij) with entries in HomR(A, B). The dual Rop-linear morphism Φ∨ : B∨n → A∨m is given by thematrix with entries (bij), where

bij = a∨ji ∈ HomRop(B∨, A∨).

Proof. Let sni : R→ Rn and pn

j : Rn → R denote the ith injection and jth projection maps for Rn,with σn

i : A→ An and πnj : An → A defined similarly. Then we have

bij = (pmi )A∨ Φ∨ (sn

j )A∨

= ((smi )A)

∨ Φ∨ ((pnj )A)

∨

= ((pnj )A Φ (sm

i )A)∨

= a∨ji .

Since Am ∼= Rm⊗R A and Bn ∼= Rn⊗R B, the above lemma can be interpreted as describingthe dual of a morphism between abelian schemes arising from Serre’s tensor construction withfree R-modules. Our eventual goal is to describe the dual of more general morphisms f ⊗ φ :M⊗R A → N ⊗R B, for general R-linear maps f : M → N and φ : A → B. The relations (1.5)give us such a description in the case where A = B, φ = idA and f = si or pi. They can besuggestively reformulated as

(si ⊗R idA)∨ = s∨i ⊗Rop id∨A, (pi ⊗R idA)

∨ = p∨i ⊗Rop id∨A.

Our next proposition is a step towards generalizing these relations.

Proposition 10. Let (A, ι) be an R-module scheme and f : Rn → Rm a right R-linear map. Then

( fA)∨ = ( f ∨)A∨


Proof. Let the map f : Rn → Rm be determined by a matrix with coordinates rij = pj f si.Then if the coordinates of fA are given by the matrix aij, we have

aij = πi fA σj

= (pi)A fA (sj)A

= (pi f sj)A = (rij)A.

The coordinates of the dual map f ∨ are given by

r′ij = pi f ∨ si = s∨i f ∨ p∨i = (pi f sj)∨ = r∨ij .

Then ( f ∨)A∨ has coordinates(r′ij)A∨ = (r∨ji )A∨ .

On the other hand by the previous lemma, ( fA)∨ has coordinates

a∨ji = ((rji)A)∨.

We are thus reduced to proving the case m = n = 1. Let `r : R → R be left multiplicaton byr ∈ R. we must show

(`∨r )A∨ = ((`r)A)∨.

For t ∈ A(T), we have

(`r)A(T) : s⊗ t 7→ `r(s)⊗ t = rs⊗ t = r(s⊗ t),

therefore (`r)A = ι(r).

The dual map `∨r : R∨ → R∨ is defined as g 7→ g `r. As a map Rop → Rop, identifyingg ∈ R∨ with g(1) = s ∈ Rop, we obtain

`∨r (s) = (g `r)(1) = g(`r(1)) = g(r) = g(1)r = sr = r ∗ s.

Hence `∨r : Rop → Rop is left multiplication by r ∈ Rop. We therefore have

(`∨r )A∨ = ι∨(r) = ι(r)∨ = ((`r)A)∨

as claimed.

As an application of the previous proposition, we can give a description of the abelianscheme dual to M⊗R A. Recall the presentation (1.1) of the finite module M,

0 // M s// Rm

t// Rm

p// M

syy

// 0 ,


and the resulting exact sequence of abelian schemes

0 // M⊗R AsA // Am tA // Am //pA // M⊗R A // 0 .

Applying duality of abelian schemes to above, we obtain another exact sequence,

0 // (M⊗R A)∨(pA)

∨// A∨m (tA)

∨// (A∨)m //(sA)

∨// (M⊗R A)∨ // 0 .

Now applying Hom−R(−, R) to the presentation of M will preserve exactness since M isprojective:

0 // M∨p∨// (R∨)m

t∨// (R∨)m

s∨// M∨

p∨vv

// 0 ,

and since M∨ is again projective, after identifying R∨ with Rop and applying −⊗Rop A∨,

0 // M∨ ⊗Rop A∨p∨A∨ // A∨m (t∨)A∨ // A∨m //

s∨A∨// M∨ ⊗Rop A∨ // 0 .

By comparing the above exact sequences we obtain

Proposition 11. Let (A, ι) be an abelian R-module scheme, and M a finite projective right R-module.There is an isomorphism of abelian schemes

(M⊗R A)∨ ∼= M∨ ⊗Rop A∨

Proof. By the previous proposition we have, with the same notation as above, under canonicalidentifications

(tA)∨ = (t∨)A∨ ,

therefore(sA)

∨ = coker((tA)∨) ∼= coker((t∨)A∨) = (s∨)A∨ .

The assertion now follows by comparing

(sA)∨ : (M⊗R A)∨ → (An)∨

and(s∨)A∨ : M∨ ⊗Rop A∨ → (A∨)n.

Finally, we consider the case of a general morphism of the form

f ⊗ φ : M⊗R A→ N ⊗R B


between abelian schemes arising from Serre’s tensor construction.

Theorem 12. Let A and B be two R-module schemes, M and N finite projective right R-modules,φ : A→ B and f : M→ N R-linear maps. Then we have

( f ⊗ φ)∨ = f ∨ ⊗ φ∨.

Proof. Taking a presentations of N analogous to (1.1), we obtain

0 // M s// Rm

t// Rm

g

p// M

syy

f

// 0

0 // N r// Rn u // Rn q // N

rbb// 0

,

where g def= r f p, and the maps s and r are sections for p and q, respectively. The map f can

be recovered from g by

q g s = (q r) f (s p) = idN f idM = f .

By applying −⊗R A to the first row, −⊗R B to the second, and −⊗ φ to the morphisms inbetween the rows, we get the diagram

0 // M⊗R A sA// Am

tA

// Am

g⊗φ

pA// M⊗R A

sAww

f⊗φ

// 0

0 // N ⊗R BrB // Bn uB // Bn qB // N ⊗R B

rB

ee// 0

.

As in the proof of Proposition 11, up to canonical identifications we have

(sA)∨ = (s∨)A∨ ,

and by a similar argument(qB)

∨ = (q∨)B∨ .

Now since f = q g s, we have

f ⊗ φ = qB (g⊗ φ) sA = qB gA (idN ⊗ φ) sA,


so

( f ⊗ φ)∨ = (sA)∨ (idN ⊗ φ)∨ gA (qB)

∨

= (s∨A∨) (idN ⊗ φ)∨ (g∨A∨) (q∨B∨).

Now let ψ = g⊗ φ. Then as matrix, the ijth entry of ψ : Am → Bn is given by

ψij = (pj)B (g⊗ φ) (si)A

= (pj ⊗ idB) (g⊗ φ) (si ⊗ idA)

= (pj g si)⊗ φ = gij ⊗ φ = φ ι(gij(1))

where gij : R → R is the ijth entry of g : Rm → Rn. Then by Lemma 9, ψ∨ has ijth entryequal to

ψ∨ji = (φ g∨ji ),

which by linearity of φ, hence φ∨, is equal to

φ∨ ∨(gji(1)).

The above is the jith entry of the matrix corresponding to g∨ φ∨. This shows that

ψ∨ = g∨ φ∨.

Now using this we have

( f ⊗ φ)∨ = (qB (g⊗ φ) sA)∨ = (s∨)A∨ (g⊗ φ)∨ (q∨)B∨

= (s∨)A∨ (g∨ ⊗ φ∨) (q∨)B∨

= (s∨ g∨ q∨)⊗ φ∨ = (q g s)∨ ⊗ φ∨

= f ∨ ⊗ φ∨

1.1.3 The Lie algebra

Let now (A, ι) consist of an abelian scheme A over a scheme S, and an action ι : R→ EndS(A)

of a ring R. Let M be a finite projective right R-module, and let M ⊗R A denote the abelianscheme obtained from this data by by Serre’s construction. In this section we will investigateLieS(M⊗R A) as an OS-module.

By Z[ε] let us denote the ring obtained from adjoining to Z an element ε with ε2 = 0.Denote by ν : Z[ε] → Z the ring map that sends ε to zero. For any scheme T, let T[ε] denotethe scheme T ×Z Z[ε] obtained by T via base change along ν.


The relative Lie algebra of A may be defined functorially as follows. [GD, Exp. II, 3.9].

LieS(A)(U) = ker(A(U[ε])→ A(U)),

where U is any S-scheme.

Lemma 13. There is a natural isomorphism of OS-modules

LieS(M⊗R A) ∼= M⊗R LieS(A)

Proof. We have

LieS(M⊗R A)(U) = ker(M⊗R A)(U[ε])→ (M⊗R A)(U[ε])

= ker

M⊗R A(U[ε])→ M⊗R A(U)

= M⊗R ker

A(U[ε])→ A(U)

= M⊗R LieS(A)(U),

where the first equality follows from the definition, the second from the definition of Serre’sconstruction, the third since M is projective, and the fourth again from the definition.

If R is commutative, we can consider M as a bimodule so that M⊗R A is endowed with anaction : R→ EndS(M⊗ A). The following proposition allows us to compare the characteris-tic polynomial of an element of R acting via with that of ι.

Proposition 14. Let (A, ι, λ) be a triple. If R a Dedekind domain, and M has rank m, then for alla ∈ R,

char((a)|LieS(M⊗R A), T) = char(ι(a)|LieS(A), T)m.

Proof. Let K = R⊗Z Q and M0 = M⊗Z Q so that M0 is isomorphic to Km as a vector space.By the corollary above we have

Q⊗Z LieS(M⊗R A) ∼= Q⊗Z (M⊗R LieS(A)),

which is naturally isomorphic to

M0 ⊗K Lie0S(A) ' Km ⊗K Lie0

S(A) ∼= Lie0S(A)m,

from which the proposition immediately follows.


1.2 Polarisation

1.2.1 Definitions

We will briefly review several essentially equivalent ways to define a polarization. The refer-ences for this section are [FC90], [MFK94, Ch. 7], and [CCO12, Appendix A].

Let A/S be an abelian scheme with identity section e : S → A. Given a relatively ampleline bundle L on an abelian scheme A/S, one can define a homomorphism Λ(L ) : A → A∨

given on T-points by a construction of Mumford:

Λ(L )T : A(T)→ A∨(T),

a 7→ Ta(LT)⊗L −1T ⊗ a∗(LT)

−1 ⊗ e∗T(LT),

where T is an S-scheme, and LT is the pullback of L under the projection AT → A. Thatthis map is a homomorphism is a consequence of the theorem of the cube for abelian schemes[FC90, §1.1].

Remark: The factor a∗(L )−1 ⊗ e∗T(LT) in the above expression is a line bundle on T,considered as a line bundle on AT via pushforward by the closed immersion eT : T → AT.It guarantees that the pullback of Λ(LT) under eT : T → AT is trivial, which is necessaryfor Λ(L )T to land in A∨(T). When the base S is a field this factor is trivial, hence absent indiscourses on abelian varieties.

Definition 15. A polarization of an abelian schemes A/S is a homomorphism λ : A → A∨ suchthat for every geometric point s ∈ S, λs = Λ(Ls) for some ample line bundle Ls on As. If λ is anisomorphism, it is called a prinicpal polarization.

It follows from this definition that every polarization is a symmetric isogeny.

Polarizations may also be defined in terms of correspondences. We give three separatedefinitions for a correspondence, each appropriate for a certain degree of generality. All threeare used in the literature, and they all differ from others commonly used in algebraic geometry.

If A and B are abelian varieties over an algebraically closed field k, a correspondence may bedefined as a line bundle on A× B, that is trivial when restricted to the subvarieties A× 0and 0 × B. This is what Mumford calls a divisorial correspondence in [Mum08].

A more careful definition is needed when the base field k is not algebraically closed. LeteA : Spec k → A and eB : Spec k → B denote the identity sections of abelian varieties A and Bover Spec k. A correspondence between A and B is a triple (L , α, β), consisting of a line bundleL on A × B, and trivializations α : OA

∼−→ (1A × eB)∗(L ), and β : OB → (eA × 1B)

∗(L ),which are compatible in the sense that e∗A(α) = e∗B(β). This is the definition used in [CCO12,Appendix A].

If A and B are abelian schemes over an arbitrary base scheme S, the sheaf of correspondences


CorrS(A, B) is the fppf cokernel sheaf of the canonical map

Pic(A/S)× Pic(B/S)→ Pic(A×S B)

given by pull-backs via the projections A ×S B → A, A ×S B → B. Here Pic(A/S) is therelative Picard functor. The correspondences CorrS(A, B) can then be defined as global sectionsof the sheaf CorrS(A, B). This is the definition espoused in [FC90, §1.1].

The Poincare correspondence is a universal line bundle on A × A∨, which provides acanonical group isomorphism

HomS(B, A∨) ∼= CorrS(A, B),

by associating to a homomorphism φ : A → B∨, the line bundle (1A × φ)∗(PB) on A× B. Ifs : B× A → A× B is the coordinate flip homomorphism, L 7→ s∗(L ) establishes a bijectionbetween Corr(A, B) and Corr(B, A), which in terms of homomorphisms corresponds to thedual φ∨ : A→ B∨ of a map φ : B→ A∨.

A homomorphism λ : A → A∨ corresponds to an element L of CorrS(A, A), and λ is asymmetric homorphism if and only if s∗(L ) ∼= L , where s : A× A → A× A is the flip map.Such L is called a symmetric correspondence. The following well-known result is an alternatecharacterisation of polarizations in terms of correspondences:

Proposition 16. The morphism λ : A → A∨ associated to a correspondence L on A × A is apolarization if and only if L is symmetric and ∆∗(L ) is ample, where ∆ : A→ A× A is the diagonalmap.

A choice of a polarization λ on the abelian variety A induces a Rosati involution ρ on thealgebra End0(A) = Endk(A)⊗Z Q, determined by the following commutative diagram in theisogeny category:

A λ //

ρ(s)

A∨

s∨

A λ // A∨

where s ∈ End0(A).Suppose we are given a pair (A, ι) consisting of an abelian variety A/k, with action by a

ring R given by an injective ring homomorphism ι : R → Endk(A). Also suppose the ring R isequipped with an involution r 7→ r∗. We say the polarization φ is compatible with the actionof R, if for any r ∈ R ⊂ Endk(A) ⊂ End0(A) we have ρ(ι(r)) = ι(r∗). We will denote suchdata by (A, ι, λ).

Note that for any r ∈ R we get a dual element r∨ ∈ Endk(A∨). Since (rs)∨ = s∨r∨, thisgives a right R-module structure on the functorial points of A∨. The involution on R allowsus to define an injective ring homomorphism ι∗ : R → Endk(A∨) by ι∗(r) = ι(r∗)∨. Now


A∨ obtains the structure of a left R-module variety. In this notation, the compatibility of apolarization λ with the involution on R is simply R-linearity. Explicitly,

λ ι(r) = ρ(ι(r))∨ λ = ι(r∗)∨ λ = ι∗(r) λ,

Now let (A, ι, λ) be as above, M a finitely generated projective right R-module, and M⊗R Athe abelian variety formed by Serre’s construction. By proposition 11 the dual abelian varietyof M ⊗R A is isomorphic to M∨ ⊗R A∨. Given a right R-linear map f : M → M∨ one canform the tensor product map f ⊗ λ : M⊗R A → M∨ ⊗R A∨ by the usual tensor product mapon functorial points. Since λ is R-linear, f ⊗ λ is a well-defined homomorphism of abelianvarieties.

Recall that a Z-bilinear function F : M×M→ R is called a sesquilinear form if F(mr, nr′) =r∗F(m, n)r′ for all m, n ∈ M, r, r′ ∈ R. The following lemma is a standard consequence oftensor-hom adjunction.

Lemma 17. For f ∈ HomR(M, M∨), let F : M×M → R be defined by F(m, n) = f (m)(n). ThenF is sesquilinear and f 7→ F is a bijection between HomR(M, M∨) and sesquilinear forms on M×M.

Proof. Since f is right R-linear we have f (mr)(n) = r∗ f (m)(n). The fact that f takes valuesin M∨ implies f (m, nr) = f (m, n)r. Thus F as defined is sesquilinear. Conversely given suchan F we can recover a linear f : M → M∨ by f (m) = (n 7→ F(m, n)). Evidently, these twoconstructions are inverses.

Let p : Rn → M be a surjective R-module homomorphism, and let σi : R → Rn be the ithembedding of R in Rn. Then the elements ei = p σi(1) generate M. A sesquilinear form F onM is determined by its n× n matrix of values fij = F(ei, ej) ∈ R. When R is equipped withan involution r 7→ r∗, the form F is called hermitian if F(m, n)∗ = F(n, m). This is the case ifand only if f (ei, ej)

∗ = f (ej, ei) for all i, j. Thus F is hermitian if and only if the matrix ( fij) ishermitian, i.e f ∗ij = f ji.

In the remainder of this section we will formulate necessary and sufficient conditions forf ⊗ φ to be a polarization on M⊗R A.

1.2.2 Symmetry

Let (A, ι, λ) be a polarized abelian R-module scheme. Fix a finite projective right R-module M,and let f : M → M∗ be R-linear. From this data, one can construct a homomorphism f ⊗ λ :M⊗R A → M∗ ⊗R A∨ of abelian schemes. We investigate exactly when this homomorphismis symmetric. Now, consider the diagram


An pA //

ψ

M⊗R A

f⊗λ

//

sAww

0

(A∨)n

s∨A

66M∗ ⊗R A∨p∨Aoo 0oo

where the map ψ is defined to be p∨A ( f ⊗ λ) pA. We will show that some properties off ⊗ λ can be formulated in terms of properties of ψ, which itself can be thought of as an n× nmatrix with coefficients in HomS(A, A∨).

If σi : R → Rn is the ith coordinate embedding of R in Rn, then si = (σi)A is the ithcoordinate embedding of A in An and its dual s∨i : (A∨)n → A∨ is the ith projection from (A∨)n

to A∨. The map ψ : An → (A∨)n is then determined by its associated matrix of coordinatesψij = s∨j ψ si ∈ HomR(A, A∨).

Proposition 18. With f ⊗ λ and ψ as above, the following are equivalent:(1) f ⊗ λ is symmetric.(2) ψ is symmetric.(3) ψij = (ψji)

∨ for all i, j(4) f is hermitian

Proof. Since ψ = p∨A ( f ⊗ λ) pA, (1) implies (2). Since p s = idM we have pA sA = (p s)A = idM⊗R A, and dually s∨A p∨A = idM∗⊗R A∨ . If ψ is symmetric, then so is s∨A ψ sA = f ⊗ λ,so (2) also implies (1).

We have(ψji)

∨ = (s∨i ψ sj)∨ = s∨j ψ∨ si = (ψ∨)ij

so that (2) implies (3). Conversely if ψij = (ψji)∨ then ψij = (ψ∨)ij, so that ψ and ψ∨ have the

same matrix coordinates and must be equal. Thus (3) implies (2).Now using the definitions we can compute ψij:

ψij = s∨j ψ si = s∨j (p∨A ( f ⊗ φ) pA) si

= (σj)∨A p∨A ( f ⊗ λ) pA (σi)A

= ((p σj)A)∨ ( f ⊗ λ) (p σi)A

= ((p σj)∨)A∨ ( f ⊗ λ) (p σi)A

= ((p σj)∨ f (p σi))⊗ λ

Identifying R∨ with R, the map (p σj)∨ f (p σi) : R → R is multiplication by its

value at 1 ∈ R. Let ei = (p σi)(1) ∈ M. We have f (p σi)(1) = f (ei) ∈ M∗. Thefunction (p σj)

∨ : M∗ → R∨ is the R-dual of p σj : R → M, so it maps f (ei) ∈ M∗ tof (ei) (p σj) ∈ R∨, which is identified with f (ei) (p σj)(1) = f (ei, ej) under R∨ ∼= R. Thus


(p σj)∨ f (p σi) : R → R is multiplication by the (i, j)th matrix entry fij = f (ei, ej) of the

map f .

Let T be an S-scheme. We have isomorphisms R⊗R A(T) ∼= A(T) via r⊗ x 7→ ι(r) x, andR⊗R A∨(T) ∼= A∨(T) through r⊗ x 7→ i∗(r) x = i(r∗)∨ x. Thus as a map A(T) → A∨(T),ψijT sends x ∈ A(T) to ι( f ∗ij)

∨ λ x = λ ι( fij) x. In other words

ψij = λ ι( fij)

as a map from A to A∨.

Now we have

ψij = (ψji)∨ ⇔ λ ι( fij) = (λ ι( f ji))

∨

⇔ λ ι( fij) = ι( f ji)∨ λ∨

= ι( f ji)∨ λ (since λ is symmetric)

= λ ι( f ∗ji) (since λ is R-linear)

⇔ λ ι( fij − f ∗ji) = 0

Since λ is an isogeny, this last statement is true if and only if ι( fij− f ji∗) = 0. As ι is injective

by assumption, we getψij = (ψji)

∨ ⇔ fij = f ji∗ ∀i, j

The above statement is the equivalence of (3) and (4).

1.2.3 Ampleness

Given a hermitian form h : M → M∗, we have constructed a symmetric homomorphismh⊗ λ : M⊗R A→ (M⊗R A)∨. For this map to be a polarization, it must correspond to ampleline bundles over geometric points of S. Thus for the remainder of this section we will workwith abelian varieties defined over an algebraically closed field k.

For abelian varieties A and B, there is a natural bijection

Hom(B, A∨) ∼= Corr(A, B), φ 7→ Lφ

which to a morphism φ ∈ Hom(A, B∨) associates the line bundle

Lφdef= (1B × φ)∗(PB)

on A× B, where PB is the Poincare line bundle on B× B. In these terms, a polarizationλ : A→ A∨ is identified with a symmetric correspondence Lλ on A× A, such that its pullbackunder the diagonal


Lλdef= ∆∗A(Lλ)

is ample on A.Therefore, to a polarization λ of A we have associated two line bundles: the correspon-

dence Lλ on A × A, and the ample Lλ on A. The next lemma proves some of their basicbehaviour.

Lemma 19. Suppose A and B are abelian varieties, and f : A → A∨ and g : B → A are homomor-phisms. Let h = g∨ f g : B→ B∨. Then we have Lh = (g× g)∗L f and Lh = g∗L f .

Proof. By definition:

Lh = (1× g∨ f g)∗(PB) = (1× g)∗(1× f )∗(1× g∨)∗(PB)

In correspondence terms, the definition of the dual morphism, and double duality togetherimply (1× g∨)∗(PB) ∼= (g× 1)∗(PA). Now since (g× 1) (1× f ) (1× g) = (1× f ) (g× g)we have

Lh = (g× g)∗(1× f )∗(PA) = (g× g)∗L f

We note that ∆A g = (g× g) ∆B as morphisms B→ A× A. This is easy to see on points:for t ∈ B(T), both maps B(T)→ A(T)× A(T) are t 7→ (g t, g t). From this we get

Lh = ∆∗BLh = ∆∗B(g× g)∗L f = g∗∆∗AL f = g∗L f

which completes the proof.

For the rest of this section in order to simplify notation we will identify R with its imageι(R) in End(A). The next proposition establishes some behaviour of correspondences underpullbacks.

Proposition 20. Let (A, ι, λ) be a polarized abelian R-module variety, and let L = Lλ ∈ Corr(A, A)

be the correspondence associated to the polarization λ. The map

l : End(A)× End(A)→ Corr(A, A), (x, y) 7→ (x× y)∗L

satisfies the relations:

l(x, y + z) = l(x, y)⊗ l(x, z)

l(x + y, z) = l(x, z)⊗ l(y, z)

l(x, y ι(r)) = l(x ι(r∗), y)

for all x, y, z ∈ End(A) and r ∈ R.


Proof. A corollary of the theorem of the cube[Mum08, p. 56] states that if f , g, h are maps froman abelian variety A to B and L is a line bundle on B, then:

( f + g + h)∗L ∼=( f + g)∗L ⊗ (g + h)∗L ⊗ ( f + h)∗L

⊗ f ∗L −1 ⊗ g∗L −1 ⊗ h∗L −1

The line bundles L|A×0 and L0×A are trivial since L is a correspondence. Taking f =

(x × 0), g = (0× y), h = (0× z) in the formula above and applying it to line bundle L onA× A shows the first propetry, with the second being analogous.

To prove the third property we may assume x = y = 1, since l(x, y) = (x× y)∗l(1, 1). Thusfor all r ∈ R:

(1× ι(r))∗L = (1× ι(r))∗(1× λ)∗PA = (1× λ ι(r))∗PA

= (1× ι(r∗)∨ λ)∗PA = (1× λ)∗(1× ι(r∗)∨)∗PA

= (1× λ)∗(ι(r∗)× 1)∗PA = (ι(r∗)× 1)∗(1× λ)∗PA

= (ι(r∗)× 1)∗L

from which follows the third assertion.

Let as before, (A, ι, λ) be a polarized R-module scheme, and h : M→ M∗ a hermitian formon the finite projective right R-module M. Let L denote the line bundle on A correspondingto the morphism h⊗ λ : M⊗R A→ M∗ ⊗R A∗. We would like to relate the ampleness of L toa positivity condition on h. For this we will use an ampleness criterion due to Mumford.

Recall that for a variety X, the Neron-Severi group NS(X) group is defined as the quotient

NS(X) = Pic(X)/Pic0(X).

The Neron-Severi theorem asserts that NS(X) is a finitely generated abelian group [LN59]. Wewill denote NS(X)⊗Z Q by NS0(X). Note that it makes sense to ask if an element of NS(X)

is ample or not, since this property is invariant under tensoring with elements of Pic0(X).

Let A be an abelian variety with a polarization φ, and let L = Lφ. The map φ induces abijection

End0(A)→ Hom0(A, A∨), f 7→ φ f .

After composing the above with the injection

NS0(A)→ Hom0(A, A∨), M 7→ φM,

the algebra NS0(A) is identified with the symmetric elements of End0(A). The amplenesscriterion of Mumford [Mum08, p.221] states that the ample line bundles in NS0(A) correspond


to totally positive elements in End0(A)⊗R, considered as a formally real Jordan algebra. Westudy this in more detail in the next section.

1.2.4 Positivity

In this section we recall some of the basic definitions in the theory of Jordan algebras, andshow how several notions of positivity are equivalent. The reference for this section is [BK66].

Definition 21. Let K be a field with char(K) 6= 2. An algebra A over K, not necessarily associative, iscalled a Jordan algebra if it is commutative and if for all u, v ∈ A

(u u) (u v) = u ((u u) v),

where denotes multiplication in A.

Given any associative algebra A over a field K of odd characteristic, one can define a newmultiplication on A by

x y =12(xy + yx).

Then (A, ) is a Jordan algebra over K. We will always denote the multiplication operation ofa Jordan algebra by to avoid confusion.

Definition 22. A Jordan Algebra A is called formally real if for u, v ∈ A, u u + v v = 0 only ifu = v = 0.

Let R be an associative algebra, free of finite rank over Z, with a positive involution r 7→ r∗.Recall that this means (x, y) 7→ TrRQ/Q(y∗x) is positive definite. R can be made a Jordanalgebra as above. We claim the subalgebra S of symmetric elements in R is formally real.Suppose u u + v v = u2 + v2 = 0. Then we have

0 = TrRQ/Q(u2 + v2) = TrRQ/Q(u2) + TrRQ/Q(v2) = TrRQ/Q(uu∗) + TrRQ/Q(vv∗).

Since the involution on R is positive, the right hand side is zero if and only if u = v = 0.

Now let R be the same as above, and consider the finite associative matrix algebra Mn(R)with coefficients in R. Using the involution on R, one can define an involution A 7→ A∗ onthese matrices by sending (aij) 7→ (a∗ji). Consider the bilinear form (A, B) = Tr(B∗A) onMn(R). We have

Tr(A∗A) =n

∑i,j=1

a∗ijaij.

If Tr(A∗A) = 0, we haven

∑i,j=1

TrR/Q(a∗ijaij) = 0,


which is possible only if aij = 0 for all i, j, i.e. A = 0. Now let Hn(R) denote the subalgebraof symmetric elements in Mn(R). If U, V ∈ Hn(R) such that U U + V V = 0, the sameargument as in the case of R, this time using the matrix trace shows that U = V = 0. ThusHn(R) is also a formally real Jordan algebra.

As example, one can take R = R with the trivial involution, C with complex conjugation,or the quaternions K with the standard involution. One obtains real symmetric and complexhermitian matrices in the first two cases.

If A is formally real over R, one can define a notion of positivity for the elements of A. Foreach u ∈ A, let Lu : A→ A denote the linear transformation given by Lu(v) = uv.

Definition 23. An element u in a formally real Jordan algebra over R is called positive, denoted u > 0,if all the eigenvalues of Lu are positive.

We note that some sources use the term totally positive in place of positive. In any casethere is only one notion of positivity in a formally real Jordan algebra.

Lemma 24. Let F = R, C, or K. Then an element of Hn(F) is positive if and only if as a matrix it ispositive definite.

Proof. Let ξ ∈ R be an eigenvalue of H, and A an eigenvector:

H A = ξ A.

In other words, HA + AH = 2ξA. Let U ∈ Mn(F), U∗ = U−1 such that D = UHU−1 isdiagonal. Then we have

U−1DUA + AU−1DU = 2ξA,

from which we getDUAU−1 + UAU−1D = 2ξUAU−1.

Letting B = UAU−1 this isDB + BD = 2ξB.

If the diagonal entries of D are d1, ..., dn, then comparing the ijth entry of each side of theequation above, we get the equivalent condition that for all i, j

(di + dj)Bij = 2ξBij.

One can take B = Ejj, the matrix with 1 in the jjth entry and 0 everywhere else. Then the aboveequations all reduce to ξ = dj. Therefore, every dj is an eigenvalue of H with eigenvectorA = U−1EjjU. Hence for H to be positive, it is necessary that all di > 0. But this is alsosufficient because all eigenvalues of H are of the form (di + dj)/2 from the above equation.


If A is a polarized abelian variety, the Jordan algebra A = End0(A) ⊗ R is isomorphicto several copies of the simple algebras Mn(F) with F = R, C or K. In fact one can ar-range this isomorphism so that the Rosati involution coincides with the standard involution oneach Mn(F). Then the symmetric elements will correspond to a product of the correspondingHn(F). Identifying these with NS0(A)⊗R, ample line bundles will correspond to totally pos-itive elements. In the standard reference [Mum08], totally positive elements of End0(A)⊗R

are taken to be those whose image in each component Mn(F) is positive definite. This matchesour definition of positivity by the previous lemma and the following observation.

Lemma 25. Let A and B denote formally real Jordan algebras over R. An element (a, b) of the productalgebra A×B is positive if and only if both a and b are positive.

Proof. Suppose (a, b) > 0 and a u = ξa for u ∈ A, u 6= 0, ξ ∈ R. Then (a, b) (u, 0) = ξ(u, 0)so ξ > 0. Conversely, suppose a > 0 and b > 0, with (a, b) (u, v) = ξ(u, v). Then a u = ξuand b v = ξv. If u 6= 0 then u is an eigenvector for a so ξ > 0 since a > 0. If u = 0, then v 6= 0since (u, v) is a (non-zero) eigenvector for (a, b), in which case again ξ > 0 since b > 0. Thisshows (a, b) > 0.

Now we restrict ourselves to special cases used in later chapters. Let K ⊂ C be a numberfield stable under complex conjugation, which we take as its involution. Let R = OK. DenoteR∩R by F. Then F is the set of symmetric elements of K. K is CM if F is totally real and if K/Fhas order 2.

Let ρ1, ..., ρr denote the real embeddings of K, and σ1, σ1, σ2, σ2, ...σs, σs its complex em-beddings. Then we have an isomorphism

K⊗R∼−→ Rr ×Cs, (a⊗ x) 7→ (aρ1 x, ..., aρr x, aσ1 x, ..., aσs x).

This depends on an arbitrary choice between each σi and σi. When K is CM, r = 0 and thischoice is picking a CM-type for K. Now this isomorphism induces an isomorphism of algebras

Mn(K)⊗R ∼= Mn(R⊗ K) ∼−→ Mn(R)r ×Mn(C)s,

where an element A⊗ x ∈ Mn(K)⊗R is mapped to (Aρ1 x, ..., Aρr x, Aσ1 x, ..., Aσs x). The involu-tion on Mn(K) translates to conjugate transposition on each factor on the right. Thus lookingat the symmetric elements we have

Hn(K)⊗R∼−→ Hn(R)r × Hn(C)r.

Let A ∈ Hn(K) be considered as A⊗ 1 ∈ Hn(K)⊗R. By lemma 25 we know that A > 0if and only if each of its factors on the right hand side are positive as Jordan algebra elements.Further, by lemma 24 we know each such factor is positive if and only if it is positive definiteas a matrix. Thus we have the following:


Lemma 26. An element A ∈ Hn(K) is positive in Hn(K)⊗R if and only if the eigenvalues of A aretotally positive as algebraic numbers.

Proof. Let p(x) ∈ K[x] be the characteristic polynomial of A, and L/K the splitting field ofp(x) over K. Then there exists a unitary matrix U ∈ Un(L) such that the vectors xi = Uei ∈ Ln

form an orthogonal basis of eigenvalues for A, with eigenvalues di. Then we have di = x∗i Axi.Suppose A > 0. For every embedding τ of K into C, we must have x∗Aτx > 0 for all

x ∈ Cn. Now let σ ∈ Gal(C/Q). Then τ = σ|K is an embedding of K in C. If xi = Uei is asabove, setting y = xσ

i , we get dσi = (x∗i Axi)

σ = y∗Aτy > 0. Since σ and xi were arbitrary, wehave shown that all eigenvalues of A must be totally positive.

Conversely, if the eigenvalues di are totally positive, each Aτ is positive definite since itseigenvalues are Galois conjugates of di and therefore positive.

Corollary 27. For A ∈ Hn(K) let L/K be an extension in which p(t) = det(A− tI) splits. ThenA > 0 if and only if x∗Ax is totally positive for every x ∈ Ln.

Proof. There exists an orthogonal basis of eigenvectors xi ∈ Ln such that x∗i Axi = di, where di

is an eigenvalue of A as before. Hence if all x∗Ax 0, then A > 0 by the lemma. Converselysuppose all di 0. For every x ∈ Ln we have

x = a1x1 + a2x2 + ... + anxn,

for some ai ∈ L. Then x∗Ax = |a1|2d1 + |a2|2d2 + ...|an|2dn 0.

1.3 First Main Theorem

We apply our results so far to abelian schemes and hermitian modules. As before, let R bean algebra over Z which is free of finite rank, equipped with a positive involution. We makethe following extra assumption on R: for every invertible matrix Q in Mn(RQ), the hermitianmatrix Q∗Q ∈ Hn(RQ) is positive in the formally real Jordan algebra Hn(RQ) ⊗Q R. Thefollowing lemma shows that this property holds in all cases of interest.

Lemma 28. If there exists a triple (A, ι, λ), then R satisfies the above property.

Proof. Since for any point s in the base scheme S the specialization map EndS(A)→ Endk(s)(As)

is injective, we can assume A is an abelian variety.After replacing Q by a positive integer multiple, we can assume Q has coordinates in R,

and so defines an isogeny φ : An → An. Since the R-linear map λn is a principal polarizationon An, the line bundle Lλn is ample. Therefore the line bundle Lφ∨λnφ is also ample, since by19 it is the same as φ∗Lλn and φ is an isogeny. Since λ is R-linear, φ∨ λn φ = λn ψ, whereψ : An → An is given by the matrix Q∗Q. By the ampleness criterion, the symmetric elementψ is positive in the formally real Jordan algebra Hn(RQ)⊗Q R ⊂ End0(An)⊗Q R, where it isidentified with Q∗Q as a hermitian matrix with coefficients in R.


Definition 29. A lattice M over R is a finite projective right R-module, such that MQ is a free moduleover RQ. A hermitian lattice (M, h) over R is a lattice M over R equipped with an R-linear hermitianform h : M→ M∨.

Given a hermitian lattice (M, h), let η : RnQ → MQ be an isomorphism of right RQ-modules.

Identifying R∨Q with RQ via the involution on R, we can consider the composition T = η∨ hQ η as a hermitian matrix in Hn(RQ).

Definition 30. A hermitian lattice (M, h) is positive definite if for some isomorphism η : RnQ →

MQ, the matrix T = η∨ hQ η ∈ Hn(RQ) is a positive element of the formally real Jordan algebraHn(RQ)⊗R.

A different choice of an isomorphism η will lead to a different matrix T′ = QTQ∗ forQ ∈ GLn(RQ). The definition above is independent of this choice by the following lemma.

Lemma 31. Let R be an algebra, free of finite rank over Z, equipped with a positive involution. LetQ ∈ GLn(RQ), and T ∈ Hn(RQ). Then T is positive in Hn(RQ) ⊗Q R if and only if QTQ∗ ispositive.

Proof. The map T 7→ QTQ∗ is an isotopy of Jordan algebras [McC04, p. 14]. Its image is aJordan algebra with product defined by

X ′ Y =12

(X(QQ∗)−1Y + Y(QQ∗)−1X

),

which has unit element QQ∗. As QQ∗ is positive in the original Jordan algebra, the isotopyT 7→ QTQ∗ preserves the cone of positive elements [McC04, p. 18].

The content of the main theorem is in the following proposition.

Proposition 32. Suppose (A, ι, λ) is a triple over an algebraically closed field and (M, h) is a hermitianlattice. Then h⊗ λ is a polarization if and only if h is positive definite.

Proof. For simplicity we assume M ⊂ RnQ and that η : Rn

Q → MQ is the inverse of the iso-morphism obtained by tensoring the inclusion map with Q. Let T = η∨ hQ η ∈ Hn(RQ) asabove.

Each basis element ei ∈ Rn ⊂ (RQ)n will correspond to 1

limi for some integer li and an

element mi ∈ M. If l = lcm(li) we have lRn ⊂ M. Let ν : Rn → M denote multiplication byl. Now let m′j be a set of generators for M. For each m′j there exists some positive integer k j

such that k jm′j ∈ Rn. If k = lcm(k j) then multiplication by k gives a map κ : M→ Rn. Then wehave maps

Rn ν // M κ // Rn

each of which is multiplication by an integer. If f = ν∨ h ν is considered as a map Rn → Rn,then as elements in Hn(RQ)⊗R, f > 0 if and only if T > 0.


After passing to the tensor construction we obtain morphisms

An νA // M⊗R AκA // An .

The composition κA νA : An → An is multiplication by an integer, so it is an isogeny. Itfollows that the kernel of νA is a finite group scheme. Now we know that An and M ⊗R Ahave the same relative dimension over S, say by comparing the rank of their Tate modules orLie algebras. It follows that νA and κA are both isogenies.

Let f = ν∨ h ν and consider the commuting diagram

An νA //

f⊗λ

M⊗R A

h⊗λ

(A∨)n M∨ ⊗R A∨.(νA)

∨oo

By Lemma 19, L f⊗λ = ν∗A(Lh⊗λ). Since νA is an isogeny, L f⊗λ is ample if and only if Lh⊗λ

is ample. It follows that h⊗ λ is a polarization of M⊗R A if and only if f ⊗ λ is a polarizationof An. Now the morphism f ⊗ λ factors as the composition

An fA−→ An λn−→ (A∨)n.

Since λn is a polarization, by the ampleness criterion λn fA is a polarization if and onlyif the symmetric endomorphism fA ∈ End0(An) is positive in End0(An) ⊗ R. The proofof proposition 18 shows that the induced Rosati involution on End0(An) matches the stan-dard involution on Mn(End0(A)) induced from End0(A). Thus the algebra isomorphismMn(End0(A)) ∼= End0(An) identifies Hn(End0(A)) with the symmetric elements in End0(An).

Now we have a map of formally real Jordan algebras

Hn(RQ)→ End0(An), f ∈ Hn(R) 7→ fA.

Under the identification End0(An) ∼= Mn(End0(A)) the above map coincides with the oneinduced by the R-action ι : R → End(A). In particular it is injective. Thus considering Hn(R0)

as a subalgebra of End0(An), it follows that fA > 0 in End0(An)⊗R if and only if f > 0 inHn(R)⊗R. By the discussion preceding the proposition this is the case if and only if (M, h) ispositive definite.

The main theorem of this section is the following.

Theorem 33. Let (A, ι, λ) be an abelian scheme A over S with an R-action ι : R→ EndS(A) and anR-linear polarization λ. Let M be an R-lattice and h : M → M∗ an R-linear map. Then h⊗ λ is apolarization if and only if (M, h) is hermitian and positive definite.


Proof. By proposition 18, h⊗ λ is symmetric if and only if h is hermitian. Thus we can assumeh is hermitian.

If s ∈ S is a geometric point, (As, ιs, λs) is a polarized abelian variety over k(s) with actionby R. Furthermore (h⊗λ)s ∼= h⊗λs since the tensor construction commutes with base change.By proposition 32, h ⊗ λs is a polarization if and only if h is positive definite. If h ⊗ λ is apolarization, so is h⊗ λs, therefore h is positive definite. Conversely if h is positive definite,each h ⊗ λs is a polarization of the abelian variety As, so it corresponds to Λ(Ls) for someample line bundle Λs on As. Therefore h⊗ λ is a polarization.

Recall that a hermitian form h : M → M∗ is called non-degenerate if it is an isomorphism.It is possible for a hermitian form to be positive definite yet fail to be surjective.

Corollary 34. Let (A, ι, λ) be as in the theorem. Suppose furthermore that λ is principal, and End(A)

is faithfully flat over R. Then h⊗ λ is a principal polarization if and only if h is positive definite andnon-degenerate.

Proof. If h is hermitian, positive definite and non-degenerate, h ⊗ λ is a polarization by thetheorem, and an isomorphism since λ and h both are.

Suppose conversely h⊗ λ : M⊗R A → M∨ ⊗R A∨ is an isomorphism. By the theorem hmust be hermitian and positive-definite, so we must show it is an isomorphism. Since λ is anisomorphism, so is idM∨ ⊗ λ and from

h⊗ λ = (idM∨ ⊗ λ) (h⊗ idA),

we deduce that h⊗ idA is also an isomorphism. Now let S = End(A), and look at the inducedisomorphism on A-points:

(h⊗ idA)A : M⊗R S ∼−→ M∗ ⊗ S.

The above is equal to h ⊗ idS. Since S is faithfully flat over over R, we deduce that h is anisomorphism.

Note that the corollary applies if S = R. This occurs, for example, if A is an abelian varietywith CM by R.

Chapter 2

Tensor Product of Monoidal Categories

In this chapter we first recall the definitions of a monoidal category, and a category with anaction by a monoidal category. Next, for two categories on which a monoidal category acts,we define a tensor product over the monoidal category by an explicit construction involvinggenerators and relations. We will then show that if the monoidal category is a 2-group, i.e.if the objects are invertible, then the morphisms of the corresponding tensor product have aparticular representation in terms of the generators and relations. In the next chapter we willapply this tensor product and the obtain characterization of its morphisms to a moduli spaceof abelian schemes.

Note: The categories Hermn(R) andMnΦ are the principal examples for the general results

obtained in this chapter. We refer to the introductory chapter for their definitions.

2.1 Action by a monoidal category

Recall that a monoidal category is a category C equipped with a bilinear functor

− ∗− : C× C → C,

along with an object E, and canonical isomorphisms

αg,h,k : (g ∗ h) ∗ k→ g ∗ (h ∗ k),

λG : e ∗ g→ g, ρg : g→ e ∗ g,

31

CHAPTER 2. TENSOR PRODUCT OF MONOIDAL CATEGORIES 32

such that the pentagon diagram

((g ∗ h) ∗ k) ∗ lα−1

g∗h,k,l

((

αg,h,k∗l

vv(g ∗ (h ∗ k)) ∗ l

αg,h∗k,l

(g ∗ h) ∗ (k ∗ l)

αg,h,k∗l

g ∗ ((h ∗ k) ∗ l)g∗(αh,k,l) // g ∗ (h ∗ (k ∗ l))

commutes, as well as the triangle diagram

(g ∗ e) ∗ hαg,e,h //

ρg∗h %%

g ∗ (e ∗ h)

g∗λhyyg ∗ h

.

Definition 35. A 2-group is a monoidal category C, such that for each object g ∈ C, there exists anobject g−1 ∈ C and an isomorphism

Ig : g ∗ g−1 ∼−→ E.

We will assume that such a map has been fixed for all g. One can always pick these in sucha way as to satisfy certain adjunction properties (cf. [BL04]).

Let R be a ring equipped with a positive involution. The category Herm1(R) of rank-oneprojective modules over a ring R is a monoidal category. The identity object is the trivialhermitian structure on R, and the associators and unitors are the ones induced by canonicalproperties of tensor product of modules. Furthemore, Herm1(R) is a 2-group, since for eachrank one hermitian structure α : a→ a−1, α⊗ α−1 is isomorphic to the identity map on R.

Notation: We will denote the 2-group Herm1(R) by H1.

In the rest of this section we will suppress the symbol ∗ and write gh for g ∗ h.

Let G be a monoidal category. A left action of G on a category C, or a left-module categoryC over G, is the following data:

• For every element g ∈ G a functor g− : C → C.

• A natural transformation λ with components λC : eC ∼−→ C, called a left unitor.

• An associator natural transformation α, with components αg,h,C : (gh)C ∼−→ g(hC).


As before, the pentagon diagram associated to this data

((gh)k)Cα−1

gh,k,C

&&

αg,h,kC

xx(g(hk))C

αg,hk,C

(gh)(kC)

αg,h,kC

g((hk)C)

g(αh,k,C) // g(h(kC))

and the triangle diagram

(ge)Cαg,e,C //

λg ""

g(eC)

gλC||gC

are required to commute

A right action of G on a category C, or a right-module C over G, is similarly defined by thedata of a category D, functors −g : D → D for every g ∈ G, a right unitor ρD : De → D, andassociator αg,h : (Dg)h → D(gh). The data is required to satisfy the analogous pentagon andtriangle relations.

The diagram axioms ensure that all other expected associativity relations will hold, byvarious instances of Mac Lane’s coherence theorem [ML98]. More general versions of theseaxiom can be found for example in [Ost03].

The monoidal category Herm1(R) acts on the category Hermn(R) on the right through thetensor product of modules. It also acts on Mn

Φ on the left via Serre’s tensor construction. Allthe required relations follow from the corresponding ones for tensor product of modules.

2.2 Tensor product of categories over a monoidal category

Tensor product of categories are defined by Saavedra [SR72], and refined by Deligne [Del90]for k-linear categories over the monoidal category Veck. Tambara has defined in [Tam01] thetensor product of k-linear categories over a k-linear monoidal category, and Greenough [Gre10]has carried out a similar construction for bimodule categories. Here we closely follow Tam-bara’s concrete construction with generators and relations, stripping it of any mention of lin-earity.

Let G be a monoidal groupoid, C a right G-module category, and D a left-module categoryover G. We will denote the left-action of g ∈ G by g(−) : D → D, its unitor by λD : eD ∼−→ D,and the associators αg,h,D. The right action will be denoted by (−)g : C → C, its right unitorby ρC : Ce ∼−→ C, and its associators by αC,g,h.

We define the tensor product category C ⊗G D as follows. The objects of C ⊗G D consist of


symbols of the form C⊗ D:

Ob(C ⊗G D) = C⊗ D : C ∈ Ob(C), D ∈ Ob(D).

The generators for the morphisms in C ⊗G D are

I. Symbols of the form φ⊗ ψ:

φ⊗ ψ : C⊗ D → C′ ⊗ D′,

for all φ : C → C′, ψ : D → D′.

II. Associator isomorphismsαC,g,D : Cg⊗ D → C⊗ gD,

for every C ∈ C, g ∈ G, D ∈ D, along with their inverses

α′C,g,D : C⊗ gD → Cg⊗ D.

The relations between the above generators are

I. Functoriality relations of ⊗

(φ⊗ ψ) (φ′ ⊗ ψ′) = (φ φ′)⊗ (ψ ψ′),

1C⊗D = 1C ⊗ 1D,

for f ∈ Mor(C) and g ∈ Mor(D).

II. The naturality property of associators

Cg⊗ DαC,g,D //

φu⊗ψ

C⊗ gD

φ⊗uψ

C′g′ ⊗ D′

αC′ ,g′ ,D′// C′ ⊗ g′D′

for any φ : C → C′, ψ : D → D′, and u : g→ g′ ∈ G.

III. Isomorphism properties of associators

αC,g,D α′C,g,D = 1C⊗gD,

α′C,g,D αC,g,D = 1Cg⊗D.

IV. The pentagon diagram


((Cg)h)⊗ DαCg,h,D

((

αC,g,h⊗D

vv(C(gh))⊗ D

αC,gh,D

(Cg)⊗ (hD)

αC,g,hD

C⊗ ((gh)D)

C⊗αg,h,D // C⊗ (g(hD))

as well as the triangle diagram

(Ce)⊗ DαC,e,D //

ρC⊗D %%

C⊗ (eD)

C⊗λDyyC⊗ D

should commute for all C,D, and g, h ∈ G.

The pentagon and triangle relations above and the similar relations for the left and rightactions of G, along with associativity of the group G, imply all the expected associativity rela-tions on formal symbols Cg1g2g3....gnD by MacLane’s coherence theorem [ML98, VII.2].

Finally, we define the functor T : C × D → C ⊗G D by sending (C, D) to C⊗ D and ( f , g)to f ⊗ g.

Typically one defines a kind of bilinear functor from C × D to a linear category, and thenone defines tensor products by a universal property that states all such functors must fac-tor uniquely through T . On the other hand if one takes the trouble of creating an explicitconstruction, it is usually trivial to show it satisfies the property. We will leave out such ademonstration as it has no bearing on our application, and refer the reader to Tambara’s paper[Tam01] wherein this step is carried out for the construction that we have imitated.

2.3 Tensor product over a 2-group

In this section we assume that G is a 2-group. We aim to get a handle on the morphisms of thetensor product C ⊗G D by reducing them to a manageable form. The next lemma is our basicreduction tool.

Lemma 36. Let α1 and α2 be associators in C ⊗G D. Then any morphism of the form α±11 α±1

2 orα±1

1 (φ⊗ ψ) α±12 can be rewritten as

(φ1 ⊗ ψ1) α±1 (φ2 ⊗ ψ2),

where α is an associator.


Proof. By the pentagon diagram for the tensor product, each composition of associators α1 α2

can be written as (1⊗ ψ) α3 (φ⊗ 1). For the same reason, each composition of inverses ofassociators α′1 α′2 can be written as (φ′ ⊗ 1) α′3 (ψ′ ⊗ 1).

Given one associator and the inverse of another

αC,g,D : Cg× D → C× gD,

α′C′,h,D′ : C′ × D′h→ C′h× D′,

for α′C′,h,D′ αC,g,D to occur in a composition of generators, we must have C = C′, and gD =

D′h. In that case we have a commutative diagram

Cg⊗ D α //

1Cg⊗φ

C⊗ gD = C⊗ D′h α′ // Ch⊗ D′

=

Cg⊗ (g−1h)(h−1gD)

α // Cg(g−1h)⊗ (h−1gD)φ′⊗ψ′ // Ch⊗ D′

,

with canonical isomorphisms φ, φ′ and ψ′. Thus α′C′,h,D′ αC,g,D = φ′ ⊗ ψ′ α (1Cg ⊗ φ).

Morphisms of type α1 (φ⊗ ψ) α2 can be rewritten using the diagram

Cg⊗ Dα1 //

1⊗µ1

C⊗ gDφ⊗ψ // C′h⊗ D′

α2 // C′ ⊗ hD′

Cg⊗ g−1(gD)φg⊗g−1ψ// (C′h)g⊗ g−1D′

µ2⊗1g−1D′// C′(hg)⊗ g−1D′α3 // C′ ⊗ (hg)(g−1D′),

1C′⊗µ3

OO

where µ1, µ2, and µ3 are canonical isomorphisms.

Those of type α′1 (φ ⊗ ψ) α′2 can also be written in the desired form by reversing thearrows in the above relation. The remaining morphisms are of type α1 (φ⊗ ψ) α′2 or α′1 (φ⊗ ψ) α2. For the latter type, we have a similar commutative diagram as above

Cg⊗ Dα2 //

1Cg⊗µ1

C⊗ gDφ⊗ψ // C′ ⊗ hD′

α′1 // C′h⊗ D′

Cg⊗ g−1(gD)φg⊗g−1ψ// C′g⊗ g−1(hD′)

1⊗µ2 // C′g⊗ (g−1h)D′ α′ // (C′g)(g−1h)⊗ D′,

µ3⊗1D′

OO

with canonical isomorphisms µ1, µ2 and µ3. For morphisms of the form α1 (φ ⊗ ψ) α′2we can again take the same diagram as above with the arrows reversed. This establishes thelemma.

The reductions allowed by the previous lemma will be applied by a combinatorial processcaptured in the following lemma.


Lemma 37. Every element of the monoid with two generators a, b and the relations

a2 = bab = aba, b2 = b,

has a representative in the seta, b, ab, ba, bab

Proof. Given a wordar1 bs1 ar2 bs2 ...arn bsn ,

we can replace all si with 1 since b2 = b. Letting a0 denote the blank word, for r > 0 we havethe relation

arbar′ = ar−1abaar′−1 = ar−1babar′−1.

The sum of the powers of a on the right hand side is one less than the sum on the left. Thusby repeatedly replacing expressions of the form on the left with those on the right and thenreplacing any new resulting subwords bb with b we arrive at one of the elements in the set.

Proposition 38. Let C and D be categories with a right and left G-action respectively. Then everymorphism τ : C⊗ D → C′ ⊗ D′ in C ⊗G D can be written in the form

ω (φ⊗ ψ)

whereω = (1C′ ⊗ (λD′ IgD′ α−1

g,g−1,D′)) αC′,g,g−1D′ ,

for some element g ∈ G. In particular, for each g there is at most one representation of τ having thisform.

Proof. Let τ be a morphism in C ⊗ D. Denote by P the set of all representations of τ as acomposition of generating symbols. Let M be the set of words freely generated by two symbolsa and b, and define a function ρ : P→ M by setting

ρ(α±1C,g,D) = a, ρ(φ⊗ ψ) = b,

andρ( f g) = ρ( f )ρ(g).

Let M′ be the monoid in lemma 37 with π : M → M′ the obvious surjection. By lemma 36each reduction step in the proof of lemma 37 has a counterpart in C ⊗ D, so that the entirereduction can be carried out in P instead of M′. In other words, for p ∈ P and m ∈ M, if wehave π ρ(p) = π(m), then there is a word q ∈ P such that ρ(q) = m. Then by lemma 37,there is a word p0 ∈ P such that ρ(p0) ∈ a, b, ab, ba, bab. This shows that any morphism τ in


C ⊗D can be written in the form

(φ1 ⊗ ψ1) α (φ2 ⊗ ψ2),

where α is an associator, and where φi ⊗ ψi may be identity maps depending on ρ(p0).

Now suppose α = αg,C,D. Then we have a commutative diagram

Cg⊗ D α //

1⊗µ

C⊗ gDφ1⊗ψ1 // C′ ⊗ D′

Cg⊗ g−1(gD)φg⊗g−1ψ// C′g⊗ g−1D′

αC′ ,g,g−1D′// C′ ⊗ g(g−1D′)

1C′⊗µ′

OO

So that(φ1 ψ1) α = (1C′ ⊗ µ′) αC′,g,g−1D′ (φg⊗ g−1ψ) (1⊗ µ).

Here µ : D → g−1(gD) is a composition of λ−1D : D → eD, the isomorphism I−1

g−1 : e → g−1g ,and the associator αg−1,g,D : (g−1g)D)→ g−1(gD) from the G-action of D, so

1Cg ⊗ µ = 1Cg ⊗ (αg−1,g,D I−1g−1 λ−1

D ).

Similarlyµ′ = λD′ IgD′ α−1

g,g−1,D′ .

Thus τ = ω (φ0 ⊗ ψ0), where

ω = (1C′ ⊗ (λD′ IgD′ α−1g,g−1,D′)) αC′,g,g−1D′ ,

andφ0 ⊗ ψ0 = (φg⊗ g−1ψ) (1Cg ⊗ (αg−1,g,D I−1

g−1 λ−1D )) (φ2 ⊗ ψ2).

with φ′ ⊗ ψ′ = (φ1 ⊗ ψ1) (1⊗ µ) (φg ⊗ g−1ψ). This proves the proposition for mor-phisms of the form (φ1 ⊗ ψ1) α (φ2 ⊗ ψ2). For those of the form (φ1 ⊗ ψ1) α−1 (φ2 ⊗ ψ2)

using the analogous diagram one gets the same final expression as above with g replaced byg−1.

If τ had two representations ω′ (φ′ ⊗ ψ′) and ω (φ⊗ ψ) with the same g, then ω = ω′,therefore (φ′ ⊗ ψ′) = (φ⊗ ψ) and the two representations are equal.

2.4 The case of a category fibred in groupoids

We wish to apply the tensor product we have defined to moduli spaces. To do this, we mustdeal with stacks. We will briefly give the definitions we require, which are numerous bythe nature of the subject. The procession of definitions below are only meant to be a quickreference. The curious reader looking for an introduction to the topic is advised to peruse a


more elaborate treatise such as [LMB00].Let p : C → D be a functor. Given a morphism f : X → Y in C, we can represent the

application of p to f in a diagram

Xf //

p

Y

p

p(X)p( f ) // p(Y)

If by analogy with commutative diagrams, we consider whether the above diagram hasthe properties of a cartesian square, i.e. a fibre product, we obtain the definition of a stronglycartesian morphism in C. A morphism f : X → Y in C is called strongly cartesian if for everypair of morphisms g : Z → Y and η : p(Z) → p(X) such that p(g) η = p(g), there exists aunique morphism h : Z → X such that p(h) = η, and f h = g.

Extending the analogy above with commutative diagrams, if we consider the propertyanalogous to the existence of fibre products, we arrive at the definition of a fibred category.Given a functor p : C → D, C is called a fibred category overD if for every morphism φ : U →V inD, and every object Y in C such that p(Y) = V, there exists a strongly cartesian morphismf : X → Y such that p( f ) = φ.

Given a functor p : C → D, for each object U in D, the fibre category CU is defined asthe subcategory of C whose objects lie over U, and whose morphisms lie over the identitymorphism 1U . A fibred category C over D is said to be fibred in groupoids if for every objectU of D, the fibre category CU is a groupoid.

Now we will consider the setting in which a groupoid is acting on a fibred category.Let G be a monoidal category, and C a groupoid with a right G-action. We say that G acts

faithfully on the objects of C if whenever Cγ ∼−→ Cγ′ in C, then γ ∼−→ γ′ in G. Clearly we canmake the same definition for left actions.

For example H1 acts faithfully on the objects ofMnΦ. The key property is EndR(A) = A for

(A, ι, λ) ∈ MnΦ. Then if φ : I⊗R A ∼−→ J⊗R A for any R-modules I and J, on A-points we have

φA : I ⊗R R ∼−→ J ⊗R R.

The following definitions have been borrowed from [Gra76].If p : C → D is a functor and G acts on C, we say that G acts fibrewise on C if p is G-

invariant. That is to say, p(γφ) = p(φ) for all morphisms φ of C. If p : C → D is fibred, we saythat the action of G is cartesian if it it preserves strongly cartesian morphisms.

Now let p : X → S be a category fibred in groupoids, equipped with a cartesian G-actionon the left. Let C ⊗G X denote the tensor product over G, and define the functor π : C ⊗G X →S by

π(C⊗ X) = p(X), π( f ⊗ φ) = p(φ), π(αC,g,D) = p(1D)


Proposition 39. Let C be a groupoid with a right-action of G, and let p : X → S be a category fibredin groupoids with a cartesian left-action of a 2-group G. Suppose further that G acts faithfully on theobjects of X . Then the category C ⊗G X is fibred in groupoids over S via the functor π defined above.

Proof. Let f : T → S be a morphism in S . Suppose that C ∈ C and X ∈ X with p(X) = S.Then there is an object f ∗(X) over T in X , and a strongly cartesian morphism fX : f ∗(X)→ Xin X lying over f . We show that 1C ⊗ fX is strongly cartesian in C ⊗G X .

Let g : U → T be a morphism in S , and ξ : D⊗ Y → C⊗ X a morphism in C ⊗G X lyingover f g. We must show there exists a unique morphism φ : D⊗ Y → C⊗ f ∗(X) lying overg such that fX φ = ξ.

By proposition 38, ξ can be written as

ξ = (1⊗ ρ) α (h⊗ ψ),

where α = αC,γ,γ−1X : Cγ⊗ γ−1X → C⊗ γ(γ−1X) is an associator, and ρ : γ(γ−1X) ∼−→ X is acomposition of canonical isomorphisms from the left action of X , with γ ∈ G. Here

h : D → Cγ, ψ : Y → γ−1X.

Since π maps α and 1⊗ ρ to 1S, we have

f g = π(ξ) = π(1⊗ ρ) π(α) π(h⊗ ψ) = π(h⊗ ψ) = p(ψ).

Let fγ−1X : f ∗(γ−1X) → γ−1X be strongly cartesian in X lying over f . Then there existsζ : Y → f ∗(γ−1X) lying over g such that fγ−1X ζ = ψ. Thus we have

h⊗ ψ = (1Cγ ⊗ fγ−1X) (h⊗ ζ).

Now, noting that (1C)g = 1Cg, by naturality of associators we have a commutative square

Cγ⊗ f ∗(γ−1X)

αC,γ, f ∗(γ−1X)

1Cγ⊗ fγ−1X // Cγ⊗ γ−1X

αC,γ,γ−1X

C⊗ g f ∗(γ−1X)1C⊗γ f

γ−1X // C⊗ g(g−1X)

So that we can write

ξ = (1⊗ ρ) α (1Cγ ⊗ fγ−1X) (h⊗ ζ) = (1⊗ ρ) (1C ⊗ γ fγ−1X) α′ (h⊗ ζ)

= (1⊗ (ρ γ fγ−1X)) α′ (h⊗ ζ),

with α′ = αC,γ, f ∗(γ−1X) as in the diagram above.

We recall that the action of G on X is cartesian, so that γ fγ−1X is a strongly cartesian mor-


phism in X . The morphism ρ : γ(γ−1X) ∼−→ X is also strongly cartesian, since it lies above 1X.Thus the composition

ρ γ fγ−1X : γ f ∗(γ−1X)→ X,

is strongly cartesian in X , lying over f , and pulling back X. Since fX : f ∗(X) → X is alsostrongly cartesian over f , there exists a unique map µ : γ f ∗(γ−1X)→ f ∗(X) over 1T such thatfX µ = ρ γ fγ−1X. Thus we have

ξ = (1C ⊗ (ρ γ fγ−1X)) α′ (h⊗ ζ) = (1C ⊗ fX) (1C ⊗ µ) α′ (h⊗ ζ) = (1C ⊗ fX) φ,

with φ = (1C ⊗ µ) α′ (h⊗ ζ) lying over g, as required. This shows existence.

Now suppose φ, φ′ : D ⊗ Y → C ⊗ f ∗(X) both lie over g and ξ = (1C ⊗ fX) φ = (1C ⊗fX) φ′. We will show that φ = φ′ necessarily.

By the existence result we have just proved, the maps φ and φ′ both factor through 1C ⊗g f ∗(X) : C ⊗ g∗( f ∗(X)) → f ∗(X), where g f ∗(X) : g∗( f ∗(X)) → f ∗(X) pulls back f ∗(X) overg. Furthermore, the map fX g f ∗(X) : g∗( f ∗(X)) → X is strongly cartesian over f g. We cantherefore reduce to the case where U = T, and g = 1T.

Thus we have ξ = (1C ⊗ fX) φ = (1C ⊗ fX) φ′, where φ and φ′ are maps D ⊗ Y →C⊗ f ∗(X) lying over 1T. Applying prop 38, we get

φ = ω0 ( f0 ⊗ ψ0).

We have f0 is an isomorphism since C is a groupoid, and from 1T = π(φ) = π( f0 ⊗ ψ0) =

p(ψ0), we get that ψ0 is also an isomorphism. Therefore so is φ, and the same argument appliesto φ′. Now we apply prop 38 again, this time to φ−1 and φ′−1:

φ−1 = ωγ (k⊗ η), φ′−1

= ωγ′ (k′ ⊗ η′),

where γ, γ′ ∈ G. Hereωγ = (1D ⊗ ρ) αD,γ,γ−1Y,

andk⊗ η : C⊗ f ∗(X)→ Dγ⊗ γ−1Y.

In particular η : f ∗(X)→ γ−1Y is an isomorphism, since it lies over 1T. From the analogousexpression for φ′ we get η′ : f ∗(X) ∼−→ γ′−1Y. Therefore, γ′−1Y ∼−→ γ−1Y. Now using theassumption that G acts faithfully on the objects of X , we obtain γ ∼−→ γ′. Thus without loss ofgenerality, we can assume ωγ = ωγ′ .

Now taking inverses of the expressions for φ and φ′ we have

φ = (k−1 ⊗ η−1) ω−1γ , φ = (k′−1 ⊗ η′

−1) ω−1

γ .


And from the two expressions for ξ

(k−1 ⊗ ( fX η−1)) ω−1γ = (k′−1 ⊗ ( fX η′−1)) ω−1

γ ,

thereforek−1 ⊗ ( fX η−1) = k′−1 ⊗ ( fX η′−1).

It follows that k = k′ and fX η−1 = fX η′−1. Now since fX is strongly cartesian in X , wehave η = η′. Thus φ = φ′ and so 1C ⊗ fX is strongly cartesian. This shows that C ⊗G X isfibred in groupoids over S .

Chapter 3

Application to moduli problems

In this chapter we apply the results obtained to moduli spaces of abelian schemes. Following[KR09], we define the moduli stack Mn

Φ parametrizing certain principally polarized abelianschemes of relative dimension n. We then construct objects inMn

Φ(S) by applying Serre’s con-struction to objects inM1

Φ(S) and Hermn(R). Using the tensor product of categories outlinedin the previous chapter we can package these constructions into a substack Hermn(R)⊗H1M1

Φ

ofMnΦ. We show that this substack is all ofMn

Φ(S) over certain base schemes S. This is truefor instance, if S is finite type over C, or Spec K with K ⊂ C algebraically closed. We will usethe the case S = Spec Q in the subsequent chapter to describe the Galois action of Gal(Q/L)onMn

Φ(Q).

3.1 The moduli stackMnΦ

We briefly recallMnΦ from the introduction. In [KR09], Kudla and Rapoport define a moduli

problemMk(n− r, r) of polarized abelian schemes with suitable action by the ring of integersof a quadratic imaginary number field k, satisfying signature conditions on the Lie algebra.These are represented by Deligne-Mumford stacks, and for 0 < r < n are integral models ofcertain Shimura varieties. The moduli problems we consider generalize the quadratic imag-inary field k to a CM-type (K, Φ), but otherwise coincide with the r = 0 case of the Kudla-Rapoport spaces. For CM-types of a special form, the same spaces appear in [How12].

3.1.1 Definitions

Let R denote the ring of integers of a CM field K ⊂ C, and let Φ a CM-type for K, with L itsreflex field. We equip R with the involution given by complex conjugation. Let [K : Q] = 2g,and fix a positive integer n. LetMn

Φ → Sch/OL denote the category fibred in groupoids definedas follows. For each locally noetherian scheme S over SpecOL, Mn

Φ(S) consists of triples(A, ι, λ) where:

• A is an abelian scheme over S

43

CHAPTER 3. APPLICATION TO MODULI PROBLEMS 44

• ι : R → EndS(A) is a ring homomorphism such that the Rosati involution on End0(A)

matches complex conjugation on K.

• λ : A→ A∨ is an OL-linear principal polarisation.

In addition, we require that the action of R on LieS(A) satisfy the signature condition:

charpoly(ι(a)|LieS(A), T) = ∏φ∈Φ

(T − φ(a))n, a ∈ R

The expression on the right hand side above is a polynomial with coefficients in OL. Hereit is considered as a global section of the sheaf OS[T] via the map OL[T] → OS[T] induced bythe structure morphism of the scheme S/ SpecOL. Note that this condition forces A to be ofrelative dimension ng over S.

Morphisms in MnΦ(S) are defined to be R-linear isomorphisms φ : (A, ι, λ) → (B, , ν)

such that φ∨ ν φ = λ. The pullback functor inMnΦ is given by base change. ThenMn

Φ isrepresented by a Deligne-Mumford stack over SpecOL.

When K is quadratic imaginary over Q, this definition is the case r = 0 of the Kudla-Rapoport moduli spacesM(n− r, r) defined in [KR09]. When n = 1, this is complex multipli-cation stack denoted CMΦ by Ben Howard in [How12].

3.1.2 Constructing objects ofMnΦ(S)

The objects of the moduli stackMnΦ can be constructed fromM1

Φ via Serre’s construction in asystematic way.

Proposition 40. Let M be a projective module of rank n over R, equipped with a non-degeneratepositive definite R-hermitian form h : M→ M∨. If S is a locally noetherian scheme over SpecOL and(A, , ν) is an object of M1

Φ(S), then the triple

(M⊗R A, M⊗ , h⊗ ν),

is an object of MnΦ(S). Here M⊗ denotes the R-action given by

(M⊗ )(a) = 1M ⊗ (a) : M⊗R A→ M∗ ⊗OK A, for all a ∈ R.

Proof. By construction (M ⊗R A, M ⊗ ) is an abelian scheme over S with an R-action. Bytheorem 33, h⊗ ν is a polarization. Since h and ν are both isomorphisms, so is h⊗ ν, thereforeas a polarization it is principal. It is also R-linear since for a ∈ R we have

(h⊗ ν) (M⊗ (a)) = h⊗ (∨(a) ν) = (M∨ ⊗ (a∗)∨) (h⊗ ν)

= (M⊗ (a∗))∨ (h⊗ ν)

= (M⊗ )∨(a) (h⊗ ν),


where we have used theorem 12 on the second line.

The signature condition follows directly from 14. Thus the triple (M ⊗R A, M ⊗ , h ⊗ ν)

belongs toMnΦ(S).

Let Hermn(R) denote the category of pairs (M, h) consisting of a finite projective R-moduleM of rank n along with a non-degenerate positive definite hermitian structure h. As the propo-sition shows, we can combine objects inM1

Φ(S) with objects in Hermn(R) to obtain objects inMn

Φ(S). In the next section we will study morphisms between the objects inMnΦ constructed

in this way, and later we will introduce a categorical tensor product containing all such com-binations. Finally, we investigate which objects inMn

Φ can be constructed this way.

3.2 Homomorphisms between constructed objects inMnΦ

The purpose of this section is to characterize the homomorphisms in HomMnΦ(M⊗R A, N ⊗R

B) in terms of maps in Hermn(R) and M1Φ. We are therefore given objects (A , i, λ′) and

(B, j, µ′), such that there exist (M, h), (N, k) in Hermn(R), and (A, ι, λ),(B, , µ) in M1Φ, and

isomorphismsα : (M⊗R A, ιM, λM) ∼−→ (A , i, λ′),

β : (N ⊗R B, N , µN)∼−→ (B, j, ν′).

We are interested in HomMnΦ(A , B). Any such map is uniquely of the form β Ψ α−1,

Ψ : (M⊗R A, ιM, λM) ∼−→ (N ⊗R B, B, µB),

where Ψ is an R-linear isomorphism of abelian schemes such that Ψ∗(µB) = λM. Thus it’senough to determine all such Ψ.

First considering all R-linear maps, recall that we have

HomR(M⊗R A, N ⊗R B) ∼−→ HomR(M, N)⊗R HomR(A, B).

If HomR(A, B) is the zero module, necessarily Ψ = 0, so we assume this is not the case.Since A, B are CM abelian schemes, the non-zero maps in HomR(A, B) are isogenies. Letus fix an R-linear isogeny ψ : B → A. Then f 7→ ψ f gives an injection of R-modulesHomR(A, B) → EndR(A). Since we have ι : R ∼−→ EndR(A), the module HomR(A, B) can beidentified with an ideal of R. It is finitely generated and torsion-free, hence it is a projectiveR-module, necessarily of rank one.

Given f ∈ HomR(A, B), and g ∈ HomR(B, A) we let

f ′ def= λ−1 f ∨ µ ∈ HomR(B, A), g′ def

= µ−1 g∨ λ ∈ HomR(A, B).


Similarly for r ∈ EndR(A) and s ∈ EndR(B) we have the involutions

r′ def= λ−1 r∨ λ ∈ EndR(A), s′ def

= µ−1 s∨ µ ∈ EndR(B).

Our overlapping choice of notation is justified by the relations

( f g)′ = g′ f ′, (g f )′ = f ′ g′.

Now with A and B as before let us fix an isogeny ψ : B → A. We obtain an injection ofmodules

α : HomR(A, B) → R, f 7→ ι−1(ψ f ),

as well asβ : HomR(B, A) → R, g 7→ ι−1(g ψ′).

Let a and a′ denote the ideals in R corresponding to the images of α and β. Since R is aDedekind domain, multiplication in R induces an R-linear isomorphism

a′ ⊗R a ∼−→ R, a′ ⊗ a 7→ a′a

Similarly, composition of morphisms gives us an R-linear map

κ : HomR(B, A)⊗R HomR(A, B)→ EndR(A), g⊗ f 7→ g f .

Then we can draw the commutative diagram

HomR(B, A)⊗R HomR(A, B) κ //

β⊗α

EndR(A)

ι−1ψ′ψ

a′ ⊗R a // R

,

which is checked by

ι−1 ψ′ ψ κ(g⊗ f ) = ι−1(ψ′ ψ g f ) = ι−1(g ψ′ ψ f )

= ι−1(g ψ′)ι−1(ψ f ) = β(g)α( f ).

Note that in the second equality we have used the fact that ψ′ψ ∈ EndR(A). Now since thebottom arrow is an isomorphism onto its image a′a ⊂ R, it follows that

ι−1 ψ′ ψ κ : HomR(B, A)⊗R HomR(A, B)→ a′a.

is also an isomorphism.

Furthermore. recall that ι, are assumed to identify complex conjugation on R with the


Rosati involutions induced by λ and µ. In other words for r ∈ R,

ι(r) = ι(r)′ = λ−1 ι(r) λ, (r) = (r)′ = µ−1 (r)∨ µ.

Given an element r ∈ a ⊂ R, we have ι(r) = f ψ for some f ∈ HomR(A, B), so that

ι(r) = ( f ψ)′ = ψ′ f ′ ∈ a′.

Since f 7→ f ′ is a bijection between HomR(A, B) and HomR(B, A) the above implies that asideals in R,

a = a′.

These facts will be useful in combination with the following observation, for which we willrequire that n is prime to the class number of R. We will make this assumption for the rest of thissection.

Lemma 41. Suppose there is an R-linear isomorphism M⊗R A ∼−→ N ⊗R B, where A, B ∈ M1Φ and

M, N have rank n. Suppose further (n, h) = 1 where h is the class number of R. Then HomR(A, B)⊗R

HomR(B, A) is free of rank 1.

Proof. Let Ψ : M⊗R A ∼−→ N ⊗R B be an R-linear isomorphism with inverse Ψ′. Then on thelevel of A-points we have

ΨA : M⊗R EndR(A) ∼−→ N ⊗R HomR(A, B),

and similarlyΨ′B : N ⊗R EndR(B) ∼−→ M⊗R HomR(B, A).

Using the isomorphisms ι : R ∼−→ EndR(A) and : R ∼−→ EndR(B) and canonical identificationsM⊗R R ∼= M and N ⊗R R ∼= N we obtain isomorphisms

M ∼−→ N ⊗R HomR(A, B), N ∼−→ M⊗R HomR(B, A).

Since M and N are rank n projective modules over a Dedekind domain, there exist rank-oneprojective modules I and J, uniquely determined up to isomorphism, and isomorphisms ofmodules

M ∼−→ Rn−1 ⊕ I, N ∼−→ Rn−1 ⊕ J.

Denoting HomR(A, B) and HomR(B, A) by a and a′, we obtain

Rn−1 ⊕ I ∼−→ (Rn−1 ⊕ J)⊗R a, Rn−1 ⊕ J ∼−→ (Rn−1 ⊕ I)⊗R a′.

From this, using again the characterization of finite rank projective modules over a Dedekind


domain, we haveI ∼−→ an ⊗ J, J ∼−→ (a′)n ⊗ I.

These together imply I ∼−→ (a⊗ a′)n ⊗ I, whence (a⊗ a′)n ∼−→ R. Thus the class in Pic(R) ofthe rank one projective module a⊗R a′ has order dividing n. The same order must also divideh = #Pic(R), so by the assumption (n, h) = 1, it is equal to 1. In other words a⊗R a′ ∼−→ R.

Remark: The condition (n, h) = 1 is rather artificial, and is only used in the above lemma.There is good reason, by way of deformation theory, to expect that the conclusion of this lemmaholds without the condition. In other words, that

HomR(A, B)⊗R HomR(B, A) ∼−→ R.

However, we shall have to make do with this restriction in place.Recall that we would like to determine all isomorphisms

Ψ : (M⊗R A, ιM, λM) ∼−→ (N ⊗R B, N , µN),

which we have identified with a subset of

HomR(M, N)⊗R HomR(A, B).

If HomR(A, B) should be free of rank one, Ψ would be representable as a pure tensor, and theproblem would become simpler. In fact we have

Proposition 42. Suppose Φ : (M ⊗R A, ιM, h ⊗ λ) → (N ⊗R B, N , µ) is a morphism in MnΦ.

Suppose also that HomR(A, B) is free of rank one. Then there exists a positive-definite hermitianstructure k : N → N∨, an R-linear principal polarization ν : B→ B∨, and isomorphisms

f : (M, h) ∼−→ (N, k) ∈ Hermn(R), and φ : (A, ι, λ) ∼−→ (B, , ν) ∈ M1Φ,

such that µ = k⊗ ν and Φ = f ⊗ φ. Furthemore, the other choices for f , φ, k and ν equal r−1 f , rφ,rr∗k and (rr∗)−1ν for r ∈ R×.

Proof. The polarization µ is an element of

HomR(N ⊗R B, N∨ ⊗R B∨) ∼= HomR(N, N∨)⊗R HomR(B, B∨).

Since HomR(B, B∨) is free of rank one, it is generated by some element ξ, and µ = k0 ⊗ ξ forsome R-linear map k0 : N → N∨. Since HomR(A, B) is also free of rank one, fixing for it a baseφ, we have similarly Φ = f ⊗ φ, where f : M→ N is R-linear.

Since Φ preserves polarizations,

h⊗ λ = ( f ⊗ φ)∨ (k0 ⊗ ξ) ( f ⊗ φ) = ( f ∨ k0 f )⊗ (φ∨ ξ φ),


so that for some element r ∈ R,

φ∨ ξ φ = r λ, h = f ∨ k0 f r,

where we have abused notation for multiplication by r. Now since h is an isomorphism, themap f ∨ must be surjective. On the other hand, we can also write h = r f ∨ k0 f by linearity.Comparing images, we have

M = im(h) = r im( f ∨) = rM.

By Nakayama’s lemma r must be a unit of R.

Thus r λ = φ∨ ξ φ is injective, hence so is φ. Since the non-zero map φ : A → B isan isogeny, it must be an isomorphism. Therefore, since A admits a principal polarization, sodoes B. We can then assume that the base ξ for HomR(B, B∨) is a principal polarization. Nowwe can consider (B, , ξ) as an object ofM1

Φ. Since µ = k0⊗ ξ is a principal polarization, by themain theorem of chapter 1, k0 : N → N∗ must be a non-degenerate positive-definite hermitianform.

Now since ξ is a polarization, so is r λ = φ∨ ξ φ. Since λ is also a polarization, theunit r must be totally positive. Now if we let ν = r−1ξ and k = rk0, then ν is also a principalpolarization, and k is also a non-degenerate positive definite hermitian form. Furthermoreν⊗ k = ξ ⊗ k0 = µ, and

φ∨ ν φ = λ, f ∨ k f = h.

This shows that φ : (A, ι, λ) ∼−→ (B, , ν) and f : (M, h) ∼−→ (N, k) are morphisms in M1Φ

and Hermn(R) respectively.

Now if Φ = f ′⊗φ′ for any other isomorphisms f ′, φ′, then φ′ φ−1 must be a unit, since it isan invertible element of EndR(A). Conversely, given any such unit r, one can set ν′ = (rr∗)−1ν

and k′ = rr∗k. Because rr∗ is a totally positive unit, (N, k′) ∈ Hermn(R) and ν′ is again aprincipal polarization. Then again we have

(φ′)∨ ν′ φ = λ, ( f ′)∨ k′ f ′ = h,

so f : (M, h) → (N, k′), and φ : (A, ι, λ) → (B, , ν′) are isomorphisms of triples. This showsthat all possibilities f r−1 ⊗ rφ can occur.

It turns out that the general case of an isomorphism

Ψ : (M⊗R A, ιM, h⊗ λ) ∼−→ (N ⊗R B, N , ν⊗ k)

can be reduced to the case considered in the proposition. Defining

a = HomR(A, B), N′ = N ⊗R a, B′ = a−1 ⊗R B,


we have

Lemma 43. With notation as above, the R-modules HomR(A, B′) and HomR(B′, B′∨) are free of rankone.

Proof. For the first assertion,

HomR(A, B′) = HomR(A, a−1 ⊗R B)

' HomR(R, a−1)⊗R HomR(A, B)

' a−1 ⊗R a ' R.

For the second, we note that since B admits a principal polarization, HomR(B, B∨) is free ofrank one. Then we have

HomR(B′, B′∨) = HomR(a−1 ⊗R B, (a−1)∨ ⊗R B∨)

' HomR(a−1, (a−1)∨)⊗R HomR(B, B∨)

' HomR(a−1, R)⊗R (a−1)∨

Now in our notation, the ring R acts on a dual module via its involution, i.e. complex conju-gation, so that for any fractional ideal b ⊂ R, we have b∨ ∼= b

−1. Therefore

HomR(a−1, R)⊗R (a−1)∨ ∼= HomR(a

−1, R)⊗R a ∼= a⊗R a ' R,

where the last isomorphism follows from lemma 41. This shows that HomR(B′, B′∨) is free ofrank one.

We have an isomorphism ω : N′ ⊗R B′ → N ⊗R B, given on T-points by

ωT : N′ ⊗R B′(T) = (N ⊗R a)⊗ (a−1 ⊗ B(T))) −→ N ⊗R B(T).

(n⊗ a)⊗ (a′ ⊗ t) 7→ n⊗ ((I(a⊗ a′)) t),

where I is any isomorphism I : a⊗R a−1 ∼−→ R.

Let ′ denote the R-action on B′, and ′N′ the induced action on N′ ⊗R B′. It is evident thatthe map ω is R-linear. Equipping N′ ⊗R B′ with the principal polarization

µ′ = ω∗(k⊗ ν) = ω∨ (k⊗ ν) ω,

we get an isomorphism of triples

ω : (N′ ⊗R B′, ′N′ , µ′) ∼−→ (N ⊗R B, N , k⊗ µ).


Thus we can write our original isomorphism of triples

Ψ : (M⊗R A, ιM, h⊗ λ) ∼−→ (N ⊗R B, N , ν⊗ k),

as Ψ = ω Φ, where

Φ = ω−1 Ψ : (M⊗R A, ιM, h⊗ λ) ∼−→ (N′ ⊗R B′, ′N′ , µ′).

By lemma 43, we can apply proposition 42 to Φ. It follows that there exist k′ and ν, withµ′ = k′ ⊗ ν′, such that (N′, k′) ∈ Hermn(R) and (B′, ′, ν′) ∈ M‘

Φ, and further Φ = f ⊗ φ,where f : M → N′ and φ : A → B′ are structure-preserving isomorphisms. In other words, Φarises as a tensor product of maps inM1

Φ and Hermn(R).

It now remains to identify the map ω. We note first that ν′ : B′ → (B′∨) is a principalpolarization. Since B also admits a principal polarization and

HomR(B′, B′∨) ∼= HomR(a−1, (a−1)∨)⊗R HomR(B, B∨),

we must have ν′ = α⊗ ν, with α : a−1 → (a−1)∨. Again by the main theorem of chapter 1,we deduce that α must be a positive-definite non-degenerate hermitian form on the rank oneprojective module a−1. This shows that the object (B′, ′, ν′) can be written as

(B′, ′, ν′) ∼= (α, a−1)⊗ (B, , ν).

We can also write ω as a composition

ω = (idN ⊗ (λB (I ⊗R B) α−1a,a−1,B)) αN,a,B′ , (Ω)

of canonical isomorphisms,

αN,a,B′ : (N ⊗ a)⊗R B′ ∼−→ N ⊗ (a⊗R B′),

αa,a−1,B : (a⊗ a−1)⊗R B→ a⊗ (a−1 ⊗R B),

λB : R⊗R B ∼−→ B.

I : a⊗ a−1 ∼−→ R,

The first three isomorphisms above are canonical, and expected to exist for any definitionof tensor products of categories. We will show that a canonical choice can also be made forI : a⊗ a−1 → R.

Lemma 44. Let A,B ∈ M1Φ, and a = HomR(A, B). If A is R-linearly isomorphic to B′ = a−1⊗R B,


then the canonical map

κ : HomR(A, B)⊗R HomR(B, A)→ EndR(A),

induced by composition is an isomorphism of modules.

Proof. We have a canonical map

HomR(A, B′ ⊗R B)⊗R HomR(B′, A)→ EndR(A), f ⊗ g 7→ g f ,

that is always injective. Since A and B′ are isomorphic, the image ideal contains idA, thereforethe map is also surjective. We have canonical isomorphisms

HomR(A, B′)⊗R HomR(B′, A) ∼= HomR(R⊗R A, a−1 ⊗R B)⊗R HomR(a−1 ⊗R B, R⊗R A)

∼= HomR(R, a−1)⊗R HomR(A, B)⊗R HomR(a−1, R)⊗R HomR(B, A)

∼= HomR(A, B)⊗R HomR(B, A).

Tracing the above isomorphisms reveal that it is compatible with the composition map. Wecarry this out below for the sake of completeness. Let

α⊗ φ⊗ f ⊗ ψ ∈ HomR(R, a−1)⊗R HomR(A, B)⊗R HomR(a−1, R)⊗R HomR(B, A).

Set a = α(1). The element

a⊗ φ ∈ a−1 ⊗R HomR(A, B) ∼= HomR(R, a−1)⊗R HomR(A, B)

corresponds to the map in HomR(A, B′) sending t ∈ A(T) to r⊗ φ t. Similarly,

f ⊗ ψ ∈ HomR(a−1, R)⊗R HomR(B, A)

is the homomorphism in HomR(B′, A) that sends a⊗ t′ ∈ a−1 ⊗R B(T) to f (a)⊗ ψ (t′). Thecompositions of the two maps corresponds to the element in HomR(A, R⊗R A) given on T-points by

t 7→ f (a)⊗ ψ φ(t),

which is simply f (a) ψ φ. Now the map

a−1 ⊗R HomR(a−1, R) 7→ R, a⊗ f 7→ f (t)

is an isomorphism, therefore the element α⊗ φ⊗ f ⊗ ψ is mapped to

f (t) φ⊗ ψ ∈ R⊗R HomR(A, B)⊗R HomR(B, A),


which is mapped to f (t) φ ψ.

This shows that the two compositions have the same image. Since B′ ' A, this image is allof R.

This shows that there is a canonical choice for all the ingredients of the map ω.

Thus we see that up to canonical isomorphisms, any Φ can be identified with a tensorproduct of isomorphisms f ⊗ φ. In fact, since ω depends only on the domain and codomainof Φ, the map f ⊗ φ is uniquely determined. We have thus proved the following:

Theorem 45. Let (A, ι, λ), (B, , µ) ∈ M1Φ and (M, h), (N, k) ∈ Hermn(R). Then there exists a

morphism ofMnΦ

Φ : (M⊗R A, ιM, h⊗ λ) ∼−→ (N ⊗R B, N , k⊗ µ),

if and only if there exist morphisms inM1Φ and Hermn(R)

f : (M, h) ∼−→ (N ⊗R a, k), φ : (A, ι, λ) ∼−→ (a−1 ⊗R B, ′, ν),

such that Φ = ω ( f ⊗ φ), where a = HomR(A, B), a−1 = HomR(B, A), and ω is the canonicalisomorphism

ω : (N ⊗ a)⊗R (a−1 ⊗R B) ∼−→ N ⊗R B,

ω = (idN ⊗ (λB (I ⊗R B) α−1a,a−1,B)) αN,a,B′

given on T-points byωT : (n⊗ f )⊗ (g⊗ t) 7→ n⊗ (g f t).

In that case all other choices for ( f , φ, k, µ) are (r−1 f , rφ, rr∗k, (rr∗)−1ν), with r ∈ R×.

3.3 Second Main Theorem

As previously mentioned the 2-group H1 acts on the categories Hermn(R) and M1Φ, on the

right and left respectively. Since the formation of Serre’s tensor construction commutes withbase change, the action of H1 onM1

Φ is cartesian. It is also faithful on objects, since EndR(A) =

R for (A, ι, λ) ∈ M1Φ. Thus by proposition 39 the tensor product Hermn(R)⊗H1M1

Φ → SchOL

is again a category fibred in groupoids over Sch/OL . For a map of schemes f : S→ S′ in Sch/OL

the pullback functor

f ∗ : Hermn(R)⊗H1 M1Φ(S

′)→ Hermn(R)⊗H1 M1Φ(S),

can be defined by sending M⊗R A to M⊗R f ∗(A).

Now to avoid confusing the formal construction of tensor objects with the Serre tensorconstruction, we change notation slightly. Given (M, h) ∈ Hermn(R) and (A, ι, λ) ∈ M1

Φ we


will denote the image of the pair under

T : Hermn(R)×M1Φ → Hermn(R)⊗H1 M1

Φ

by M A, and image of morphisms ( f , φ) by f φ.The functor S : Hermn(R)×M1

Φ →MnΦ which sends two objects (M, h) and (A, ι, λ) to

the Serre construction (M⊗R A, ιM, h⊗ λ) factors through T. Thus we get

Hermn(R)×M1Φ

T ))

S //MnΦ

Hermn(R)⊗H1 MnΦ

Σ

77 ,

with S = Σ T. Evidently, Σ is a functor between categories fibred in groupoids over Sch/OL .The main theorem of this chapter is the following.

Theorem 46. Suppose n is prime to the class number of K. Then the functor Σ : Hermn(R) ⊗H1

M1Φ →Mn

Φ is fully faithful.

Proof. It is enough to show that Σ is fully faithful when restricted to each fibre (the generalstatement following from the existence of strongly cartesian morphisms). Thus we pick S ∈Sch/OL , and let (M, h),(N, k) ∈ Hermn(R) and (A, ι, λ),(B, , ν) ∈ M1

Φ(S). Let

Φ : (M⊗R A, ιM, h⊗ λ)→ (N ⊗R B, N , k⊗ ν)

be a morphism inMnΦ(S). We wish to show that Φ is in the image of the functor Σ. By theorem

45, we have Φ = ω ( f ⊗ φ) for some

f : (M, h) ∼−→ (N ⊗R a, k), φ : (A, ι, λ) ∼−→ (a−1R ⊗ B, ′, ν),

where a = HomR(A, B), a−1 = HomR(B, A), and ω is an isomorphism

ω : (N ⊗R a)⊗R (a−1 ⊗R B) ∼−→ N ⊗R B.

We have f ⊗ φ = Σ( f φ), so we must show ω is in the image of Σ. From equation (Ω) wehave

ω = (idN ⊗ (λB (I ⊗R B) µ−1a,a−1,B)) αN,a,B′ .

The map αN,α,B′ is the image under S of an associator morphism, and

U = λB (I ⊗R B) αa,a−1,B

is a morphism inM1Φ so that idN ⊗U = Σ(idN U). This shows that Φ is in the image of Σ,

so that Σ is surjective on homomorphisms.


Now suppose that Σ(τ) = Σ(τ′) for two morphisms τ and τ′ in MnΦ. By prop 38 every

morphism τ : M A→ N B of Hermn(R)⊗H1MnΦ can be represented in the form ωI ( f

φ) for some fractional ideal a ∈ Hermn(R) and

ωI = (1N ⊗ (λB α−1a,I−1,B)) IaB αN,a,a−1B

Then we have τ = ωI ( f φ) and τ′ = ωI′ ( f ′ φ′), where

f : M ∼−→ N ⊗R a, φ : A ∼−→ a−1 ⊗R B,

andf ′ : M ∼−→ N ⊗R a′, φ′ : A ∼−→ a′

−1 ⊗R B.

Then on B-points we have two isomorphisms

φB : HomR(A, B) ∼−→ a−1 ⊗R EndR(B), φ′B : HomR(A, B) ∼−→ a′−1 ⊗R EndR(B),

from which we get that a is isomorphic to a′. Thus we can replace a′ by a in the representationsof τ and τ′. By the last part of proposition 38, we must have f φ = f ′ φ′.

3.4 Stackification of Hermn(R)⊗H1M1Φ

AlthoughMnΦ is a stack, Hermn(R)⊗H1 M1

Φ as we have defined it may not be one. It does,however, have one of the required properties.

Corollary 47. For each T ∈ Sch/OL , and x, y in X = Hermn ⊗H1 M1Φ over T, the presheaf of

morphisms MorXT (x, y) : Sch/T → Set is a sheaf.

Proof. By the proposition, X → Sch/OL can be identified with a full subcategory ofMnΦ. Thus

MorXT (x, y) is identified with MorMnΦT(x, y), which is a sheaf sinceMn

Φ is a stack.

As above, let X denote the tensor product Hermn(R) ⊗H1 M1Φ. Then X admits a stack-

ification X ′, and there exists a morphism ξ : X → X ′ over Sch/OL which is universal asa morphism of X to stacks over S. In particular, there exists a unique morphism of stacksΞ : X ′ → Mn

Φ over S , such that Σ = Ξ ξ. In general, if x and y are objects of X overU ∈ S , then the functor MorX (x, y) → Mor′X (ξ(x), ξ(y)) is the sheafification of the presheafMorX (x, y). By the corollary above, MorX (x, y) is already a sheaf, therefore

MorX (x, y) ∼−→ Mor′X (ξ(x), ξ(y)).

Then X ′ can be constructed as a category whose objects consist of descent data relative toa family of etale covers, with appropriate morphisms [LMB00, Ch.3, Lemma 3.2]. In otherwords X ′ consists of those objects in Mn

Φ which are etale locally isomorphic to an object in


Hermn(R) ⊗H1 M1Φ. When speaking of Hermn(R) ⊗H1 M1

Φ as a substack of MnΦ, we mean

the stackification X ′.

3.4.1 The essential image

It then remains to measure the essential image of Hermn(R)⊗H1 M1Φ → Mn

Φ. The followingproposition shows that if the abelian scheme in a triple inMn

Φ arises from the tensor construc-tion, the rest of the data does as well.

Proposition 48. Let (B, , ν) and (A, ι, λ) be objects inMnΦ(S) andM1

Φ(S) respectively, and M afinite projective R-module. An isomorphism B ∼−→ M⊗R A of abelian schemes extends to an isomor-phism ofMn

Φ(S)(B, , ν) ∼−→ (M⊗R A, ιM, λM)

for a unique non-degenerate positive definite hermitian form h on M.

Proof. By transport of structure, we can assume B = M⊗R A. Let µ : R→ End(M) denote themodule structure on M. For r ∈ R, we have

(r) = µ(r)⊗ idA = M⊗ ι.

The abelian scheme dual to M⊗R A is isomorphic to M∨ ⊗R A∨, so the polarization λ canbe viewed as an element of

HomR(M⊗R B, M∨ ⊗R B∨),

which by corollary 8 is isomorphic to

HomR(M, M∨)⊗R HomR(A, A∨).

The principal polarization λ : A → A∗ is a base for the free R-module HomR(A, A∨) of rankone. Therefore, λ = h⊗ ν for a unique R-linear map h : M → M∨. By corollary 34, h must bea positive definite non-degenerate hermitian form.

Thus if for some S we succeed in showing that every abelian scheme inMnΦ(S) is of the

form M⊗R A, then functor above induce an equivalence of categories

Hermn(R)⊗H1 M1Φ 'Mn

Φ.

In the next section, we show this for S = Spec K, for K any algebraically closed subfield ofC.

3.4.2 Complex points ofMnΦ

Let (A, ι, λ) ∈ MnΦ(C). In this section, we will show that in fact A is, up to isomorphism, an

object of Hermn(R)⊗H1 M1Φ.


Let Λ = H1(A, Z). As a real manifold, A is isomorphic to (Λ ⊗Z R)/Λ. The space Λbecomes an R-module via the maps induced by the various ι(a) on homology, for a ∈ OK. Itis finite and torsion-free over OK, hence it is projective. The OK-action on Λ determines theOK-action on the tangent space Λ⊗Z R, thereby recovering ι.

Recall that the choice of a CM type (K, Φ) induces an isomorphism

OK ⊗Z R∼−→ Cg, a⊗ r 7→ (rφ1(a), rφ2(a), ..., rφg(a)),

where Φ = φ1, ..., φg are embeddings of OK into C. The ring OK acts on Cg through thismap, and for a ∈ OK, each factor C(i) on the right hand side is an eigenspace, with eigenvalueφi. Thus the characteristic polynomial of the R-linear action of a on Cg is

(T − φ1(a))(T − φ2(a))...(T − φg(a)) ∈ OL[T],

where OL is the ring of integers of the reflex field L ⊂ C.

Now we have

Λ⊗Z R∼−→ (Λ⊗OK OK)⊗Z R

∼−→ Λ⊗OK (OK ⊗Z R) ∼−→ Λ⊗OK Cg.

Now, we can write Cg as C(1) × C(2) × ...C(g) where a ∈ OK acts on C(i) by φi(a) via theisomorphism OK ⊗Z R

∼−→ Cg. Then we have

Λ⊗OK Cg ∼−→ (Λ⊗OK Cg)⊗Z Q∼−→ Kn ⊗K ⊗Cg ∼−→ Cn

(1) ×Cn(2) × ...Cn

(g).

Each Cn(i) factor is an eigenspace for ι(a) with eigenvalue φi(a). Now, let J denote the

complex structure of A on the tangent space Λ⊗Z R. By the signature condition on Lie(A),J matches the action of OK via the embeddings φi. Since K contains negative square roots,the action of OK via φi determines J completely on each Cn

(i) factor, and therefore on the en-tire tangent space. Thus J is identified with the ordinary complex structure on (Cg)n via theisomorphism Λ⊗Z R→ Cg.

Now consider the exact sequence of OK-modules

0 // OKΦ // Cg // A0 // 0,

where A0 = Cg/OK. Now tensoring the above sequence by Λ we obtain another sequence

0 // Λ⊗OK OKΛ⊗Φ // Λ⊗OK Cg // Λ⊗OK A0 // 0 ,

which is exact because Λ is projective.

On the other hand,

A ' (Λ⊗Z R)/Λ ∼= (Λ⊗OK Cg)/(Λ⊗OK OK).


ThereforeA ' Λ⊗OK A0

as complex tori, and hence as abelian varieties.

Now, if A0 admits anOK-linear principal polarization, then by appeal to prop. 48 we wouldbe done. In general, Cg/OK may not admit a principal polarization at all. We can replace OK

by an arbitrary fractional ideal a by noting:

A ' Λ⊗OK A0 ' (Λ⊗OK a−1)⊗OK (a⊗OK A0) ∼= Λ′ ⊗OK Cg/a,

with Λ′ = Λ⊗OK a−1.

Then we have:

Proposition 49. Let (K, Φ) be a CM type, with Φ = φ1, ..., φg. Then there exists a fractional ideala of OK such that the abelian variety Cg/a admits a principal polarization.

Proof. If A = Cg/a, then the dual abelian variety A∨ is isomorphic to Cg/b where b = (da)−1

and d is the different ideal of K/Q. Furthermore in this uniformization, any polarization mapA→ A∨ is given by a diagonal matrix

T[ζ] = diag(φ1(ζ), φ2(ζ), ..., φg(ζ)),

for some ζ ∈ K such that ζ = ζ, and Im φi(ζ) > 0 for φi ∈ Φ (cf. [Shi98][§§14.2]).

Thus Cg/a admits a principal polarization if and only if for some ζ ∈ K as above

ζa = (da)−1.

Since every complex abelian variety admits a polarization, there exists some α ∈ K suchthat α = −α and Im φi(α) > 0. Then necessarily K = F(α), and we can assume further that α isintegral. Let β = −α2. Then α has minimal polynomial f (X) = X2 + β over OF, therefore thedifferent element δL/K(α) = f ′(α) = 2α belongs to the different ideal d (cf. [Neu90][§III, 2.5].)

Now let ζ = 2α, and observe that ζ = −ζ and Im(φi(ζ)) > 0) as required. We first claimthat ζd can be written as d0OK for some fractional ideal d0 of OF. Note that d = dK/FdF/Q,so it suffices to show this for ζdK/F. The different dK/F is generated by elements x − x forx ∈ OK, so ζdK/F is generated by ζ(x − x) ∈ F, from which the claim follows. Thus therelation ζa = (da)−1 can be rewritten as

d0 = NK/F(a−1),

where d0OK = ζd.

Let q be a prime of F in the prime factorization of d0. If p is a prime of K over q, thenp|d0OK = ζd. In fact, since (ζ) ⊂ d, we have p|d, hence p is ramified over F. Since [K : F] = 2,


p is totally ramified, and we have NK/F(p) = q. Thus if

d0 = qr11 q

r22 ...qrk

k ,

is the prime factorizaiton of d in F, we can take

a = p−r11 p−r2

2 ...p−rkk ,

where pi lies over qi. Then we have NK/F(a−1) = d0 as required. In other words for a as above,

Cg/a admits a principal polarization given by the map

Cg/a→ Cg/(da)−1, (z1, ..., zg) 7→ (2φ1(α)z1, ..., 2φg(α)zg).

Given an abelian variety A0 = Cg/a with a principal polarization λ0, we can always choosean action of OK on A0 with respect to which λ0 is OK-linear. Thus we have shown that for any(A, ι, λ) ∈ Mn

Φ(C), there exists a triple (A0, ι0, λ0) inM1Φ(C) such that

A = M⊗OK A0,

for some finite projective OK-module M. Now by proposition 48 we can conclude that thereexists a non-degenerate positive-definite hermitian form h : M→ M∨ such that

(A, ι, λ) ' (M⊗OK A0, ι0M, h⊗ λ0).

Hence we have shown the following

Proposition 50. The tensor construction induces an equivalence of categories

Hermn(OK)⊗H1 M1Φ(C) ∼−→Mn

Φ(C).

Corollary 51. The same statement holds with C replaced by any algebraic closed field K of characteristiczero.

Proof. Let (A, ι, λ) ∈ MnΦ(K). Then (A, ι, λ) is defined over a subfield K0 ⊂ K which is finitely

generated over Q, and so embeds in C. By the proposition, A is isomorphic over C to an objectM⊗R A0 ∈ Hermn(OK)⊗H1 ⊗M1

Φ(C). Both the isomorphism and the corresponding object(A0, ι0, λ0) ∈ M1

Φ(C) can be defined over a number field contained in C, hence also over afinite extension K1 of K0. The field K1 is isomorphic to a subfield of K since K is algebraicallyclosed. Therefore (A, ι, λ) is isomorphic to an object in Hermn(OK)⊗H1Mn

Φ(K). The corollarynow follows from proposition 48 and theorem 46.


3.4.3 Schemes of finite type over C

The following theorem is due to to Grothendieck [Gro66].

Theorem 52. Let S be a connected reduced scheme of finite type over a field k of characteristic zero.Suppose A and B are two abelian schemes over S, l is a prime number, and ul : Tl(A)→ Tl(B) is a ho-momorphism. Suppose also that there exists a point s ∈ S such that (ul)s comes from a homomorphismus : As → Bs of abelian schemes over k(s). Then there exists a homomorphism u : A → B such thatTl(u) = ul .

Suppose as in the theorem above, that S is a reduced connected scheme of finite type overC. Let (A, ι, λ) be an object ofM(S).

Proposition 53. Any closed point s ∈ S has an etale neighbourhood U such that the base change(AU , ιU , λU) ∈ M(U) is given by Serre’s construcction. In other words, (AU , ιU , λU) lies in theessential image of the functor

Hermn(OK)⊗OK A(U)→M(U).

Proof. If f : A → S is the structure map of A, then the sheaf R1 f∗(Z) is locally constantof finite rank, so it can be considered as the π1(S, s)-module H1(As, Z). One observes thatπ1(S, s) must preserve the hermitian structure of H1(As, Z). We claim the image of π1(S, s) isfinite in GL(H1(As, Z)).

Given any positive definite hermitian module (M, h) of rank m, one can considerM as alattice inside km. An automorphism of (M, h) extends linearly to an automorphism of km. Suchan automorphism must preserve M, so it must lie in a discrete subgroup of GLn(k). On theother hand it must also preserve the hermitian structure h, so it must also lie in the subgroupof matrices unitary with respect to h. This subgroup is compact since h is positive-definite. Theintersection of compact and discrete subgroups being finite, so is the automorphism group ofthe hermitian module (M, h).

Now it follows that the action of π1(S, s) factors through a finite quotient. Therefore, thereexists an etale neighbourhood g : U → S of s with g(u) = s, such that the action of π1(U, u)on g∗(R1 f∗(Z)) is trivial. The map g : U → S is smooth, and f : A → S is quasi-compactbecause it is proper. By the smooth base change theorem, the sheaf g∗(R1 f∗(Z) is the sameas R1( fU(Z)), where fU : AU → U is the structure sheaf of AU = A×S U. This shows thatthe action of π1(U, u) on H1(Au, Z) is trivial. Hence, the sheaf R1( fU∗(Z)) is constant. Itfollows that the sheaf Tl(A) is also constant, since for an abelian variety A0 over C, Tl(A0) =

H1(A, Z)⊗Z Zl .

Now, A is an abelian variety over C so there is an isomorphism φ : A → M ⊗OK E forsome (any) complex elliptic curve E with OK-action. Let EU denote the constant elliptic curveE⊗SpecOK U over U.


The tate modules Tl(A) and Tl(M ⊗OK EU) = M ⊗OK Tl(EU) are constant, the latter bydesign and the former by the discussion above. Hence the isomorphism of Tate modulesφlTl(Au)→ Tl(M⊗OK E) extends to an isomorphism of Tl(A) and Tl(M⊗OK E). By Grothendieck’stheorem mentioned above, φ extends to a map Φ : AU → M ⊗OK Eu. By the same ar-gument, the inverse ψ of φ also extends to a map Ψ going the other way. Now applyingGrothendieck’s theorem again, by the uniqueness property Φ Ψ and Ψ Φ must both bethe identity maps. We have shown that Φ : AU → M ⊗OK EU is an isomorphism. By theresults of the previous sections this suffices to show (AU , ιU , λU) is in the essential image ofHermn(OK)⊗OK A(U)

The above proposition can be interpreted as an isomorphism of stacks.

Corollary 54. Let Hermn⊗H1M1Φ denote the stackification of Hermn ⊗H1 M1

Φ, and let

Ξ : Hermn⊗H1M1Φ →Mn

Φ

be the induced morphism of fibred categories. Then Ξ is an isomorphism of stacks over the category offinite type schemes over C.

Proof. We have already shown that the functor Ξ is fully faithful over Sch/OL . The previousproposition shows that over the restricted base, the objects inMn

Φ are etale locally isomorphicto objects in Hermn ⊗H1 M1

Φ, so they are isomorphic to an object in the stack Hermn⊗H1M1Φ.

This shows that the functor Ξ is essentially surjective over the restricted category. Thus Ξ is anequivalence of fibred categories, hence an isomorphism of stacks.

To be able to prove the same statement as above with less restrictions on the base category,we must show that the the functor Ξ is essentially surjective. This amounts to detecting whenthe abelian scheme of a triple inMn

Φ can be obtained, etale locally at least, via Serre’s tensorconstruction. In the next section we provide some useful criteria for this purpose.

3.5 Criteria for being in the image of Hermn(R)⊗H1M1Φ

We wish to investigate when an abelian scheme B/S is isomorphism to another of the formM ⊗R A. Here as usual R is a ring, and M is a finite projective right R-module, and A is anabelian scheme over S equipped with a ring homomorphism : R→ EndS(A).

Suppose B = M⊗R A. Then HomS(A, B) is by definition M⊗R HomS(A, A). We see thatif EndS(A) = R, then HomS(A, B) ∼= M. Thus M can be recovered from B and A.

Now suppose R is commutative, and EndR(A) = R, that is, (R) is its own commutant inEndS(A). Then by corollary 8 in chapter 1 we have

HomR(A, B) = HomR(A, M⊗R A) ∼= HomR(R, M)⊗R HomR(A, A) ∼= M,


with the isomorphism M ∼−→ HomR(A, B) mapping m to the homomorphism A→ M⊗R Awhich on T-points sends t ∈ A(T) to m⊗ t ∈ M⊗R A(T).

Now we suppose R is a Dedekind domain.

Lemma 55. Let A and B be abelian schemes over S with action by a Dedekind domain R. If HomR(A, B)is non-zero, it is a finite projective R-module.

Proof. Since HomR(A, B) is a free Z-module of finite rank, it is finitely generated as a Z-module, hence also finitely generated over R. Over a Dedekind domain, a finitely generatedmodule is projective if and only if it’s flat, and a module is flat if and only if it’s torsion free.Now let r ∈ R. Then since R⊗Q is a field, r has an inverse in R⊗Q of the form r′n−1, withr′ ∈ R and non-zero integer n, so that rr′ = n. Now if for x ∈ HomR(A, B) we have rx = 0,then nx = r′rx = 0. But nx = 0 implies x = 0, since HomR(A, B) is free over Z. This showsHomR(A, B) is torsion-free over R, hence it is flat, and so projective.

By this lemma, whenever HomR(A, B) is non-zero, we can form the abelian scheme HomR(A, B)⊗R

A by Serre’s tensor construction. As R is commutative, M is naturally an R-bimodule, and sowe can equip HomR(A, B) ⊗R A with an action of R. Then we have natural R-linear homo-morphism

Φ : HomR(A, B)⊗R A→ B,

given on T-points byΦT( f ⊗ t) = f t, t ∈ A(T).

The following proposition gives us a criterion for an abelian scheme to be of Serre-Type.

Proposition 56. Let R be a Dedekind domain, A,B abelian schemes over S with action by R. SupposeHomR(A, B) is non-zero. Then there exists an R-linear isomorphism from B to M⊗R A for some M,if and only if Φ is an isomorphism.

Proof. One direction is obvious: if Φ is an isomorphism, then it is an R-linear isomorphismfrom B to M⊗R A for M = HomR(A, B).

Suppose we have an R-linear isomorphism ψ : B→ M⊗R A. Then in particular

ψA : HomR(A, B)→ M⊗R HomR(A, A),

is an isomorphism of R-modules. Let λ denote the natural isomorphism

λ : M⊗R R→ M, λ(m⊗ r) = mr.

Then silently identifying HomR(A, A) with R, we obtain an isomorphism

λ ψA : HomR(A, B) ∼−→ M,


and after tensoring with idA, we get yet another isomorphism

Λ : HomR(A, B)⊗R A→ M⊗R A,

Λ = (λ ψA)⊗ idA

. Note since λ and ψA are R-linear, so is Λ. We show the following diagram commutes:

Hom(A, B)⊗R AΛ

((Φ

Bψ

// M⊗R A

.

Then since Λ and ψ are isomorphisms, so is Φ = ψ−1 Λ. We show the commutativity onT-points. Suppose f ⊗ t belongs to Hom(A, B)⊗R A(T). Then

(ψ Φ)T( f ⊗ t) = ψT ΦT( f ⊗ t) = ψT( f t),

andΛT( f ⊗ t) = λ ψA( f )⊗ t.

The map t : T → A induces the following commutative diagram:

B(A)ψA//

t∗

(M⊗R A)(A)

t∗

B(T)ψT

// (M⊗R A)(T)

,

where t∗ is precomposition with t. We first note that for any m⊗ r ∈ M⊗R R, we have

t∗(m⊗ r) = m⊗ rt = mr⊗ t = λ(m⊗ r)⊗ t.

Then for f ∈ HomR(A, B), on the one hand

ψT(t∗( f )) = ψT( f t) = (ψ Φ)T( f ⊗ t),

on the othert∗(ψA( f )) = λ(ψA( f ))⊗ t = ΛT( f ⊗ t).

Then by the commutativity of the above diagram ΛT = (ψ Φ)T for any T, thereforeΛ = ψ Φ, and so Φ = ψ−1 Λ is an isomorphism as desired.

We apply the proposition to the case S = Spec k for a field k.

Corollary 57. Suppose that (B, ), and (A, ι) are abelian varieties over k with action by a Dedekind


domain R. Let k be an algebraic closure of k and suppose that all maps in Homk,R(A, B) are definedover k. Then if B is isomorphic to some M⊗R A over k, it is also isomorphic to it over k.

Proof. If such an isomorphic exists over k, by proposition 56, the map

Homk,R(A, B)⊗R A→ B,

f ⊗ t 7→ f t.

is an isomorphism. But if all maps in Homk,R(A, B) are defined over k, the definition abovemakes sense over k as well.

Groups with Compact Factors

Introduction

In this short interlude we first quickly recall the definition of a Shimura variety and that ofa canonical model. Next we consider a weakening of the axioms and show that if G is aconnected group with Gad(R) semi-simple and compact, the associated double coset spacehas a unique canonical model as a kind of zero-dimensional Shimura variety.

In the next chapter, we will consider the special case of G = GU(n), which corresponds toa moduli space of the kind studied in the previous chapter.

Background

We briefly recall the basic definitions and facts of the theory of Shimura varieties, followingMilne’s exposition. We refer to Milne’s article [Mil05] for more detail. Standard references forShimura varieties are the two articles of Deligne ([Del71, Del79]). Let S denote the real groupResC/R(GmC), so that S(R) ' C× and S(C) ' C×C. We fix these isomorphisms such that theinclusion R → C induces z 7→ (z, z) on S(R)→ S(C).

Let G be a reductive group over Q, and X a G(R)-conjugacy class of real homomorphismsh : S→ GR. The pair (G, X) is called a Shimura datum if it satisfies the following axioms:

S1: The representation of S on Lie(Gad)C induced by h contains only the characters z/z, 1,and z/z.

S2: adh(i) is a Cartan involution on Gad.

S3: Gad has no Q-factors H with H(R) compact.

Let (G, X) be a Shimura datum, and K an open compact subgroup of G(A f ). The corre-sponding double coset space

ShK(G, X) = G(Q)\X× G(A)/K

has the structure of a variety. A choice of such a structure is called a canonical model if it isdefined over an explicitly determined number field E = E(G, X) (defined below), and satisfies

65


certain reciprocity laws. It is known that such models exist uniquely. We will make this defini-tion precise below, following Milne’s formulation. First we need a few constructions attachedto (G, X):

For any x = hx ∈ X, let the cocharacter µx(z) of GC be defined by µx(z) = hC(z, 1). Notingthat any such homomorphism can be defined over Qal, we denote its field of definition by E(x).The reflex field E(G, X) ⊂ Qal is the fixed field of those elements of Gal(Qal/Q) that stabilizethe G(Qal)-conjugacy class µxx∈X of cocharacters of GQal . It follows that E(x) ⊂ E(G, X) forevery x, and it is known that if G is quasi-split over some field k ⊃ E(G, X), then there existsan x ∈ X such that µx is defined over k (first lemma in [Kot84]).

For a given x ∈ X, if T ⊂ G is a torus over Q such that hx(C×) ⊂ T(R), then the data(T, x) is called a special pair. In this case one can associate an algebraic homomorphism rx :(Gm)E(x)/Q → T over Q, induced by the map

P ∈ E(x)× 7→ ∑σ:E(x)→C

σ(µ(P)),

where σ ranges over all complex embeddings of E(x). Note that the right hand side a prioribelongs to T(E(x)), but because of Galois invariance lands in T(Q). Note also that sum opera-tion on the right hand side, being multiplication in G, is independent of the choice of the torusT, so that rx depends only on x if any such torus T exists. On A f points, rx induces a grouphomomorphism AE(x) → G(A f ), which we also denote by rx by abuse.

A canonical model MK(G, X) for ShK(G, X), is a variety structure on ShK(G, X) defined overE(G, X) such that for every special pair (T, x) and all a ∈ G(A f ), the points

[x, a]K ∈ G(Q)\X× G(A f )/K,

have coordinates in E(x)ab, and the action of Gal(E(x)ab/E(x)) on those coordinates is givenby

σ[x, a]K = [x, rx(s)a]K.

Here s ∈ A×E(x) is any idele that maps to σ under the reciprocity map of class field theory:artE(x)(s) = σ|E(x)ab . Note that by convention artE(s) = recE(s)−1. Elsewhere in this articlewe have referred to the global reciprocity map, by which we mean the map recE : A×E →Gal(Eab/E) which is normalized to take uniformizers to arithmetic Frobenius elements.

A canonical model M(G, X) for the system Sh(G, X) of varieties is a compatible system ofcanonical models MK(G, X), endowed with a regular right-action of G(A f ) compatible withthat induced by right multiplication via the isomorphism

lim←−K

ShK(G, X) ∼= G(Q)\X× G(A f ).


Groups with G(R) compact

In [Del71] the condition S3 on the pair (G, X) is necessary in order to apply the theorem ofstrong approximation to G. It would be interesting to investigate the degree to which one mayrelax this condition while maintaining the main results of the theory.

In what follows we extend all the definitions of the previous sections to pairs (G, X) whichsatisfy S1 and S2, but not necessarily S3. One may ask whether canonical models exist for suchShK(G, X). We show that if G is connected, with Gad semi-simple and compact, this is alwaysthe case. The following lemma shows that in such cases h is trivial on Gad.

Lemma 58. Let G be a reductive group over Q with Gad semi-simple and Gad(R) compact. Then anyalgebraic homomorphism h : S → GR has its image contained in the centre of the identity componentof G.

Proof. Since Gad is semi-simple and compact, the Killing form κ on g = Lie(Gad) is negativedefinite. Let x ∈ gC be a non-zero vector and suppose S acts through h on x ∈ gC by thecharacter χ. Then we have

κC(Ad h(z)(x), Ad h(z)(x)) = κC(χ(z)x, χ(z)x) = χ(z)2κC(x, x).

On the other hand, κ is Aut(g)-invariant, so

κC(x, x) = χ(z)2κC(x, x).

Since κC(x, x) < 0, we have χ(z) = ±1. Now, S(R) is connected and so is its image under χ,therefore χ ≡ 1. In other words Ad h is trivial. If G0 denotes the identity component of G,we have

im(h) ⊆ ker Ad = ZG(G0).

Furthermore, im(h) is connected, therefore it is contained in G0. From G0 ∩ ZG(G0) =

Z(G0), we conclude that im(h) ⊂ Z(G0).

Note that if Gad(R) is compact, any pair (G, X) automatically satisfies condition S2 fora Shimura datum, since the subgroup of fixed points of adh(i) is closed inside the compactgroup Gad(R), and is hence itself compact. If Gad is further semi-simple, the lemma impliesthat condition S1 is also satisfied. Condition S3 is clearly violated so (G, X) is not a Shimuradatum.

Nevertheless one may form the associated double coset spaces and ask the same questions.In particular, one may ask about the existence of canonical models.

Proposition 59. Let G be a connected reductive group G over Q with GadR semisimple and compact,

and h : S → GR a homomorphism defined over R. Then the associated inverse system Sh(G, X) ofdouble coset spaces admits a unique canonical model.


Proof. By the lemma, h is trivial on Gad so that the G(R)-conjugacy class X consists of a singlepoint. Therefore for any compact open subgroup K of G(A f ),

ShK(G, X) = G(Q)\X× G(A f )/K

is a finite set.To give the structure of a variety over a number field E to a finite set is to equip it with an

action of the Galois group Gal(E/E). For a canonical model for ShK(G, X), this Galois actionis prescribed on special points by a reciprocity law. Recall that these are points [x, a]K such thathx(C×) is contained in T(R) for some torus T. When the image of h lies in the centre of G(R),every point in ShK(G, X) is special, since hx(R) is contained in a torus, e.g. any maximal one.In this case, the prescribed reciprocity law completely determines the action of Gal(E/E), andhence the canonical model itself. We need only check that the definition is consistent.

Let r, µ be the norm and weight maps associated to h, and let E be the field of definition ofµ. Since h is central, so is µ, and the associated G(Qal) conjugacy class of cocharacters of GQal

consists of a single element. This shows that the reflex field of (G, X) is the field of definitionof µ, i.e. E = E(G, X).

A canonical model for each ShK(G, X) must match the unique action of Gal(Qal/E) on theset ShK(G, X), prescribed by the formula

σ[x, a]K = [x, r(s)a]K, ∀σ ∈ Gal(Qal/E), ∀a ∈ G(A f ),

where s ∈ A×E is any idele with artE(s) = σ|Eab .For any k ∈ K, we have [x, a]K = [x, ak]K, as well as [x, r(s)a]K = [x, r(s)ak]K, so that the

formula above is well-defined after passing to the right-quotient by K. For any q ∈ G(Q), wehave x = qx, since X is a single element, therefore [x, a]K = [qx, qa]K = [x, qa]K. Since r(s)belongs to the centre of G(A f ) we have [x, r(s)a]K = [x, qr(s)a]K = [x, r(s)qa]K, so that theabove formula is also well-defined after passing to the left-quotient by G(Q). This shows thatthe action is well-defined on the finite set ShK(G, X), prescribing a unique canonical model forit.

The action on ShK(G, X) described above commutes with right action of G(A f ). Thereforeright multiplication by an element of G(A f ) is defined over E and is compatible with theinverse system structure of ShK(G, X). This shows the unique system of canonical models forthe various ShK(G, X) does provide a canonical model for Sh(G, X).

In the next chapter, we will investigate the double coset space for G = GU(n) where wehave an interpretation of Sh(G, X) as isomorphism classes of abelian varieties with extra data.We will show that the Galois action on these classes obtained by base change satisfies a reci-procity law and matches what is prescribed above.

Chapter 4

Galois action and complexmultiplication

Let (K, Φ) be a CM-type and n > 0 a positive integer such that (n, h) = 1, with h the classnumber of K. By the results of the previous chapters, we have a fully faithful functor

Σ : Hermn(OK)⊗H1 M1Φ(S)→Mn

Φ(S).

The Galois group Gal(Q/L) acts on the Q points of MnΦ. When n = 1, this action is

described explicitly by the main theorem of complex multiplication. Using this descriptionand the above equivalence, we can describe the Galois action onMn

Φ.

4.1 Hermitian fractional ideals

Every rank one projective OK-module is isomorphic to an ideal of OK. We start with a fewbasic observations about these.

All OK-linear maps between fractional ideals are multiplication by an element of the field.Indeed, tensoring such a map with Q we obtain a K-linear map from K to itself. Thus for twofractional ideals a, b, the module HomOK(a, b) is isomorphic to ab−1.

For any fractional ideal a we have a bijection

a−1 ∼−→ a∨, y 7→ (x 7→ yx).

By our convention for the dual module, the above map is OK-antilinear. Therefore as OK-modules the fractional ideal a∨ is identified with a−1.

Lemma 60. Let α : a → a−1, and β : b → b−1 be multiplication by ρ, σ ∈ K× respectively. If

f : a→ b is multiplication by c, thenf ∗(β) = |c|2 σ

ρα.

69

CHAPTER 4. GALOIS ACTION AND COMPLEX MULTIPLICATION 70

Proof. We have

f ∗(β)(x) = ( f ∨ β f )(x) = ( f ∨ β)(cx) = f ∨(σcx) = cσcx = ccσ

ρα(x).

In the classic work [Shi98] on complex multiplication, a group C(K) is defined, each ele-ment of which is a class of positive Hermitian forms in K. We recall its definition here.

The elements of C consist of equivalence classes of pairs (a, ρ), where a is an ideal of Kand ρ ∈ K is a totally positive number such that aa = (ρ). Two such pairs (a, ρ), (a0, ρ0) areequivalent if, for some µ ∈ K, µa = a0 and ρ0 = ρµµ. Multiplication is given by

(a, ρ).(b, σ) = (ab, ρσ),

and (OK, 1) is the identity element.The next lemma shows that Herm1(OK) can be considered as a categorification of the

group C(K).

Lemma 61. The map that sends the pair (a, ρ) representing a class in C(K), to the hermitian structure

H : a−1 × a−1 → OK, α(x, y) = ρyx,

is an isomorphism from C(K) to the group of isomorphism classes of objects in Herm1(OK).

Proof. For each pair (a, ρ), the form H is hermitian and positive-definite since ρ is totally pos-itive. It is OK-valued and non-degenerate since aa = (ρ). If two pairs (a, ρ) and (a0, ρ0) areequivalent via µ ∈ K as above, then x 7→ µ−1x is an isomorphism of hermitian modulesa→ a−1. Thus each element in C(K) defines an object in Herm1(OK) up to isomorphism. Thisshows the described map is well-defined.

Now let α : a → a−1 be an object in Herm1(OK). If we tensor α by Q we obtain a K-linearmap K → K. Hence α is multiplication by an element r ∈ K, and we have

ra = a−1,

therefore aa = (r−1). Thus the linear map α corresponds to a bilinear form H given by

H(x, y) = α(y)(x) = ryx.

Since H is positive-definite, r|x|2 must be totally positive for x ∈ K, hence r must be totallypositive. This shows that every hermitian form in Herm1(OK) can be obtained, up to isomor-phism, from an element of C(K).

Finally, suppose α : a → a∨ and β : b → b are objects in Herm1(OK) and φ : α → b is anisomorphism. We can take a and b to be fractional ideals and identify a∨, b∨ with a−1, b−1 so


that α, β are by multiplication by totally positive elements ρ, σ ∈ K. Since φ is OK-linear, itis also multiplication by some µ ∈ K. The condition φ∗β = α, by lemma 60, is equivalent toρ = σµµ. Thus µ gives the corresponding equivalence between (a−1, ρ) and (b−1, σ) in C(K),so that the stated map has trivial kernel.

We note that for a pair (a, ρ) in C(K), the element ρ is unique, since aa as a principal idealcan have at most one totally positive generator. Thus the significance of ρ in the pair (a, ρ) isthat it indicates the existence of a (unique) positive-definite non-degenerate hermitian form ona−1 (and so on a itself). In other words, the group described in the lemma is the kernel of thehomomorphism from the class group of K to its narrow class group.

Finally, we enter a remark about the action of Herm1(OK) on Hermn(OK). Let (M, h) bean object of Hermn(OK) and α : a→ a∨ in Herm1(OK). We can form the tensor product

h⊗ α : M⊗OK a→ M∨ ⊗OK a∨,

which corresponds to a positive-definite OK-valued bilinear form

H : M⊗OK a×M⊗OK a→ OK,

given byH(m⊗ s, m′ ⊗ s′) = H(m, m′)α(s, s′).

Now if MK denotes the K-vector space M ⊗OK K, the hermitian form H extends to a K-linear form HK : MK × MK → K. On the other hand, the inclusion i : a → K induces aninjection

i : M⊗OK a → MK,

through which the hermitian form HK can be restricted to M ⊗OK a. Thus from a hermitianform on M, we have two ways of constructing a K-valued hermitian form on M⊗OK a. Thesetwo forms are related as follows.

Lemma 62. With notation as above

H(x, y) =1r

HK(x, y), x, y ∈ M⊗OK a

Proof. We have

HK(i(m⊗ s), i(m′ ⊗ s′)) = HK(ms, m′s′)

= HK(m, m′)ss′ = H(m, m′)η(s, s′)r

= H(m⊗ s, m′ ⊗ s′)r


4.2 Galois action on triples over Q

4.2.1 Complex multiplication

In this section we will rephrase the main theorem of complex multiplication in the languageof Serre’s tensor construction, so that we may apply our results. In this we closely follow[CCO12, Appendix A.2], with only a slight alteration in form. In later sections we will applythis formulation to the problem of Galois action onMn

Φ(Q).Now suppose that (K, Φ) is a CM-type with reflex (L, Ψ). Then the reflex norm induces a

homomorphism of ideal groups

NΦ : IL → IK, NΦ(a) = ∏σ∈Ψ

aσ.

Then for any b in the image of this map

bb = NΦ(a)NΦ(a) = ∏σ∈Ψ

aσ ∏σ∈Ψ

aσ = ∏σ:L→C

a = NL/Q(a).

In particular bb = (r) for a unique positive rational number. This implies that (b, r) ∈ C(K).In other words, we can consider the reflex norm as a map

NΦ : IL → Ob(Herm1(OK)).

By hΦ(a) : NΦ(a) → NΦ(a)∨ we will denote the OK-linear form corresponding to (b, r) as

above. Thus for a ∈ IL, the reflex norm ideal NΦ(a) acts on a triple (A, ι, λ) by sending it to(NΦ(a)⊗OK A, hΦ(a)⊗ λ, NΦ(a)⊗OK ι). We denote this triple by NΦ(a)⊗OK (A, ι, λ).

Lemma 63. Let (A, ι, λ) be a triple as usual, and suppose a, b ∈ IL. If f : NΦ(a) → NΦ(b) ismultiplication by NΦ(c) ∈ K× for some c ∈ L×, and φ = f ⊗ idA, then

φ∗(hΦ(b)⊗OK λ) = r(hΦ(a)⊗OK λ),

where r is the totally positive generator of NK∗/Q(cba−1)

Proof. We have

φ∗(hΦ(b)⊗ λ) = φ∨ (hΦ(b)⊗ λ) φ = ( f ⊗ idA)∨ (hΦ(b)⊗ λ) ( f ⊗ idA)

= ( f ∨ ⊗ idA∨) (hΦ(b)⊗ λ) ( f ⊗ idA)

= ( f ∨ hΦ(b) f )⊗ λ.

By lemma 60, we have

f ∨ hΦ(b) f = NΦ(c)NΦ(c)σ

ρhΦ(a) = NL/Q(c)

σ

ρ,


where ρ, σ are the totally positive generators of NΦ(a)NΦ(a), NΦ(b)NΦ(b) respectively. Thusσρ is the totally positive generator of the ideal NL/Q(b

−1a), and NL/Q(c) σρ is equal to r as in the

statement.

Now let s ∈ A×L be an idele of L, and a ∈ IL the corresponding ideal. We set

NΦ(s) = NΦ(a), hΦ(s) = hΦ(a),

so that now we have a homomorphism

A×L → Ob(Herm1(OK)), s 7→ (NΦ(s), hΦ(s)),

with multiplication in A×L mapping to the tensor product in Herm1(OK).

The properties of the polarizations hΦ(s)⊗ λ shown in lemma 63 characterize the uniquesystem of polarizations arising from the Galois action on the triple (A, ι, λ). This is provedin the appendix of [CCO12, Lemma A2.7.1], as a step along the way to an algebraic proof ofthe main theorem of complex multiplication. The following is the statement of the theoremproved in [CCO12, A.2.8.8], in the special case where the CM order is OK.

Theorem 64. (Main Theorem of Complex Multiplication) Let (A, ι, λ) be a CM abelian variety overQ with multiplication by OK, and polarization λ. Suppose that σ ∈ Gal(Q/L) and s ∈ A×K such thats maps to σ|Lab under the global reciprocity map. There is then a unique isomorphism

Φs,σ : (A, ι, λ)σ → NΦ(s−1)⊗OK (A, ι, λ),

such that the diagram

Tf (A)

NΦ(s−1)

σ // Tf (Aσ)

Tf (Φs,σ)

NΦ(s−1)⊗OK Tf (A) // Tf (NΦ(s−1)⊗OK A)

ψ = ψs,σ : (NΦ(s), hΦ(s))⊗Ok (A, ι, λ)→ (A, ι, λ)σ

is commutative. Here the arrow on the left is multiplication by NΦ(s−1) ∈ A×K , and the bottom arrowis a canonical isomorphism.

That the isomorphisms Φs,σ are natural in the triple (A, ι, λ) may be interpreted as theGalois action on triples factoring through the action of Herm1(OK) via the reflex norm onideles.


4.2.2 Description of the action onMnΦ(Q)

The main theorem of complex multiplication applies to the triples inM1Φ(Q), which are CM

abelian varieties. Now suppose n > 1 and let (A, ι, λ) ∈ MnΦ(Q). For σ ∈ Gal(Q/L) as before

we obtain a new triple by base change

(A, ι, λ)σ = (Aσ, ισ, λσ) ∈ MnΦ(Q),

with ισ(a) = ι(a)σ. We would like to obtain a description of (A, ι, λ)σ similar to that providedby theorem 64 in the n = 1 case.

By corollary 51 we have

(A, ι, λ) ∼−→ (M⊗OK A0, idM ⊗ ι, h⊗ λ0) = (M, h)⊗OK (A0, ι0, λ0),

for (A0, ι0, λ0) ∈ M1Φ(Q), and (M, h) ∈ Hermn(OK). It is then enough to describe triples of

the form(M⊗OK A0, idM ⊗ ι, h⊗ λ0)

σ.

Since Serre’s construction commutes with arbitrary base change, the above is canonically iso-morphic to

(M⊗OK Aσ0 , idM ⊗ ισ, h⊗ λσ

0 ) = (M⊗ h)⊗OK (A0, ι0, λ0)σ.

By theorem 64 we have

(A0, ι0, λ0)σ = (NΦ(s−1), hΦ(s−1))⊗OK (A0, ι0, λ0),

where s ∈ A×K∗ such that NΦ(s−1) ∈ A×K corresponds to σ|(K∗)ab under the reciprocity map.

Thus we have, by associativity of Serre’s tensor construction and the commutativity oftensor products of OK-modules,

(A, ι, λ)σ ∼= (M, h)⊗OK ((NΦ(s−1)⊗OK (A0, ι0, λ0))

∼= NΦ(s−1)⊗OK ((M⊗OK h)⊗OK (A0, ι0, λ0))

∼= NΦ(s−1)⊗OK (A, ι, λ)

Therefore we can claim the same statement for triples inMnΦ(Q) as in theorem 64.

Proposition 65. Let (A, ι, λ) ∈ M(Q). Suppose that σ ∈ Gal(Q/L) and s ∈ A×K such that s mapsto σ|Lab under the global reciprocity map. Then there are canonical isomorphisms

Φs,σ : (A, ι, λ)σ → NΦ(s−1)⊗OK (A, ι, λ),


such that the diagram

Tf (A)

NΦ(s−1)

σ // Tf (Aσ)

Tf (Φs,σ)

NΦ(s−1)⊗OK Tf (A) // Tf (NΦ(s−1)⊗OK A)

ψ = ψs,σ : (NΦ(s), hΦ(s))⊗Ok (A, ι, λ)→ (A, ι, λ)σ

is commutative.

The commutativity of the diagram follows from 64 and the fact that taking Tate modulescommutes with Serre’s construction.

We note that since

NΦ(s−1)⊗OK ((M, h)⊗OK (A0, ι0, λ0)) ∼= ((M, h)⊗OK NΦ(s−1))⊗OK (A0, ι0, λ0),

the proposition may be interpreted as stating that the action of the Galois group onMnΦ(Q) '

Hermn(OK)⊗OK M1Φ(Q) factors through the action of Herm1(OK) on Hermn(OK).

Note that by lemma 62 of the previous section, under the inclusion

M⊗OK NΦ(s) → M⊗OK K = MK,

the hermitian structure h ⊗ hΦ(s) on M ⊗OK NΦ(s) coincides with the restriction to M ⊗OK

NΦ(s) ofNL/Q(s)−1HK(x, y) : MK ×MK → K.

4.3 Double cosets of the unitary group

We have shown that the Galois action on MnΦ(Q) is given by tensoring with an object in

Herm1(OK) given by the reflex norm. There is another description of MnΦ(Q) as a double

coset spaces associated to U(n). The theory of canonical models for Shimura varieties thensuggests another Galois action on these sets. For the double coset spaces of U(n) to behavelike a Shimura variety these two actions should agree.

In this section we show this is the case, by explicitly computing the Galois action onMn

Φ(Q) which is prescribed by its canonical model, and comparing it with the action in propo-sition 65.

Let (K, Φ) be a CM-type, with [K : Q] = 2g. Let n > 0 and fix an n-dimensional hermitianK-vector space (V, H), with H a positive-definite. The general unitary group GU = GUV

associated to V is the algebraic group over Q whose functor of points is given by

GU(R) = x ∈ EndK(V)⊗Q R)|xx∗ ∈ R×,


for R any Q-algebra. Here x 7→ x∗ is the adjoint involution of the hermitian form HR onV ⊗Q R.

The unitary group U = UV associated to V is the subgroup of GU given by

U(R) = x ∈ GU(R)|xx∗ = 1.

Note that EndK(V)⊗Q R ∼= EndK⊗QR(V ⊗Q R). When R = R we have

V ⊗Q R ∼= V ⊗K (K⊗Q R) ' V ⊗K (CΦ) = V ⊗K (C(φ1) × ...×C(φg)) ∼= VC(φ1) × ...×V

C(φg) ,

where C(φ) is the K-algebra φ : K → C. Each V(Cφ) is isomorphic to V ⊗K C with complexstructure induced by φ ∈ Φ. Thus we see that GU(R) is isomorphic to g copies of the generalunitary group associated to the complex hermitian space V⊗K C. For any a ∈ K, the character-istic polynomial of multiplication by a on each VCφ is (T− φ(a))n. Therefore the characteristicpolynomial on all of V is

∏φ∈Φ

(T − φ(a))n.

Now let L be a a self-dual OK-lattice L in V of rank n. By this we mean a finite OK-submodule L ⊂ V of rank n, equipped with a positive definite OK-valued hermitian form.Fix also a fractional ideal a ⊂ K, and a principal polarization on A0 = Cg/a. Such an idealexists by roposition 49.

Then L⊗OK A0 is an element of M1Φ(C), by theorem 33. Conversely, we have the following

lemma.

Lemma 66. The set of isomorphism classes of objects inM1Φ(C) is a principal homogeneous space for

the group C(K) of hermitian ideal classes.

Proof. Recall as in the proof of proposition 49, that an object in M1Φ(C) is of the form Cg/a,

with a ⊂ K a fractional ideal, and the principal polarization given by an element ζ ∈ K withζ = −ζ, and Im(φ(ζ)) > 0 for φ ∈ Φ, such that

aa = ζd−1,

where d is the different ideal of K/Q. Suppose we are given another pair a′ and ζ ′ with thesame properties:

a′a′ = ζ ′d−1, ζ ′ = −ζ ′, Im(φ(ζ ′)) > 0, φ ∈ Φ

Let b = aa′−1, and β = ζζ ′−1 then a pair (a′, ζ ′) satisfies the above conditions if and only if

bb = (β), β = β, φ(β) > 0, φ ∈ Φ.

In other words, (a′, ζ ′) defines an element ofM1Φ(C) if and only if (b, β) is an element of C(K).

An isomorphism class inM1Φ(C) determines b up to isomorphism, and vice versa.


Using this lemma, we can describe all objects inMnΦ(C) by hermitian lattices.

Proposition 67. Let (A0, ι0, λ0) ∈ MnΦ(C) be a fixed object. Then the map

Hermn(OK)→MnΦ(C), L 7→ L⊗OK A0,

is a bijection on objects.

Proof. We know that any object (B, , ν) in MnΦ(C) can be written as M ⊗OK A for some her-

mitian OK-lattice M and (A, ι, λ) ∈ M1Φ(C). By the lemma (A, ι, λ) itself is of the form

(a, α)⊗OK (A0, ι0, λ0), for some fixed (A0, ι0, λ0), and (a, α) in Herm1(OK). Then letting L =

M ⊗ a, we obtain a hermitian OK-lattice L, such that B ∼−→ L ⊗OK A0 as objects in MnΦ(C).

If L ⊗OK A0 and L′ ⊗OK A0 are isomorphic, then by looking at the A0-points of the isomor-phism, we can deduce that L ' L′ as OK-modules. Assuming L = L′ as OK-modules, thetwo hermitian structures h and h′ on L, will satisfy h ⊗ λ = h′ ⊗ λ, therefore h = h′ by thecharacterization of morphisms inMn

Φ(C) given in corollary 7.

Now let L be a fixed OK-hermitian lattice inside a K-vector space V. The group UV(A f )

acts on the set of such lattices by sending L to gL∩V(Q), for g ∈ U(A f ), where the intersectionis taken in V(Q). Let K be the compact open subgroup of U(A f ) defined by

K = g ∈ U(A f )|gL = g.

Then if g = g0hk, for k ∈ K, and g0 ∈ U(Q), then

g0hkL ∩V(Q) = g0(hkL ∩ g−10 V(Q)) = g0(hL ∩V(Q)) ∼−→ hL ∩V(Q).

Thus the action of U(A f ) on the OK-hermitian lattices factors through the double coset

U(Q)\U(A f )/K.

Up to isomorphism, there is only one positive-definite K-vector space V of dimension n,and given an OK-lattice L in V, there is only one complex structure on the real torus VR/Lsatisfying the signature condition

char(ι(a)|V , T) = ∏φ∈T

(T − φ(a))n,

for the action ι of OK on V/L. Under the isomorphism

V ⊗Q R ' VC(φ1) × ...×V

C(φg) ,


the unique element J ∈ EndK⊗R(VR) corresponding to this complex structure is given by

(iI × iI × ...× iI).

If κ ∈ K⊗Q R is the element that maps to (i, i, ..., i) under

K⊗Q R∼−→ CΦ,

then the homomorphismh : S→ UR,

given on real points by h(a + bi) = aI + bJ, can also be described by h(i) = κ I. In particular,the image of h lies in the centre of U(R), and so the U(R)-conjugacy class of h consists of asingle point. Therefore the same double coset space described above can be written as

U(Q)\X×U(A f )/K,

where X is a single point x, with x corresponding to h.

We note that our description of MnΦ(C) in terms of a double coset differs slightly from

the usual setup, as in [KR09], of fixing a K-hermitian space V and an OK-lattice L ⊂ V. Thecrucial difference is that if L ⊂ V is a self-dual OK-valued hermitian OK-lattice, then the com-plex torus V/L is not the abelian variety we have associated to L above. This is because it isunhelpful to express V/L as L ⊗ Cg/OK, since Cg/OK is not always a principally polarizedabelian variety. Instead one must choose, as we have, an ideal a such that A0 = Cg/a admitsa pricipal polarization, then form the abelian variety L⊗OK A0 inMn

Φ(C). This correspondsto taking OK-lattices L′ = L ⊗OK a−1 in V with hermitian forms taking values in a−1. Thetwo descriptions match when the different ideal dK/Q is principal, such as when K is quadraticimaginary over Q. In any case the action of U(A f ) on lattices is the same in both descriptions.

4.3.1 Computation of the reflex norm

Let [x, a]K be a point inU(Q)\X×U(A f )/K,

The point x ∈ X is unique and corresponds to the map h : S→ UR described above. Under theisomorphism C× ' S(R), h is given by a+ bi 7→ aI + bJ ∈ EndK⊗R(VR), where I is the identity,and J the complex structure on VR which satisfies the signature condition. To h is associateda cocharacter µ of GC given by µ(z) = hC(z, 1), where the isomorphism S(C×) ' C× ×C× ischosen so that S(R) ⊂ S(C) corresponds to z 7→ (z, z). The cocharacter µ can be calculatedexplicitly as follows.


Lemma 68. We haveµ(z) =

(z + 1

2

)I +

(z− 1

2i

)JC

Proof. For a real algebra R, if we make the identification

S(R) =

(z w−w z

)∈ M2(C) : z2 + w2 ∈ R×

.

Then the isomorphism S(C) ∼−→ C× ×C× can be given by(z w−w z

)7→ (z + iw, z− iw),

and S(R) ∼−→ C× by (a b−b a

)7→ a + bi.

Then hC : S(C)→ End(VC), a priori given by(z w−w z

)7→ zI + wJC

can be identified with(z1, z2) 7→

(z1 + z2

2

)I +

(z1 − z2

2i

)JC,

from which the lemma follows by evaluating at (z, 1).

We are interested in the field of definition of µ. That is to say, we want to find a numberfield K0 such that µ is the base change from K0 to C of an algebraic homomorphism K×0 → VK0 .The expression obtained for µ in the lemma involves the endomorphism J, which in general isdefined only on VR. The signature condition on Lie(A) ' VR relates J with the K-structure onVR induced from V. This will allow us to descend J to a number field.

To keep notation separate, we will denote the action of an element a ∈ K on V by ι(a) andthe complex action that J induces by (z). Thus

(a + bi) = aI + bJ, for a, b ∈ R.

As in the lemma, ordinary multiplication will be used to denote the complex structure on VC

coming from the tensor product.The compatibility of polarization with action of K implies that ι(a) is a normal operator

on VR with respect to the hermitian form H, and therefore diagonalizable. Let α ∈ K bea primitive element over Q. By the signature condition, the eigenvalues of ι(α) are givenby φj(α) for φj ∈ Φ. Therefore, since every element of K is a polynomial in α, we have an


eigenspace decomposition

VR =n⊕

i=1

Vj, ι(a)|Vj = (φj(a)), ∀a ∈ K.

Let now β ∈ F be a primitive element that generates F over Q. Thus β has degree d overQ, and its conjugates β1, ..., βd are given by β j = φj(β). Since β is totally real, all β j are realnumbers, so that (β j) = β j. Thus for v ∈ Vj, ι(β)v = β jv, so that Vj can be characterized asthe eigenspace for ι(β) corresponding to the eigenvalue β j. Let F denote the Galois closure ofF/Q. Then the minimal polynomial of ι(β) ∈ EndQ(V) splits over F and we have

VF =d⊕

i=1

Uj,

where ι(β)u = β ju for u ∈ Uj, and Uj ⊗F R = Vj. Therefore the projection maps pi : V → Vj

are also defined over F, i.e., pi = πi ⊗F R, where πi : VF → Uj is the projection map onto Uj.

Let α ∈ K now be any totally imaginary number, set αj = φj(α). Since the αj are also totallyimaginary, we have αj = iaj where aj ∈ R. Now if v ∈ Vj, we have ι(α)v = (αj)v = aj J(v). Inother words,

J(vj) = ι(α)/aj, ∀v ∈ Vj.

Then from the expression for µ(z) given above, it follows that

µ(z)|(Vj)C =

(z + 1

2

)+

(z− 1

2

)ι(α)

αj

Let p(X) be the polynomial ∏dj=1(X − αj)(X − αj). Let the constants cj be given by p(X)

X−αj

evaluated at αj. Now define the polynomial Q(X) by

Q(X) =d

∑j=1

p(X)

αjcj(X− αj)+

p(X)

αjcj(X− αj).

Note that Q(αj) = Q(αj) =1αj

for all j.

Recall that the reflex field of the reflex type (K, Φ) is defined as the subfield of Q fixedby those elements σ ∈ Gal(Q/Q) such that σ Φ = Φ. Each such σ permutes the numbersα1, ..., αd. It follows that σ permutes the summands of Q(X), so that Q(X) itself is invariant.Therefore Q(X) is defined over the reflex field L. Then we have

Q(X) = r1X2d−1 + r2X2d−2 + r3X2d−3... + r2d−1X + r2d, rj ∈ K

Since each αj is totally imaginary αj = −αj. Each number cj is a product of 2d− 1 imaginary


numbers, so is itself totally imaginary. We have

Q(X) =d

∑j=1

p(X)

αjcj(X− αj)− p(X)

αjcj(X + αj)

=d

∑j=1

p(X)(X + αj)− (X− αj)

αjcj(X2 − α2

j )

=d

∑j=1

2p(X)

cj(X2 − α2j )

Since p(X) = ∏dj=1(X2 − α2

j ), we deduce that Q(X) has only even powers of X, i.e. r2k−1 = 0.

Now define an element R of K⊗Q L by

R = α2d−1 ⊗ r2 + α2d−3 ⊗ r4 + ... + α⊗ r2d

If v ∈ (Vj)C, then

(α⊗ 1)v = (1⊗ aj)Jv =Ji(1⊗ αj)v,

where iaj = αj as before. It follows that for all r ∈ C,

(α2k+1 ⊗ r)v =

(Ji

)2k+1

(1⊗ rα2k+1j )v =

Ji(1⊗ rα2k+1

j )v.

Using the above, since all the powers of α occuring in R are odd, we obtain

Rv =Ji(1⊗ r2α2d−1

j + 1⊗ r4α2d−3j + ...1⊗ r2dαj)v

Rv =Ji

(1⊗ r1α2d−1

j )v + (1⊗ r3α2d−3j )v + ...(1⊗ r2d−1αj)v

=

Ji(1⊗ αjQ(αj)v) =

Ji(1⊗ αj

1αj)v =

(Ji

)v

Since neither R nor J/i depend on the index j, we have

Rv =

(Ji

)v, ∀v ∈ VC.

Using the above, we can write

µ(z) =(

z + 12

)I +

(z− 1

2

)R.

Since the terms in R involve only the rational action of α and multiplication by elements in K,we have proved:

Proposition 69. The weight µ(z) is defined over the reflex field L of the CM-type (K, Φ).


For P ∈ L×, let

r(P) = ∑σ:L→C

σ(µ(P))

The function r : ResLQGm → GQ, occurs in the formula prescribed by the theory of canonical

models for the action of Gal(Lab/L) on the special points of a double coset space. We willexplicitly compute it for our case.

Recall that the field L can be generated by the reflex traces

TrΦ(a) = ∑φ∈Φ

φ(a),

of all elements a ∈ K. To obtain the reflex type (L, Ψ) one first extends Φ to a CM-type Φ forthe Galois closure K of K. In other words

Φ = φ ∈ Aut(K/Q) : φ|K ∈ Φ.

Then one forms Ψ by restricting the inverse of the elements of Φ to L:

Ψ = ψ : L → C : ∃φ ∈ Φ, ψ = φ−1|L.

Recall also that the reflex norm NΦ : L→ K can be defined by

NΦ(a) = ∏ψ∈Ψ

ψ(a).

The next proposition concludes our calculations.

Proposition 70. The map r : ResLQGm → G(Q) is given by

P 7→ ι(NΨ(P)), P ∈ L×.

Proof. We want to show that for v ∈ V, r(P)v = (NΨ(P) ⊗ 1)v. It suffices to show this forv ∈ VC.

Now let v ∈ (Vj)C for a fixed j. Let φj be an extension of φj to K. Define the CM-type Ψj asφj Ψ. In other words, the restriction to L of φj Φ−1. Thus an element ψ ∈ Ψj is characterizedby this property: for any extension ψ of ψ to E, ψ−1(αj) ∈ α1, ..., αd = Φ(α).

We haver(P) = ∏

ψ∈Ψj

ψ(µ(P))ψ(µ(P))

From our expression for µ(z), we can write, for ψ ∈ Ψ:

ψ(µ(P)) =(

ψ(P) + 12

)I +

(ψ(P)− 1

2

)ψ(R)


Whereψ(R) = α2d−1 ⊗ ψ(r2) + α2d−3 ⊗ ψ(r4) + ... + α⊗ ψ(r2d),

considered as an element of K⊗Q ψ(L).

Recalling how R is obtained from the polynomial XQ(X), we see that ψ(R) correspondssimilarly to XQψ(X), where Qψ is obtained from Q by applying ψ to each of its coeffcients. Letσ be an extension of ψ from L to K. We have

Qψ(αj) = ψ(Q(ψ−1(αj)) = ψ(1/ψ−1(αj)) = 1/αj,

where we have used the fact that Q(αk) = 1/αk for any k, and ψ−1(αj) = αk for some k. Wealso have

Qψ(αj) = ψ(Q(ψ−1(αj)) = ψ(1/ψ−1(αj)) = −1/αj.

Thus by a similar calculation as in the proof of proposition 69, we get that on (Vj)C:

ψ(R) =Ji, ψ(R) = − J

i,

from which it follows that, setting ψ(P) = w = x + iy, we get

ψ(µ(P))ψ(µ(P))

=

(w + 1

2

)I +

(w− 1

2

)Ji

(w + 1

2

)I −

(w− 1

2

)Ji

.

=|w|2 + w + w + 1

4I − |w|

2 − w− w + 14

I

− |w|2 − w + w− 1

4

(Ji

)+|w|2 + w− w− 1

4

(Ji

)=

w + w2

I +w− w

2

(Ji

)= xI + yJ = (w) = (ψ(P)).

From this we getr(P) = ∏

ψ∈Ψj

(ψ(P)) = (Nψj(P)).

Now recalling that Ψj = φj Ψ,

NΨj(P) = ∏ψ∈φjΨ

ψ(P) = ∏ψ∈Ψ

φj ψ(P) = φj(∏ψ∈Ψ

ψ(P))

= φj(NΨ(P)) = φj(NΨ(P)).

Thus we have that on v ∈ (Vj)C,

r(P) = (Nψj(P)) = (φj(NΨ(P))) = ι(NΨ(P)),


where the last equality follows by definition of Vj. Since the final expression does not dependon j and is defined for v ∈ V, we have

r(P)v = ι(NΨ(P))v, ∀P ∈ K×, ∀v ∈ V,

which is the statement of the proposition.

Note the statement is consistent with our expectation from the main theorem of complexmultiplication. The non-appearance of an inverse sign over the reflex norm is due to Milne’sconvention of using the Artin map artE(s) in defining a canonical model, whereas the reci-procity map recE has been used to state the theorem of complex multiplication.

Thus the proposition shows that the action of the Galois group prescribed by the definitionof a canonical model on the double coset space

U(Q)\U(A f )/K

agrees with the natural Galois action onMnΦ(Q) given by Galois base change of abelian va-

rieties. We have demonstrated that in the same manner as Shimura varieties of PEL type, thereis arithmetic information encoded in the canonical model of double coset spaces associated tocompact unitary groups. This is proposed as evidence that the requirement in [Del71] and[Del79] for a Shimura variety to arise from a group with no compact factors may be relaxed.

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