Canonical models of Shimura varieties of PEL type

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MSc Mathematics

Master Thesis

Canonical models of Shimura varieties of

PEL type

Author: Supervisor:

Yachen Liu

dr. Arno Kret

Examination date:

August 12, 2020

Korteweg-de Vries Institute for Mathematics

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Abstract

The final goal of this thesis is to prove that the canonical model of a PEL type Shimura variety is the solution to a PEL type moduli problem of abelian varieties. The most important tools in the proof are Hodge theory and the fundamental theorem of complex multiplication. We also develop the necessary preliminary knowledge required in under-standing Shimura varieties and their canonical models, in formulating the final goal and in the proof of our final goal.

Title: Canonical models of Shimura varieties of PEL type Author: Yachen Liu, yachen.liu@student.uva.nl, 12148261 Supervisor: dr. Arno Kret

Second Examiner: dr. Mingmin Shen Examination date: August 12, 2020

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

Shimura varieties are higher-dimensional analogues of modular curves. Analytically, a Shimura variety is a disjoint union of quotients of a Hermitian symmetric space (for example, complex upper-half plane) by congruence subgroups of a reductive algebraic group defined over Q (for example, GLn,Q, , GSp2n,Q). A Shimura variety has a unique

canonical model (up to isomorphism) defined over some number field (Deligne’s canonical model).

A Shimura variety of PEL type comes from some linear-algebraic datum called PEL datum. Shimura varieties of PEL type make up a small but very important class in the totality of Shimura varieties. The final goal of this thesis is to show that the canonical model of a Shimura variety of PEL type is moduli of abelian varieties plus polarizations, endomorphism structures and level structures, that’s also where PEL comes from.

This thesis consists of four chapters. The first chapter is the study of quotient of the complex upper half plane by congruence subgroup, which provides some motivation behind the theory of Shimura varieties and their canonical models. The goal of the first chapter is to show that quotient of the complex upper half plane by congruence subgroup has a model defined over some number field. First, we will show that quotient of the complex upper half plane by congruence subgroup (which has the structure of a Riemann surface) has the structure of a complex algebraic curve by studying the compactification of it. Then we will show that the compactified quotient of the complex upper half plane by congruence subgroup has a model over some number field by studying the corresponding function field (equivalently, by studying the corresponding field of meromorphic functions).

The second chapter is the study of CM abelian varieties. The final goal of this chapter is to prove the fundamental theorem of complex multiplication, which is an important tool in proving our final result in chapter four. Firstly, we develop the general theory of CM types, CM abelian varieties and a-multiplications of CM abelian varieties. Then we prove the Shimura-Taniyama formula, which is one of the key ingredients in proving the fundamental theorem of complex multiplication. Then we introduce some key concepts in global class field theory, for example, ray class groups and ray class fields. At the end of this chapter, we formulate and prove the fundamental theorem of complex multiplication. The third chapter is an introduction to Shimura varieties and their canonical mod-els, including some inspiring examples. Firstly, we define Shimura datum and Shimura varieties associated to a Shimura datum. We also introduce Shimura varieties associated to a torus as our first example of Shimura varieties. Then we define the reflex field of a Shimura datum, the spacial points of a Shimura datum and the canonical model of a Shimura variety. And we apply the fundamental theorem of complex multiplication to construct canonical models of Shimura varieties associated to a ”CM” torus, which is our first example of canonical models of Shimura varieties. At the end of this chapter, we introduce Hodge structures and Siegel modular varieties.

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over some number field), the study of PEL type Shimura varieties and the proof of our final result. First, we compare lattices in Anf (where Af is the ring of finite ad`eles of Q)

and lattices in Qn, which allows us to reformulate N -level structure in terms of ad`eles. Then we introduce moduli problem of abelian varieties with polarizations plus K-level structure (such that K is a compact open subgroup of an ad`elelized reductive group over Q), which is called Siegel moduli problem. And we prove that the canonical model of a Siegel modular variety is the solution to a Siegel moduli problem. At the end of this chapter, we introduce PEL type Shimura varieties and moduli problem of abelian varieties with polarizations, endomorphism structure and K-level structure (which is called PEL moduli problem). And we prove our final result, that is, the canonical model of a PEL type Shimura variety is the solution to a PEL moduli problems.

Acknowledgement

I thank Arno Kret, for introducing me this fascinating topics, answering many questions of mine and giving me suggestions on doing and writing mathematics. I thank Mingmin Shen for being the second reviewer of this thesis and answering my questions. I thank Hessel Postume, Rob de Jeu and Lenny Taelman for their answers to my questions. I thank my parents for their support and I thank Xiaoying Cao for her accompany.

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1 Quotients of complex upper plane

It is unusual that a complex algebraic variety has a model defined over some number field, but a special class of algebraic varieties which arise from disjoint unions of quo-tients of Hermitian symmetric domains (together with some other properties) have unique ”canonical” models (which will be defined in Chapter 3) defined over number fields. In this chapter, we will provide some motivation behind the theory of Shimura varieties and their canonical models: We will introduce the arithmetic theory of meromorphic functions on quotients of complex upper half plane by congruence subgroups. We will also introduce the geometric theory and the analytic theory of quotients of the complex upper half plane needed for the arithmetic theory.

The goal of this chapter is to show that certain quotients of complex upper half plane can be defined over some number fields. Naturally, the first step is to show that certain quotients of complex upper half plane has the structure of a complex algebraic variety, which will be introduced in section 1.1. Then in section 1.2, we will show that quotients of the complex upper half plane by principle congruence subgroups of SL2(Z) can be

defined over some number fields. Finally in section 1.2 we will show that quotients of the complex upper half plane by congruence subgroups of SL2(Z) can be defined over some

number fields.

1.1 Γ\H as complex algebraic variety

In this section, we will sketch on how to give certain quotients (which will be made precise later) of the complex upper half plane the structure of a Riemann surface and the structure of a complex algebraic variety. Then we will collect some important facts about the analytic theory of meromorphic functions on quotients of the complex upper half plane.

Let k be a field. In this thesis, we define algebraic varieties over k to be separated and geometrically reduced schemes which are of finite type over k. Algebraic varieties over k of dimension 1 are called algebraic curves over k.

We define a complex manifold of dimension n to be a Haustroff, second countable topological space with an atlas of charts to the open disk in Cn. When n = 1, we call such complex manifolds Riemann surfaces. We use (−)an to denote the analytification functor from the category of complex algebraic varieties to the category of complex manifolds as introduced in Serre’s GAGA (pp.7-9 of [Ser56]).

Definition 1.1.1. Given a topological space X and a group G acting on X, we say that the action of G on X is properly discontinuous if for every x1, x2 ∈ X with Gx1 6= Gx2

there exist open neighbourhoods x1 ∈ U1 ⊂ X and x2 ∈ U2 ⊂ X such that gU1 ∩ U2 = ∅

for all g ∈ G.

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induced from C. We consider the action of SL2(Z) on H defined as follows: [a b c d] · z := az + b cz + d with [a b

c d] ∈ SL2(Z) and z ∈ H. The action of SL2(Z) on H is properly discontinuous

and that the quotient SL2(Z)\H is Hausdorff (see pp.46-47 of [DS]).

We define H∗ _{to be H ∪ P}1_{(Q) = H ∪ Q ∪ {∞}. We define the action of SL}2(Z) on

P1(Q) as follows:

[a b

c d] · (x : y) := (ax + by : cx + dy)

for [a b

c d] ∈ SL2(Z) and (x : y) ∈ P1(Q). Together with the action of SL2(Z) on H, we

obtain an action of SL2(Z) on H∗. We define the topology on H∗ as follows: Define

NM to be {x ∈ H : Im(x) > M }, and take the topology generated by α(NM ∪ {∞})

(α ∈ SL2(Z)) and the open sets of H.

Definition 1.1.2. For N ∈ Z>0, define

Γ(N ) := {[a b

c d] ∈ SL2(Z) : [a bc d] ≡ [1 00 1] mod N },

then a subgroup Γ ⊂ SL2(Z) is called a congruence subgroup if Γ ⊃ Γ(N) for some

N ∈ Z>0.

Let Γ ⊂ SL2(Z) be a congruence subgroup. Then Γ\P1(Q) is finite since SL2(Z) acts

transitively on P1_{(Q) and [SL}

2(Z) : Γ] is finite. We also have that Γ\H∗ is compact,

Hausdorff and connected (Proposition 2.4.2 of [DS]). And pp.59-62 of [DS] constructs a unique structure of Riemann surface on Γ\H (the resulting Riemann surface is denoted by YΓ) such that the quotient map H → Γ\H is holomorphic, and a structure of Riemann

surface on Γ\H∗ (the resulting Riemann surface is denoted by XΓ) such that YΓ is a

sub-complex manifold XΓ.

Now we will give a description of the complex charts on XΓ where Γ is a congruence

subgroup (We need Γ to be a congruence subgroup because the condition Γ ⊃ Γ(N ) for some N ∈ Z>0 is crucial if we want to show that Γ\H∗ can be defined over some number

fields, as we will see in the proof of Proposition 1.3.3). Before doing that, we need the following definition:

Definition 1.1.3. Let Γ ⊂ SL2(Z) be a congruence subgroup, and let x ∈ H∗ with

π : H∗ → Γ\H∗ _{be the quotient map. We define Γ}

x to be {g ∈ Γ : gx = x} and we define

I ∈ SL2(Z) to be the identity matrix. We call x (or π(x)) an elliptic point for Γ if the

containment {±I}Γx⊃ {±I} is proper. And we call x (or π(x)) a cusp if x ∈ Γ\P1(Q).

Now we give the charts on Γ\H∗ as described in pp.59-62 of [DS], we will just sketch the construction, see pp.59-62 of [DS] for more detail:

For an arbitrary x0 ∈ Γ\H∗, choose y0 ∈ H∗ such that π(y0) = x0, choose a

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always exists: Since Proposition 1.19 of [Shi69] tells us that the set of all elliptic points for Γ has no limit point in H∗; and the set of all cusps for Γ has no limit point in H∗ because α(NM ∪ {∞}) is open in H∗ (α ∈ SL2(Z), NM = {x ∈ H : Im(x) > M }).

• If y0 ∈ H, define δ : U → C by δ(y) =

1 −y0

1 −¯y0 y, and define ρ : C → C by

ρ(z) = zh_{, ρ(0) = 0 where h = #|{±I}Γ}

y0/{±I}|. Then the chart is given by

ϕ : π(U ) → ρ ◦ δ(U )

where ϕ ◦ π = ρ ◦ δ.

• If y0 ∈ P1(Q), define δ : U → C by δ(y) = Ay with A ∈ SL2(Z) satisfying

that Ay0 = ∞, and define ρ : C → C by ρ(z) = e2πz/h, ρ(∞) = 0 where h =

#|SL2(Z)y0/{±I}Γy0|. Then the chart is given by

ϕ : π(U ) → ρ ◦ δ(U )

where ϕ ◦ π = ρ ◦ δ.

By Chow’s theorem (pp.29-30 of [Ser56]), we know that XΓ = X(Γ)an for a unique

projective complex variety X(Γ) because XΓ is a compact Riemann surface. Note that

XΓ− YΓ = Γ\P1(Q) is finite, which is a Zarski closed set of X(Γ). Therefore X(Γ) −

Γ\P1(Q) is a complex algebraic variety, and we have YΓ = Y (Γ)an with Y (Γ) = X(Γ) −

Γ\P1(Q) as a sub-complex algebraic variety of X(Γ).

A meromorphic function on Γ\H is a meromorphic function on H which is invariant under the action of Γ.

Meromorphic functions on H∗ _{are more complicated, a C-value function on H is a} meromorphic function on Γ\H∗ if and only if (pp.28-29 of [Shi69])

• f is meromorphic on H.

• f is invariant under the action of Γ. • f is meromorphic at every cusp of Γ.

The precise meaning of the last condition is as follows: If s ∈ Γ\P1_{(Q), take an element}

of ρ of SL2(Z) so that ρ(s) = ∞, put Γs= {g ∈ Γ : g(s) = s}, we have

ρΓsρ−1· {±1} = {± [1 h0 1] m

|m ∈ Z}

with a positive real number h (pp.17-18 of [Shi69]). Then f (ρ(−)) is invariant under the action of ρΓρ−1, which can be expressed as a function f∗(q) with q = e2πi/h. Such f∗ is defined on a punctured disk 0 < |q| < , then f is meromorphic at cusp s means that f∗

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as defined above is meromorphic at 0. We call such functions modular functions with respect to Γ (For more detail, see pp.28-29 of [Shi69]).

We are also interested in meromorphic functions on H satisfying certain condition under the action of SL2(Z):

Definition 1.1.4. Let Γ be a congruence subgroup of SL2(Z), a C-valued function f on

H is called a meromorphic modular form of weight k with respect to Γ if • f is meromorphic on H.

• f (ρz) = (cz + d)k_{f (z) for ρ = [}a b c d] ∈ Γ.

• f is meromorphic at every cusp of Γ.

If we replace meromorphic by holomorphic in the definition, then such functions are called modular forms of weight k with respect to Γ. When Γ = Γ(N ) for some N ∈ Z>0, we may replace ”with respect to Γ(N )” by ”of level N”.

Definition 1.1.5. Let k be a field. Then elliptic curves over k are defined to be irreducible smooth projective curves over k of genus 1 together with a base point e.

In the following, we will give some important examples of modular forms and modular functions which play an important role in the study of elliptic curves, and we will use them in studying the function fields of X(Γ):

Example 1.1.1. (i) Define G2k : H → P1(C) by

τ 7→ X

m,n∈Z−{0}

(mτ + n)−2k,

then G2k is a meromorphic modular form of weight 2k with respect to SL2(Z) for k ≥ 2

(Example 3.4.1 of [Sil94]).

(ii) Write g2 = 60G4, g3 = 140G6, the modular discriminant is the function ∆ :=

g3

2−27g32. And ∆ is a modular form of weight 12 with respect to SL2(Z) (Example 3.4.3 of

[Sil94]). For the usage of ∆, g2, g3 in the study of elliptic curves, see IV .3.5.1 of [Sil09].

(iii) The j-invariant is the function given by j := 1728g2

∆, then j is a modular

function with respect to SL2(Z). And j gives an isomorphism between Riemann surface

XSL2(Z) and P

1_{(C) (Theorem 4.1 of [Sil94]). Note that j(τ ) is the j-invariant of the}

elliptic curve E : y2 _{= 4x}3_{− g}

2(τ )x − g3(τ ) (which is denoted by jE, for more detail, see

IV .3.6 of [Sil09]).

(iv) Given τ ∈ H, we define Weierstrass ℘ function with respect lattice Z ⊕ Zτ (we define Λτ := Z ⊕ Zτ ) by ℘τ(z) := 1 z2 + X ω∈Λτ−{0} (( 1 z − ω) 2₋ 1 ω2),

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then ℘τ is a meromorphic function on complex torus C/(Z⊕Zτ ) (Section IV .3 of [Sil09]).

For every a = (a1, a2) ∈ Q2−Z2and for every z ∈ H, we define fa(z) :=

g2(z)g3(z)

∆(z) ℘z(a2z/a1).

Then fa is a meromorphic modular form of level N (pp.133-134 of [Shi69] or p.279 of

[DS]). We can check that for every g ∈ SL2(Z), we have that fa◦ g = fag and that

fa = fb if and only if a ≡ ±b mod Z2. From these two properties, we can conclude that

if N a ∈ Z2, then fa◦ g = fa for all g ∈ {±1}ΓN.

Note that every complex elliptic curve is isomorphic to a complex tori as complex Lie group via an isomorphism defined in terms of ℘, for more detail, see pp170-171 of [Sil09]. Remark 1.1.1. We may also regard j as function on the set of all lattices in C: We may set j(Za ⊕ Zb) to be j(a/b).

Definition 1.1.6. Given a complex elliptic curve E given by Weierstrass equation y2 = 4x3− ax − b with base point O = [0 : 1 : 0], we define the discriminant of E, which is denoted by ∆E, to be a3− 27b2. And we define the j-invariant of E, which is denoted by

jE, to be 1728a

3

∆E. And we define function hE on E given by (x, y) 7→ (ab/∆E) · x, O 7→ 0.

Given an N ∈ Z>0, we use E[N ] to denote the group of N -torsion points of E.

1.2 The function field of X(Γ

N

)

This section is based on pp.133-141 of [Shi69].

Fix an N ∈ Z>0. In this section, we will study the function field of X(ΓN), and we

will show that X(ΓN) has a model defined over Q(ζN) (such that ζN ∈ C is a primitive

N-th root of unit). First, we introduce the following notations: For N ∈ Z>0, we define

• XN := XΓN, YN := YΓN,

• X(N ) := X(ΓN), Y (N ) := Y (ΓN).

Definition 1.2.1. Let X be an irreducible algebraic variety over a field k. Now let k(X) := {(U, f ) : U ⊂ X open, f ∈ OX(U )}/∼ such that (U, f ) ∼ (V, g) if and only

if there exists open W ⊂ V ∩ U such that f |W = g|W. We call k(X) the function field

of X.

By Chow’s theorem (pp.29-30 of [Ser56]), we have that

C(X(Γ)) = {complex variety morphisms from X(Γ) to P1(C)}

= {holomorphic functions from XΓ to P1(C)} = {meromorphic functions on XΓ}.

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Example 1.2.1. Recall that the j-invariant as defined in Example 1.1.1 gives us an Riemann surface isomorphism between X1 and P1(C) (Theorem 4.1 of [Sil94]). For an

arbitrary f ∈ C(X(1)), we have that f ◦ j−1 is a meromorphic function on P1(C). Then f ◦j−1_{must a rational function (pp.31-33 of [Mir95]). As a result, we have that C(X(1)) =} C(j).

Fix an N ∈ Z>0. We have the following proposition (Proposition 6.1 of [Shi69]).

Proposition 1.2.1. We have C(X(N)) = C(j, fa: a ∈ N−1Z2, a 6∈ Z2) (fa as introduced

in Example 1.1.1, (iv)).

Proof. Define ϕ : SL2(Z) → AutC(j)(C(X(N))) by g 7→ ϕ(g) such that ϕ(g) is defined

as follows: fϕ(g) := f ◦ g. We can check that ker(ϕ) ∼= {±I}Γ(N ). Now ϕ(SL2(Z)) is a

subgroup of Aut_C(j)_{(C(X(N))) and the subfield it fixes is C(j) ∼}= C(X(1)). As a result, we know that C(X(N ))/C(j) is Galois with Galois group ϕ(SL2(Z)) ∼= SL2(Z)/{±I}ΓN.

Also note that

C(j) ⊂ C(j, fa : a ∈ N−1Z2, 6∈ Z2) ⊂ C(X(N)),

and that given a g ∈ SL2(Z), we have that fa◦ g = fa for all a ∈ N−1Z2, a 6∈ Z2 implies

g ∈ {±I}ΓN. Then the proposition follows.

Let L/k be a field extension and let X be an algebraic variety over L, we call an algebraic variety X0 over k a model of X over k if there exists an L-algebraic variety isomorphism ϕ : X0×Spec(k)Spec(L) → X.

Let k be a field. We call a field K a function field over k if it satisfies the following properties: (i) k is algebraically closed in K. (ii) K is a finite extension of a field k(t) where t is transcendental over k. In order to find a model of X(Γ) over a smaller field, we will need the following theorem (Theorem 7.2.5 of [DS]):

Theorem 1.2.1. Let k be a field, then the category of irreducible nonsingular projective curves over k is equivalent to the category of function fields of k (with morphisms being field embeddings).

The above equivalence of categories is given by functor F defined as follows: we define F (C) to be k(C) for every irreducible nonsingular projective curve C defined over k. And we define F (h) to be h∗ such that h∗ is defined by h∗G = G ◦ h with h being a morphism between irreducible nonsingular projective curves over k and G being a field embedding.

We define FN to be Q(j, fa : a ∈ N−1Z2, a 6∈ Z2). Then the following proposition

(Theorem 6.6 of [Shi69]) implies us that FN will give us a model of X(N ) over Q(ζN):

Proposition 1.2.2. (i) FN is a Galois extension of Q(j).

(ii) FN contains a primitive root of unit, which is denoted by ζN.

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Proof. The proof of (i): By Lemma 6.5 of [Shi69], there exists a z0 ∈ H such that the

specialization Q(j, fa)(a ∈ N−1Z2, 6∈ Z2) → Q(j(z0), fa(z0)) gives us an isomorphism of

fields. Consider the complex elliptic curve E given by y2 = 4x3− cx − c such that c/(c − 27) = j(z0), then we have that jE = j(z0). By pp.98-99 of [Shi69], there exists a complex

Lie group isomorphism ϕ from C/(Z ⊕ Zz0) to Ean. We define FN := Q(jE, hE(t), (t ∈

E[N ]) (hE as introduced in Definition 1.1.5), and define η(a) ∈ C to be ϕ(a [z10]) for

a ∈ Q2.

First, we will show that hE(η(a)) = fa(z0): Let E0 be the complex elliptic curve

defined by y2 = 4x3 − g2(z0)x − g3(z0) (with g2, g3 as defined in Example 1.1.1). Then

the map from C/(Z ⊕ Zz0) to E0an defined by z 7→ (℘(z), ℘0(z)) is a complex Lie group

isomorphism (pp.170-171 of [Sil09]), and we use ϕ0 to denote this isomorphism. Then we have that ϕ0 ◦ ϕ−1 _{is a complex elliptic curve isomorphism from E to E}0_{. By equation}

(4.5.4) in p.107 of [Shi69], we have (with hE as defined in Definition 1.1.6)

hE(ϕ(a [z10])) = hE0(ϕ

0

(a [ 1 z0])),

by definition, the latter is fa(z0). Then we can conclude that

{hE(t) : t ∈ E[N ]} = {fa(z0) : a ∈ N−1Z2, a 6∈ Z2},

therefore,

FN = Q(jE, hE(t), t ∈ E[N ]) = Q(jE, fa(z0), a ∈ N−1Z2, a 6∈ Z2).

Next, we will show that FN is a Galois extension of Q(j(z0)) = Q(jE): Note that E

is defined over Q(jE). We define ϕ : GalQ(jE)(Q(jE)) → AutQ(jE)(FN) as follows: Let σ

be an element of Gal_Q(j_E)(Q(jE)). Then Eσ = E and σ gives an automorphism of E[N ].

Note that hE is rational over Q(jE), then we have that hE(t)σ = hE(tσ). As a result,

FN is fixed by σ. We may set ϕ(σ) = σ|FN, and ϕ(GalQ(jE)(Q(jE))) is a subgroup of

Aut_Q(j_E)(FN) whose fixed field is Q(jE). Then we can conclude that FN/Q(jE) is Galois.

Then we have that FN is a Galois extension of Q(j) because the specialization map

as defined in the beginning of proof gives us the following field isomorphisms (recall that jE = j(z0)):

FN ∼= FN, Q(j) ∼= Q(j(z0)).

The proof of (ii): We only need to show that FN contains ζN. We fix a basis {t0, t1}

of E[N ]. For an arbitrary σ ∈ Aut_Q(j)_{(C), we define φ(σ) ∈ GL}2(Z/NZ) as follows:

σ(t0) = pt0 + qt1, σ(t1) = rt0 + st1, then we define φ(σ) := [p qr s] (note that φ(σ) =

[p qr s] ∈ GL2(Z/NZ) because σ sends a basis of E[N] to another basis of E[N]). Let

eN : E[N ] × E[N ] → Q(ζN) be the Weil pairing as introduced in Section 4.3 of [Shi69].

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Proposition 4.3 of [Shi69], we have that ζ_Nσ = eN(tσ0, t σ 1) = eN(t0, t1)det(φ(σ)) = ζ det(φ(σ)) N .

Note that if σ = id on FN, then φ(ϕ) ≡ ±1 mod N (p.107, (4.5.3) of [Shi69]). So that

ζ_Nσ = ζN, and we have ζN ∈ FN.

The proof of (iii): Define τ : GL2(Z/NZ) → Gal(FN/Q(j)) by β 7→ f τ (β)

a := faβ.

Note that

FN = Q(jE, hE(t), t ∈ E[N ]) = Q(jE, fa(z0), a ∈ N−1Z2, a 6∈ Z2),

and that there exists a z0 ∈ H such that specialization gives us the following isomorphisms:

FN ∼= FN, Q(jE) ∼= Q(j),

as we proved in (i). Then we get an action of GL2(Z/NZ) on FN given by hE(t)β = hE(tβ).

Also note that all elements of Gal(FN/Q(jE)) gives an automorphism of GL2(Z/NZ) via

the homomorphism φ as we defined in the proof of (ii), and for every σ ∈ Gal(FN/Q(jE)),

we have that τ (φ(σ)) = σ (note that hE(t)σ = hE(tσ) because hE is rational over jE in

our case). Thus τ is surjective, and we can check that β fixes FN if and only if β ∈ {±1}.

Then we have Gal(FN/Q(j)) ∼= GL2(Z/NZ)/{±1}.

In the proof of (ii), we show that the action of GL2(Z/NZ) on ζN via τ is given by

ζ_Nτ (σ) = ζ_Ndet(φ(τ (σ)) = ζ_Ndet(σ).

Then we have that τ (SL2(Z/NZ)) corresponds to Q(ζN, j): Because for an arbitrary

σ ∈ GL2(Z/NZ), we have that τ (σ) fixes Q(ζN, j) if and only if det(σ) = 1 if and only if

σ ∈ SL2(Z).

Put k = C ∩ FN, then every element of k is invariant under the action of SL2(Z/NZ)

via τ , thus k ⊂ Q(ζN, j). So that k ⊂ Q(ζN), which proves (iii).

Apply Theorem 1.2.1, we can conclude that X(N ) has a model defined over Q(ζN)

such that ζN is a primitive root of unit.

1.3 The models of X(Γ) over number fields

This section is based on pp.141-157 of [Shi69].

In this section, we will show that X(Γ) has a model over some number field when Γ is a congruence subgroup of SL2(Z).

Given a number field K, we use A(K) to denote the set of all places of K, and we use Af(K) to denote the set of finite places of K. Given a v ∈ A(K), we use Kv

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define OKv to be {x ∈ Kv : v(x) ≥ 0} if v is a finite place, and we define OKv to be Kv if

v is an infinite place.

Let S be a subset of A(K), we define the ring of S-a`eles of K (which is denoted by AK,S) to be

{(xv) ∈

Y

v∈S

(xv) : xv ∈ OKv for all but finitely many v}.

We define a topology on AK,S (which is called the restricted topology) as follows: The

base of open sets consists of set of the form Q

v∈S0Wv×

Q

v∈S−S0Ov such that S0 ⊂ S is

a finite subset containing all infinite places of K and Wv ⊂ Kv are open for each v ∈ S0.

And we view K as a subring of AK,S via diagonal embedding.

We define the ring of ad`_{eles of K (which is denoted by A}K) to be

{(xv) ∈

Y

v∈A(K)

Kv : xv ∈ OKv for all but finitely many v},

viewed as a topological ring equipped with the restricted product topology.

And We define the ring of finite ad`_{eles of K (which is denoted by A}K,f) to be

{(xv) ∈

Y

v∈Af(K)

Kv : xv ∈ OKv for all but finitely many v},

viewed as a topological ring equipped with the restricted product topology. We use Af, A to denote AQ, AQ,f respectively.

Definition 1.3.1. Let k be a field k. We define GLn,k to be the k-group scheme defined

as follows (which is represented by Spec(k[x1,1, x1,2..., xn,n, y]/(det(xi,j)y = 1))):

Given a k-algebra A,

GLn(A) := {m ∈ EndA(An) : m is invertible}.

A k-group scheme is called a linear algebraic group over k if it is isomorphic to a smooth closed k-subgroup scheme of GLn,k.

And we have the following theorem (p.72 of [Mil15]):

Theorem 1.3.1. Let k be a field and let X be a k-scheme. Then we have that X is a linear algebraic group over k if and only if X is an affine algebraic group over k.

Let K be a number field, let G be an algebraic group defined over K. Suppose that G is defined by f1, ..., fn ∈ K[x1, ..., xm]. We define the ad`elzation of G as

fol-lows: Define G(AK,S) := {(x1, ..., xm) ∈ AnK,S : fi(x1, ..., xm) = 0 for 1 ≤ i ≤ n}. And

G(AK,S) is equipped with the subspace topology. We view G(K) = {(x1, ..., xm) ∈ Km :

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Remark 1.3.1. The topological ring G(AK,S) as we defined above is independent of the

choice of an affine covering (that is, if we choose a different affine covering, we will get a topological ring isomorphic to G(AK,S) as we constructed above). See pp.244-246 of [PR].

And we have the following theorem (pp.9-10 of [Con12]):

Theorem 1.3.2. Fix K, G and S as defined above. We have G(AK,S) is locally compact,

Hausdorff and G(K) is discrete in G(AK,S).

Let K be a number field, let G, H be linear algebraic groups over K and let S be a subset of A(K), with A(K) as defined at the beginning of this section. Let f : G → H be a K-algebraic group morphism. Suppose that G is defined by K[x0, ..., xn]/(g0, ..., gp) and

that H is defined by K[y0, ..., ym]/(h0, ..., hq). Suppose that f is defined by (y0, ..., yn) =

(f0(x0, ..., xn), ..., fm(x0, ..., xn)), then we define f (AK,S) : G(AK,S) → H(AK,S)

(some-times still denoted by f ) by (a0, ..., an) 7→ (f0(a0, ..., an), ..., (fm(a0, ..., an)) for every

(a0, ..., an) ∈ G(AK,S). Then f (AK,S) is well-defined and continuous (p.244 of [PR]).

Given an abelian group A we use ˆA to denoted the profinite completion of A equipped with the profinite topology.

Now let us talk about reciprocity map and global class field theory, they will play an important role throughout this thesis.

Let K be a number field and let L/K be a Galois extension with Gal(L/K) being an abelian group. We use OK, OLto denote the rings of integer of K and L respectively. Let p

be a prime of K which does not ramify in L and let P be a prime lying above p. Recall that if L/K is an abelian extension, then there exists a unique element F robP/p ∈ Gal(L/K)

mapping to the arithmetic Frobenius element of Gal((OL/P)/(OK/p)) which is

given by x 7→ x#OK/p_{. And F rob}

P/p ∈ Gal(L/K) is independent the choice of the prime

lying above p. Let F robL/Kp = F robP/p, then we get a well-defined element F rob L/K p of

Gal(L/K), which is called the arithmetic Frobenius element for L/K at p.

Definition 1.3.2. Let c be an integral ideal of K which is divisible by all primes ramifying in L/K and let I(c) be the group of fractional ideals of K which are relatively prime to c. Then we can define the Artin map as following:

θL/K : I(c) → Gal(L/K) Y p pnp _7→Y p (F robL/Kp )np

For an element s of AK, define (s) to be the fractional ideal of K given by

Q

pfinite

pordpsp_.

Note that (s) is well defined, because sp is a p-adic unit for all but finitely many p.

Given a number field K, recall that we define ring of ad`_{eles of K (denoted by A}K)

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K as a subring of AK (and AK,f) via diagonal embedding. And we have the following

Theorem.

Theorem 1.3.3. (Global class field theory, pp.119-120 of [Sil94]) Let K be a number field, and let Kab be the abelian closure of K. There exists a unique continuous homo-morphism resK : A×K → Gal(K

ab_{/K) (which is called the reciprocity map), with the}

following property: Let L/K be a finite abelian extension, and let s ∈ A×K such that (s) is

not divided by any primes ramifying in L, then resK(s)|L = θL/K(s).

And that resK has the following additional properties:

(a) The map resK is surjective and that K× is in its kernel.

(b) We have that resL(x)|Kab = res_K(N m_L/K(x)) for an arbitrary finite abelian

ex-tension L/K.

(c) Let p be a prime of K, let I_pab be the inertia group of p for the Galois extension Kab/K, let πp ∈ OK,p be a uniformizer, then for an arbitrary finite abelian extension L/K

which is unramified at p, we have that resK(πp)|L= F rob L/K

p , resK(OK,p) = Ipab.

Remark 1.3.2. We have res_Q _{factors through A}f, see pp.175-176 of [Mil13] for more

details. And we will give an explicit description of res_Q in the proof of Proposition 1.3.2 (ii).

We use GL2(R)+(GL2(Q)+) to denote the subgroup of GL2(R) (GL2(Q)) with positive

determinant. Given an N ∈ Z>0, we define Uf(N ) to be the open subgroup of GL2(ˆZ)

satisfying the following property: g ∈ Uf(N ) if and only if g ≡ 1 mod N . Define Kf(N )

to be the open subgroup of ˆ_Z× _{⊂ A}×_f satisfying the following property: g ∈ Kf(N ) if and

only if g ≡ 1 mod N . In particular, we have that Uf(1) = GL2(ˆZ) and that Kf(1) = ˆZ×.

We define σ : GL2(Af) → Gal(Qab/Q) by s 7→ resQ(det(s) −1_).

Proposition 1.3.1. The morphism det : GL2(Af) → A×_f maps open subgroups of GL2(Af)

to open subgroups of A×f.

Proof. Let U be a open subgroup of GL2(Af), then U contains Uf(N ) for some N ∈ Z>0

(p.143 of [Shi69]). Note that det(Uf(N )) = Kf(N ), thus for every g ∈ det(U ), we have

that gKf(N ) ⊂ det(U ), therefore, det(U ) is open.

We put F = ∪_{N ∈Z}_>0FN. In the last section, we show that Q(ζN) ⊂ FN and that

Q(ζN) is algebraically closed in FN. Note that all finite abelian extensions of Q can be

embedded into Q(ζN) for some N ∈ Z>0. Thus Qab ⊂ F and Qab is algebraically closed

in F .

Define τ : Uf(1) → AutF1(F ) as follows: Given a u ∈ Uf(1) and f = fa ∈ FN for some

N ∈ Z>0 and for some a ∈ N−1Z2. There exists an α ∈ M2(Z) such that u ≡ α mod N

(such α are uniquely determined modulo N ), then we define fτ (u) _{to be f} au.

We have the following proposition about the structure of AutF1(F ), as well as the

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Proposition 1.3.2. (i) F /F1 is Galois with Galois group τ (Uf(1)).

(ii) For an arbitrary u ∈ Uf(1), we have that σ(u) agrees with τ (u) on Qab.

Proof. The proof of (i): From the proof of Proposition 1.2.2 (iii), we know that FN/F1

is Galois and that τN : GL2(Z/NZ) → Gal(FN/F1) defined by β 7→ f τN(β)

a := faβ is

surjective. Note that F = ∪_{N ∈Z}>0FN, and by the definition of τ , we have τ (Uf(1)) is

a subgroup of AutF1(F ) and the fixed field of τ (Uf(1)) is F1. Thus the fixed field of

AutF1(F ) is F1, therefore F /F1 is Galois.

Also note that we have a directed system of finite Galois extensions defined as follows: • For arbitrary M, N ∈ Z>0, define M ≺ N if M |N .

• A family of field extensions (FN/F1)N ∈Z>0 index by N ∈ Z>0.

• Field embedding FN ,→ FM if N ≺ M .

By tag 0BU2 of [Sta20] (we use limN to denote the inverse limit with respect to directed

poset (Z>0, ≺)), we have that

Gal(F /F1) ∼= limN(Gal(FN/F1)) = limN(τN(GL2(Z/NZ))) ∼= τ (Uf(1)),

with the isomorphism limN(τN(GL2(Z/NZ))) ∼= τ (Uf(1)) given by

(τ (aN)) ∈ limN(Gal(FN/F1)) 7→ τ (a) ∈ τ (Uf(1))

such that aN ∈ GL2(Z/NZ) and a ≡ aN mod N .

The proof of (ii): For an arbitrary u ∈ Uf(1), there exists an α ∈ M2(Z) such that u ≡

α mod N (such α are uniquely determined modulo N ). From the proof of Proposition 1.2.1 (iii), we know that the action of τN(α)|Q(ζN) is given by ζ

τN(α)

N = ζ det(α) N .

Note that Gal(Qab/Q) ∼= ˆ_Z× as topological groups with the isomorphism ψ : ˆ_{Z →} Gal(Qab/Q) defined as follows: For an arbitrary u ∈ ˆZ, define ψ(u) to be the unique σ ∈ Gal(Qab/Q) such that σ(ζN) = ζu

0

N (for every N ∈ Z>0) where u ≡ u0 mod N and

ζN is an N -th root of unit. Therefore, ψ(det(u)) = τ (u) on Qab. We also have that

ψ(det(u)) = res_Q(det(u)−1) = σ(u) (pp.175-176 of [Mil13]). Therefore, σ(u) agrees with τ (u) on Qab.

Let Γ ⊂ SL2(Z) be a congruence subgroup, then Γ contains Γ(N) for some positive

integer N . Put H = Γ/Γ(N ) ⊂ SL2(Z/NZ). Then we have

Γ = {g ∈ SL2(Z) : ¯g ∈ H}.

Where ¯g denotes the reduction of g modulo Γ(N ). We associate Γ to P (Γ)f ⊂ GL2(ˆZ)

defined as follows:

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Where ¯g denotes the reduction of g modulo Uf(N ). For every g ∈ P (Γ)f, we have

gUf(N ) ⊂ P (Γ)f, therefore, P (Γ)f is open.

Global class field theory tells us that for an arbitrary open subgroup U of A×f/Q × _of

finite index, there exists a unique finite abelian extension L/Q such that Gal(L/Q) ∼= (Af/Q×)/U . We use k(U ) to denote the field constructed from U as above. For example,

k(Uf(N )) ∼= Q(ζN). Define FΓ to be {f ∈ F : fτ (s)= f for all s ∈ P (Γ)f} (recall that we

define τ : Uf(1) → Gal(F1/F ) before Proposition 1.3.2).

The following proposition (Proposition 6.27 of [Shi69]) gives us a model of X(Γ) over some number field, we write k(Γ) = k(σ(P (Γ)f)) (with σ as defined before Proposition

1.3.1) and write Γ0 = τ (P (Γ)f).

Proposition 1.3.3. (i) CFΓ= C(X(Γ)).

(ii) k(Γ) is algebraically closed in FΓ.

Proof. The proof of (i): Apply Proposition 1.3.2, we can show that F /FΓ is Galois with

Galois group Γ0. Note that P (Γ)f ⊃ Uf(N ) for some positive integer N , thus k(Γ) ⊂

Q(ζN). Put

R = {x ∈ GL2(ˆZ) : τ (x) = id on Q(ζN)FΓ}.

We can check that ΓUf(N ) ⊂ R (p.154 of [Shi69]). Conversely, we have

R ⊂ (P (Γ)f) ∩ (GL2(Q)+Uf(N ))

by Lemma 6.17 of [Shi69] and Proposition 1.3.2 (ii). Also note that

(P (Γ)f) ∩ (GL2(Q)+Uf(N )) = (P (Γ)f ∩ GL2(Q)+)Uf(N ) = ΓUf(N )

(p.154 of [Shi69]). Thus R = ΓUf(N ).

In the proof of Proposition 1.2.1 (iii), we showed that C ∩ FN = Q(ζN). Then by

Proposition 1.3.2 (i) an pp.152-153 of [Shi69], we have

[CFN : CFΓ] = [FN : FΓQ(ζN)] = [ΓUf(N ) : Uf(N )] = [Γ : Γ(N )].

On the other hand, we have that

[CFN : C(X(Γ))] = [C(X(N)) : C(X(Γ))] = [Γ : Γ(N )].

Also note that CFΓ⊂ C(X(Γ)), therefore, CFΓ= C(X(Γ)).

The proof of (ii): Note that Gal(Qab_{/k(Γ)) = σ(P (Γ)}

f) and that every element of

FΓ ∩ Qab is invariant under the action of τ (s) for every s ∈ P (Γ)f (as we discussed

before Proposition 1.3.3). By (ii) of Proposition 1.3.2 and the definition of FΓ, we have

that Qab _{∩ F}

Γ is the fixed field of σ(P (Γ)f). So that Qab ∩ FΓ = k(Γ). Since Qab is

algebraically closed in F , then we can conclude that k(Γ) = Qab _{∩ F}

Γ is algebraically

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Proposition 1.3.3 tells us that X(Γ) has a model defined over a number field k(Γ). By the construction of k(Γ), we have that k(Γ) ⊂ Q(ζN) if Γ ⊃ Γ(N ). Such phenomenon

provides motivation behind the theory of canonical models (defined over number fields) of Shimura varieties, where Shimura varieties are higher dimensional generalizations of quotients of the complex upper half plane by congruence subgroups.

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2 Complex Multiplication

In this chapter, we will study the theory of complex multiplication of abelian varieties. The final goal of this chapter is to prove the fundamental theorem of complex multiplication. This theorem describes the Galois action on torsion points of CM abelian varieties, and we will use this theorem many times in Chapter 3 and Chapter 4.

2.1 CM algebras and CM types

This section is based on Section I.1 of [Mil20].

A number field K is said to be totally real if the image of K under every homomor-phism K ,→ C is contained in R. We say K is totally imaginary if the image under every homomorphism K ,→ C is not contained in R. And K is said to be a CM field if K is a quadratic extension of a totally real field and if K is totally imaginary.

A CM algebra is by definition a finite product of CM fields (equivalently, we have that E is a CM algebra if there exists an automorphism ιE of order 2 of E, such that

ι ◦ ρ = ρ ◦ ιE for all homomorphism ρ : E → C where ι is the complex conjugation, see

p.11 of [Mil20]. Note that ιE can be taken to be product of complex conjugation on each

factor). An ´_{etale algebra over Q is defined to be a Q-algebra isomorphic to a finite} product of finite separable extensions of Q. Given a number field K with a place v, we use Kv to denote the completion of K with respect to v.

Let A be an ´_{etale algebra over Q. We say that O is an order in A if O is a subring} of A such that O ⊗_Z_{Q = A. We say that L is a lattice in A if L is a sub-Z-module of} A such that L ⊗_Z_{Q = A}

The following proposition is based on Proposition 1.39 f [Mil20], which gives us a useful criteria to ensure that a number field is a CM field or totally real.

Proposition 2.1.1. Given a number field K, if there exists r ∈ Aut_Q(K) such that r2 _{= 1 and that the Q-bilinear map K × K → Q given by (x, y) 7→ T r}

K/Q(r(x)y) is

positive-definite, then K is totally real or K is a CM field.

Proof. If r = id, then (x, y) 7→ T r_K/Q(xy) is positive-definite. Thus (x, y) 7→ T rK⊗_QR/R(xy)

is positive-definite. Then T rK⊗_QR/R(x

2_{) ≥ 0 for every x ∈ K ⊗}

Q R. So K ⊗QR has no

complex factors. Thus, K is totally real.

If r 6= id, let K+ _{be the subfield of K given by K}+ _{= {x ∈ K|r(x) = x}. Note}

that r|_K+ = id, so K+ is totally real by the above argument. We also have that [K :

K+_{] = 2: For an arbitrary a ∈ K − K}+_{, we have that a + r(a), ar(a) ∈ K}+_{, thus}

f (x) = x2_{−(a+r(a))x+ar(a) ∈ K}+_{[x]. Also note that f (a) = 0, therefore, [K : K}+_{] = 2.}

Now we will prove that K has no real places: If not, let v be a real place of K. As r is non-trivial, then v0 := v ◦ r gives another real place of K. By the positive-definiteness of (x, y) 7→ T r_K/Q(r(x)y), we know that (x, y) 7→ T rK⊗_QR/R(r(x)y) is positive-definite.

Notice that r(Kv × Kv0) = K_v0 × K_v if we consider the action of r on K ⊗

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have

(x, y) 7→ T rKv×Kv0/R(r(x)y) = T rK⊗QR/R|Kv×Kv0(r(x)y)

is positive-definite (r swaps the factors). Which is impossible, because

T rKv×Kv0/R(r(1, 0)(1, 0)) = rKv×Kv0/R((0, 1)(1, 0)) = 0.

Then we have that K is totally imaginary. Also notice that K is a quadratic extension of a totally real field K+ _{as we defined above, then we can conclude that K is a CM}

field.

In the following, if E is an ´etale algebra, then we use Hom_Q_{(E, C) to denote the set} of non-zero Q-algebra homomorphisms from E to C.

Definition 2.1.1. Given a CM algebra E, a CM type of E is a subset Φ ⊂ Hom_Q_{(E, C)} such that Hom_Q_{(E, C) = Φ}` ιΦ (ι is the complex conjugation). A CM pair (E, Φ) consists of a CM algebra E and a CM type Φ on E.

Given a CM pair (E, Φ), a subfield F (E, Φ) of Q is called the reflex field of (E, Φ), if it satisfies the following conditions: σ ∈ Aut_Q_{(C) fixes F (E, Φ) if and only if σΦ = Φ} (also note that the reflex field of a CM type is a CM field, see Proposition 1.18 of [Mil20]). Proposition 2.1.2. Given a CM pair (E, Φ), we have that F (E, Φ) is generated by ele-ments of the form P

φ∈Φφ(a), a ∈ E.

Proof. If σ ∈ Aut_Q_{(C) satisfies that σΦ = Φ, then σ fixes all elements of the form} P

φ∈Φφ(a) for a ∈ E. Conversely, if

P

φ∈Φφ(a) = σ(

P

φ∈Φφ(a)) =

P

φ∈Φσφ(a) for all

a ∈ E, then from the Q-linear independence of elements in Φ and that P

φ∈Φφ(a) −

P

φ∈Φσφ(a) = 0 for all a ∈ E, we can conclude that σΦ = Φ.

We also have another formulation of the reflex field, which can be easily fitted into the framework of Shimura varieties, as we will see later. First let us look at the action of Aut_Q_{(C) on Q-schemes.}

Let X, Y be affine algebraic varieties defined over Q, and suppose that X is given by Spec(Q[x1, ..., xn]/(f1, ..., fm)), Y is given by Spec(Q[y1, ..., yp]/(g1, ..., gq)). Consider

a morphism from X_C to Y_C given by (h1(x1, ..., xn), ..., hp(x1, ..., xn)). For arbitrary σ ∈

Aut_Q_{(C), x}1, ..., xn∈ X(C), we define σ(h1, ..., hp)(x1, ..., xn) to be

(σ(h1(σ−1x1, ..., σ−1xn)), ..., σ(hp(σ−1x1, ..., σ−1xn))).

In another words, we have that Aut_Q_{(C) acts on the coefficients of h}i.

Let X, Y be Q-schemes, and suppose that X, Y are locally of finite presentation over Q, let f : XC → YC be a C-morphism. We define σf : XC → YC to be σ

−1

Y_C ◦ f ◦ σX_C,

where σX_C (or σY_C) is obtained as shown the in the following pull-back diagram (note that

Xσ

C = XC because X is locally of finite presentation over Q, then X is covered by affine

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Xσ C = XC XC Spec(C) Spec(C) σX_C Spec(σ−1₎ .

In the following, we will give another description of the reflex field of a CM pair. Let (E, Φ) be a CM pair.

For an arbitrary C-algebra B, the following isomorphism

(E ⊗_QB)× → Y ρ:E→C B×, a ⊗ r 7→ (ρ(a)r)ρ, gives us an isomorphism sΦ : T_CE → (Gm,C)Φ` ιΦ. Define f : SC ∼ −_{→ G}2

m,C by a ⊗ z 7→ (za, ¯za) for every a ⊗ z ∈ (A ⊗QC)

× _{(such that A}

is a C-algebra). Define hΦ : SC → T E

C to be a homomorphism such that sΦ◦ hΦ◦ f −1 _is

given by (z1, z2) 7→ (z1)φ∈Φ× (z2)φ∈ιΦ.

Therefore, the cocharacter µΦ : Gm,C → TCE := hΦ◦ µ satisfies that

(sΦ◦ µΦ(z))φ= z if φ ∈ Φ and (sΦ◦ µΦ(z))φ = 1 if φ ∈ ιΦ

with u : Gm,C → SC defined as follows: we define f ◦ u : Gm,C u − → SC ∼ −_{→ G}2 m,C to be z 7→ (z, 1).

And we have the following proposition (based on p.34 of [Mil20]): Proposition 2.1.3. The field of definition of uΦ is F (E, Φ).

Proof. For an arbitrary σ ∈ Aut_Q_{(C), and for an arbitrary z ∈ C}×, notice that sΦ◦ µΦ is

defined by

z 7→ (z)φ∈Φ× (1)φ∈ιΦ,

and σ(sΦ◦ µΦ) is defined by

z 7→ σ−1z 7→ (σ−1z)φ∈Φ× (1)φ∈ιΦ 7→ (z)φ∈Φ× (1)φ∈ιΦ.

Also note that sΦ : T_CE → (Gm,C)Φ` ιΦ is defined by

(E ⊗_Q_C)× → Y

ρ:E→C

C×, a ⊗ r 7→ (ρ(a)r)ρ,

and that σsΦ is defined by

a ⊗ r 7→ a ⊗ σ−1r 7→ (ρ(a)σ−1r)ρ7→ (σρ(a)r)ρ.

From σ(sΦ◦ µΦ) = σsΦ◦ σµΦ, we can conclude that σµΦ(z) = a ⊗ r such that σρ(a)r = z

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if and only if σΦ = Φ. Then we have

CAutQ(C)µΦ _{= C}AutQ(C)Φ _{= F (E, Φ).}

Thus F (E, Φ) is the field of definition of uΦ. Therefore, uΦ gives a homomorphism

Gm,F (E,Φ) µΦ

−→ TE F (E,Φ).

Definition 2.1.2. The reflex norm NΦ : Gm,F (E,Φ) → TE for a CM pair (E, Φ) is

defined to be the composite of the morphisms

Gm,F (E,Φ) µΦ −→ TE F (E,Φ) N m_{F (E,Φ)/Q} −−−−−−−→ TE . Notice that we have an isomorphism E ⊗_QF (E, Φ) → Q

φ∈Φ

F (E, Φ) defined by e ⊗ f 7→ (φ(e)f )φ. More generally, let k be a field containing F (E, Φ), then we have an isomorphism

E ⊗_Q k → Q

φ∈Φ

k defined by e ⊗ f 7→ (φ(e)f )φ. If we take the Q-points, then we get a

group homomorphism NΦ ◦ N mk/F (E,Φ) : k× → E×. Similarly, if we take the Af-points,

then we get a continuous group homomorphism

NΦ◦ N mk/F (E,Φ) : A×k,f = (k ⊗QAf) ×

→ A×E,f := (E ⊗QAf) ×

,

which is an extension of NΦ◦ N mk/F (E,Φ): k×→ E×. We also have that NΦ◦ N mk/F (E,Φ)

as defined above induces a morphism from ˆOk\A×k,f/k

× _{to ˆ}_O

E\A×E,f/E

× _{(that is, between}

ideal class groups. See pp.16-18 of [Mil20]).

If a is a fractional ideal of k, then we have the following formula (Which will be used many times later):

NΦ◦ N mk/F (E,Φ)(a) =

Y

φ∈Φ

φ−1(N mk/φE(a)).

Actually, for an arbitrary a ∈ k×, then we have that φ(NΦ◦N mk/F (E,Φ)(a)) = N mk/φE(a),

then the equality follows easily for principal ideals, and the equality for fractional ideals follows immediately (for more detail, see pp.16-18 of [Mil20]).

2.2 CM abelian varieties

This section is based on Section I.3 of [Mil20].

Given a field k, an algebraic variety X over k is called an abelian variety over k if X is a connected proper smooth variety over k. An abelian variety is said to be simple if there does not exists an abelian variety B ⊂ A with 0 6⊂ B 6⊂ A. A morphism between group varieties is said to be an isogeny if it is surjective and has a finite kernel.

Remark 2.2.1. Given a complex torus T = Cg_{/Λ, we have that T is projective}

(equiv-alently, T = Aan _{for some complex abelian variety A) if and only if there exists an}

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• ψ_R_{: C}g

× Cg

→ R satisfies that ψR(iv, iw) = ψR(v, w) for all (v, w) ∈ C g_.

• The associated hermitian form H(v, w) := ψ_R(iv, w) + iψ_R(v, w) is positive-definite for all (v, w) ∈ Cg.

For the proof, see Section 3, Chapter I of [Mum70].

We call an alternating bilinear form ψ : Λ × Λ → Z with the above property a Rie-mann form on Λ, and we call an alternating bilinear form α : Λ_Q× Λ_Q _{→ Q with the} above property a rational Riemann form on Λ_Q.

Given an abelian variety A over k, recall that A is isogenous to

n

Q

i=1

Ari

i such that all

Ai are simple abelian varieties and that Ai is isogenous to Aj if and only if i = j. We

also have that n, (r1, ..., rn) are uniquely determined by A, and that all Ai are uniquely

determined by A up to isogenous (pp.42-43 of [Mil08]). If n = 1, we call A an isotypic abelian variety.

We use End0_{(A) to denote End(A) ⊗}

Z Q. Let A be an abelian variety over C. If

A is simple, then End0_{(A) is a division Q-algebra. If A is isogenous to} Qn

i=1

Ari

i , then

End0_{(A) ∼}₌ Qn

i=1

Mri(Di) such that Di = End

0_(A

i) is a division Q-algebra (p.43 of [Mil08]).

It can be shown that dim_Q(Di) = 2 dim(Ai) if and only if Di is a field with degree

2 dim(Ai) over Q, and that dimQ(Di) < 2 dim(Ai) otherwise (Proposition 3.1 of [Mil20]).

Now we can formulate the definition of CM abelian varieties:

Definition 2.2.1. Given an abelian variety A over C, if A is simple, then we call A a CM abelian variety (or A has complex multiplication) if End0_{(A) is a field with}

dim_Q(End0_{(A)) = 2 dim(A). For a general abelian variety A over C, we call A a CM}

abelian variety (or A has complex multiplication) if A is isogenous to product of simple CM abelian varieties.

Example 2.2.1. Given an elliptic curve E over C (note that E is always a simple abelian variety), we know that E is complex analytic isomorphic to C/Λ for some lattice Λ ⊂ C. Write Λ = Zω1⊕ Zω2. Then we have that End(E) = {λ ∈ C|λΛ ⊂ Λ}, which equals Z

or an order in Λ ⊗_Z_{Q (the latter happens when Q[ω}1/ω2] is a quadratic imaginary field,

and End(E) is isomorphic to an order in Q[ω1/ω2]). If End(E) is bigger than Z, then E

is a CM elliptic curve. In particular, Let Λ = Z[i], and let E = C/Λ, then iΛ = Λ. And E is a CM elliptic curve with End(E) ∼_{= Z[i], End}0(E) ∼_{= Q[i]. (For more detail, see} Section 2, Chapter II of [Sil94])

Let k be a field, and let A be an abelian variety over k, we use A∨ to denote the dual abelian variety of A (as constructed in pp.123-124 of [Mum70]). Let B be another abelian variety over k, and let f : A → B be a morphism of abelian varieties, we use f∨ : B∨ → A∨ _{to denote the dual morphism of f (as constructed in p.41 of [Mil08]).}

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Hom(B∨, A∨) ⊗_Z_{Q to be f}∨⊗ q. And we define f−1 _{∈ Hom(B, A) ⊗}

ZQ to be the inverse

of f .

A polarization of A is defined to be a isogeny from A to A∨_{. Suppose k = C, and} let A(C) = Cg/Λ. Then giving a polarization of A is the same as giving a Riemann form (as defined in Remark 2.2.1) on Λ (pp.82-87 of [Mum70]).

Given an abelian variety A over C (with A ∼= Cn/Λ) together with a polarisation λ. The Rosati involution with respect to λ (denoted by rλ) is defined as follows (recall

that λ−1 is the inverse of λ in Hom(A, A∨) ⊗_Z_Q):

rλ : End0(A) → End0(A) : rλ(α) = λ−1◦ α∨◦ λ.

Definition 2.2.2. Let C be a Q-algebra, and let α : C → C be a Q-algebra morphism. We call α an involution if α2 is the identity. An involution α is said to be positive-definite if T r_C/Q(aα(a)) > 0 for all a ∈ C with a 6= 0.

Fix A, λ as above, then the associated Rosati involution is positive-definite. For abelian varieties over C, we have the following elementary proof of the positive-definiteness, which is based on [Orr12]:

Proposition 2.2.1. The bilinear form End0(A) × End0_{(A) → Q given by (α, β) 7→} T rEnd0_(A)/Q(α ◦ r_λ(β)) is positive definite.

Proof. Write A(C) = Cg/Λ, use ψ to denote the Riemann form associated to λ (p.82 of [Mum70]), and let H(x, y) = ψ_R(ix, y) + iψ_R(x, y) be the associated Hermitian positive-definite form. Then λ equals x 7→ H(x, −) (p.109 of [JoW15]). For every α ∈ End0(A), and for arbitrary x, y ∈ Λ_R, we have H(rλ(α)x, y) = H(x, αy). Actually,

H(rλ(α)x, y) = λ(rλ(α)x)(y) = (λ ◦ rλ(α))(x)(y) =

λ ◦ λ−1◦ α∨◦ λ(x)(y) = α∨◦ λ(x)(y) = α∨(λ(x))(y) = H(x, αy).

Note that α∨(λ(x))(y) = H(x, αy) because α∨ sends x 7→ H(x, −) to x 7→ H(x, α(−)) (pp.86-87 of [Mum70]).

Then we have that

H(x, αy) = H(rλ(α)x, y) = H(x, rλ(rλ(α))y),

which tells us that r2

λ(α)x = αx for all x ∈ ΛR. Thus r 2 λ = 1.

Let us consider the action of α ◦ rλ(α) on V := T gt0(A) (we define T gt0(A) to be the

tangent space of A at the identity element, viewed as a complex vector space). Let c be the eigenvalue corresponding to an eigenvector x, then we have that

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as H is positive definite, we can conclude that c ≥ 0.

Also note that α ◦ rλ(α) 6= 0 if α 6= 0. Actually, as rλ2 = 1, we have that rλ(α) 6= 0.

Then there exists y ∈ Λ_R such that rλ(α)y 6= 0. Therefore,

H(rλ(α)y, rλ(α)y) = H(α ◦ rλ(α)y, y) 6= 0,

which tells us that α ◦ rλ(α)y 6= 0. Thus if we consider the action of α ◦ rλ(α) on V , the

eigenvalues of α ◦ rλ(α) are all non-negative and at least one of them is positive.

Consider the action of α ◦ rλ(α) on EndC(V ) defined by β 7→ α ◦ rλ(α) ◦ β, then the

eigenvalues of α ◦ rλ(α) on EndC(V ) are also eigenvalues of α ◦ rλ(α) on V . Also note

that α ◦ rλ(α) 6= 0 when considered as an action on EndC(V ). Then we have that

T rEnd_C(V )/C(α ◦ rλ(α)) > 0,

which tells us that

T rEnd0_(A)⊗

QC/C(α ◦ rλ(α)) > 0,

this is because by considering the action of End0(A) ⊗_Q_{C on V , we get a faithful} rep-resentation of End0(A) ⊗_Q_{C in End}_C(V ) (pp.21-22 of [Mil20]). Then the eigenvalues of α ◦ rλ(α) on End0(A) ⊗QC must be eigenvalues of α ◦ rλ(α) on EndC(V ).

Thus T rEnd0_(A)/Q(α◦r_λ(α)) > 0. Therefore, the bilinear form End0(A)×End0(A) → Q

given by (α, β) 7→ T rEnd0_(A)/Q(α ◦ r_λ(β)) is positive definite.

And we have the following theorem about CM abelian varieties (Proposition 3.6 of [Mil20]):

Theorem 2.2.1. Let A be an abelian variety over C, then we have the following:

(i) A is a simple CM abelian variety if and only if End0_{(A) is a field of degree 2 dim(A)}

over Q if and only if End0_{(A) is a CM field of degree 2 dim(A) over Q.}

(ii) A is an isotypic CM abelian variety if and only if End0_{(A) contains a number}

field K of degree 2 dim(A) over Q. And if End0_{(A) contains a number field K of degree}

2 dim(A) over Q, then K is a CM field.

(iii) A is a CM abelian variety if and only if End0_{(A) contains an ´}_{etale algebra M of}

degree 2 dim(A) over Q. And if End0_{(A) contains an ´}_{etale algebra M of degree 2 dim(A)}

over Q, then M is a CM algebra.

To prove this theorem, we need the following lemmas:

Lemma 2.2.1. Let K be a subfield of End0_{(A) for some abelian variety A over C such that}

[K : Q] = 2 dim(A), then K has no real place and K is closed under complex conjugation. Proof. Write n = dim(A), and write Φ to be the set of all embeddings K → C. Let TR(A)

be the tangent space of A at the identity while regarding A as a real manifold, let T_C(A) be the tangent space of A at the identity while regarding A as a complex manifold. Then

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we have that T_R(A) ⊗_R_C ∼= Q

φ∈Φ

Ceφ as K ⊗Q C-module because K ⊗Q C acts faithfully

on T_R(A) ⊗_R_{C. Note that}

T_R(A) ⊗_R_{C = T}_C(A) ∪ T_C(A)

such that T_C(A) ∩ T_C(A) = ∅ (pp.16-17 of [CEZG+14]). Then we can conclude that T_C(A) ∼= Q

φ∈Φ0Ceφ

as a K ⊗_Q_{C-submodule of T}_R(A) ⊗_R_{C such that Φ = Φ}0` ιΦ0 _{(ι is the}

complex conjugation). The decomposition Φ = Φ0` ιΦ0 _{implies that K has no real place,}

and that K is closed under complex conjugation.

The following Lemma gives us a Rosati involution fixing K, the proof of which is based on [Shi69], pp.126-129.

Lemma 2.2.2. Let K be a subfield of End0_{(A) for some abelian variety A over C such}

that [K : Q] = 2 dim(A), then there exists a polarization λ of A such that rλ(K) = K.

Proof. Recall that in the proof of Lemma 2.2.1, we have a decomposition Φ = Φ0` ιΦ0

such that ι is the complex conjugation and Φ is the set of all embeddings K → C. Also note that K ⊗_Q_R∼= T_C(A) ∼= Q

φ∈Φ0Ce

φ as K ⊗QR-module.

From Lemma 2.2.1 we know that K cannot be a totally real field and that K is closed under complex conjugation.

Write A(C) = Cn_{/Λ. The field K acts faithfully on H}

1(Aan, Q) = ΛQ, so ΛQ ∼= K as

K-vector spaces. Fix an embedding K → C and consider the bilinear form E : ΛQ× ΛQ → Q

defined by (x, y) 7→ T r_K/Q(ζxy) (with the conjugation induced from K, and ζ is an element of K). We will show that there exists a ζ ∈ K such that E gives a rational Riemann form on Λ_Q (as defined in Remark 2.2.1). We can check that E is alternating if and only if ζ = −ζ. In the following, we suppose that ζ = −ζ.

We have that

E_R(ix, iy) = T rK⊗_QR/R(ζixiy) = T rK⊗QR/R(ζiixy) = T rK⊗QR/R(ζxy) = ER(x, y).

Also notice that

E_R(x, ix) = T rK⊗_QR/R(ζiixy) = X φ∈Φ0 ζφxφ(ix)φ= X φ∈Φ0 −iζφ_(xx)φ_,

with ζφ_{being totally imaginary as we supposed. Then E}

Ris positive definite if Im(ζ φ_{) > 0}

for all φ ∈ Φ0.

We may choose ζ ∈ K such that ζ = −ζ and that Im(ζφ_{) > 0 for all φ ∈ Φ}0_{: Let}

Φ0 = {φ1, ..., φn}. As φ(K) is dense in C for all embedding φ ∈ Φ0, we may choose

xi ∈ Kφi such that Im(xi) > 1 for all 1 ≤ i ≤ n. By the week approximation theorem

(Theorem 7.20 of [Mil]), there exists β ∈ K such that |βφi− x

i| < 1₂ for 1 ≤ i ≤ n. Thus 1

2 < Im(β

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With the chosen ζ, we have that E gives a rational Riemann form on Λ_Q (as defined in Remark 2.2.1). Then a suitable multiplication by some m ∈ Z>0 will give us a Riemann

form mE|Λ×Λ on Λ. Let λ be the polarization corresponding to mE|Λ×Λ, and let H be

the associated Hermitian form. Let r : End0(A) → End0(A) be the Rosati involution corresponding to λ. Then r satisfies that r(k) = k for every k ∈ K: Because for an arbitrary x ∈ Λ_R, we have the following (recall that λ equals x 7→ H(x, −) and α∨ sends x 7→ H(x, −) to x 7→ H(x, α(−)) for every α ∈ End0(A)):

r(k)x = λ−1◦k◦λ(x) = λ−1H(x, k(−)) = λ−1m(T rK⊗_QR/R(ζixk(−))+iT rK⊗QR/R(ζxk(−)))

= λ−1m(T rK⊗_QR/R(ζi¯kx( ¯−)) + iT rK⊗QR/R(ζ ¯kx( ¯−))) = λ

−1

H(¯kx, −) = kx.

Therefore, r(k) = ¯k ∈ K for every k ∈ K (note that ¯k ∈ K by Lemma 2.2.1). And we get the desired Rosati involution.

Now we can prove Theorem 1.2.1:

Proof. The proof of (i): The first ”if and only if” follows from the definition of CM abelian. To prove the second ”if and only if”, the if part is trivial. To prove the only if part, we may choose a polarization λ of A and consider the Rosati involution rλ corresponding to

λ, then the bilinear form (x, y) 7→ T r_End0_(A)/Q(xr_λ(y)) (with x, y ∈ End0(A)) is

positive-definite (Proposition 2,2,1). By Proposition 2.1.1, we have that End0_{(A) is totally real}

or CM, then Lemma 2.2.1 tells us that End0_{(A) must be a CM field.}

The proof of (ii): Suppose that A contains a number field K of deg 2dim(A) over Q. Note that A is isogenous to

n

Q

i=1

Ari

i such that all Ai are simple abelian varieties and that

Ai is isogenous to Aj if and only if i = j. And we know that End0(A) ∼= n

Q

i=1

Mri(Di) such

that Di = End0(Ai) is a division Q-algebra. As K is simple, then K can be viewed as a

subalgebra of Mri(Di) for some i.

The algebra Mri(Di) is central simple over Z(Di). Then we have that

dimZ(Di)(K) ≤

q

dimZ(Di)(Mri(Di)) = ri

q

dimZ(Di)(Di).

Because Z(Di) is a field, then the maximal subfield of Mri(Di) containing Z(Di) has

degree pdimZ(Di)(Mri(Di)) over Z(Di). Then we have that

dim_Q(K) = dim_Q(Z(Di)) dimZ(Di)(K) ≤ ri

q

dimZ(Di)(Di) dimQ(Z(Di)) ≤

ridimZ(Di)(Di) dimQ(Z(Di)) = ridimQ(Di) ≤ 2ridim(Ai) ≤ 2 dim(A).

The equality holds if and only if

n = 1, dimZ(Di)(Di) = 1, dimZ(Di)(K) =

q

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if and only if n = 1, Di is a field, dimZ(Di)(K) = ri, dimQDi = 2 dim(Ai) if and only if

n = 1 and Ai is a CM abelian variety. Therefore, A = Ar11 with A1 being a CM abelian

variety, thus A is an isotypic CM abelian variety.

Conversely, if A is an isotypic CM abelian variety, then A = Ar₁ with A1 being a CM

abelian variety and that End0(A) ∼= Mr(k) such that k ∼= End0(A1) is a field of degree

2 dim(A1) over Q. As [k : Q] = 2 dim(A1) < ∞, we can choose a prime number p such

that xr− p = 0 is irreducible over k[x]. Consider the matrix m = (mi,j)1≤i,j≤r ∈ Mr(k)

given by mi,j = 0 for i = j and mi,j = p for i = 1, j = r, then CharM = xr − p is

irreducible over k[x]. Also notice that CharM(M ) = 0, then k[M ] is a field contained in

End0_{(A) such that [k[M ] : Q] = [k[M ] : k][k : Q] = r[k : Q] = 2r dim(A}1) = 2 dim(A).

If End0_{(A) contains a number field K with [K : Q] = 2 dim(A), by Lemma 2.2.2, we} may choose a polarization λ of A such that the Rosati involution rλ corresponding to λ

satisfies rλ(K) = K. By Proposition 2.2.1, the bilinear form (x, y) 7→ T rK/Q(xrλ(y)) is

positive-definite. By Proposition 2.1.1, we have that K is totally real or CM. Then K must be a CM field by Lemma 2.2.1.

The proof of (iii): this part follows immediately from (ii).

Then we have the following corollary about abelian varieties over C:

Corollary 2.2.1. Let A be an abelian variety over C, if End0_{(A) contains a number field}

K of degree 2 dim(A) over Q, then A is an isotypic CM abelian variety and K is a CM field.

Proof. This corollary follows immediately from Theorem 1.2.1 (ii).

Given a field k, we use AV (k) to denote the category of abelian varieties over k. We use AV0(k) to denote the category of abelian varieties over k up to isogeny (p.78 of [Mil08]). Note that AV0(k) consists of the following:

• Objects: Abelian varieties over k.

• Morphisms: Let A, A0 be ableian varieties over k, then a morphism in AV0(k) from A to A0 is an element of M ork(A, A0) ⊗ZQ. Note that we use M ork(A, A

0

) to denote the abelian group of morphisms (of abelian varieties over k) from A to A0.

Let A, B be abelian varieties defined over a field k, and let f : A → B be an isogeny. Write d = deg(f ). If (d, char(k)) = 1, then there exists a unique isogeny g : B → A such that g ◦ f = [d]A (multiplication by d in End(A)) and f ◦ g = [d]B (multiplication by d in

End(B)), see Proposition 5.12 of [EVdGM]. And g is called the inverse of f in AV0_(k).

If A is a CM abelian varieties of CM type (E, Φ) defined over a field k. Write E = Qn

i=1Ei with all Ei being CM fields. Then for an arbitrary e ∈ E

×_{, we can write e =}

(r1, .., rn)(s−11 , ..., s −1

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an order in E, see p.52 of [Mil20]. Also note that (r1, ..., rn), (s1, ..., sn) are isogenies, see

p.58 of [Mil20]).

Then we may associate e ∈ E to an e0 ∈ End(A) defined by (r1, ..., rn) ◦ (s1, ..., sn)−1

with (s1, ..., sn)−1 as constructed above. Note that e0 ∈ End(A) is not well-defined. When

we want to associate e ∈ E to some e0 ∈ End(A), we just pick one of the elements in End(A) as constructed above. When k = C and A(C) = CΦ/Λ, we can check that all elements of End(A) associated to e ∈ E as constructed above induce the same action on Λ_Q.

Let X be a scheme over a ring R, and let a : Spec(R) → X be an R-point of X. We define the tangent space of X at a (denoted by T gta(X)) to be the module of

R-morphisms f : Spec(R[]/(2 = 0)) → X such that Spec(R)−→ Spec(R[]/(res 2 _{= 0))}₋_{→ X}f

equals a. Where res is given R[]/(2 = 0) → R, 7→ 0.

The R-module structure on T gta(X) is given as follows: for every p, q ∈ T gta(X), we

define p + q to be R[]/(2 = 0)−→ R[g 1, 2]/(21 = 2 2 = 12 = 0) f − → X

where f is given in the following push-out diagram (Tag 07RS of [Sta20]), and g is given by 1 7→ , 2 7→ : Spec(R) Spec(R[]/(2 _{= 0))} Spec(R[]/(2 = 0)) R[1, 2]/(12 = 22 = 12 = 0) X 7→0 7→0 _p q f

And for every r ∈ R, we define rp to be

Spec(R[]/(2 = 0))−−−→ Spec(R[]/(7→r 2 _{= 0))}₋_{→ X.}p

Then our above constructions give T gta(X) the structure of an R-module (Tag 0B2B of

[Sta20]).

Definition 2.2.3. Given a CM type (E, Φ) and an abelian variety over a field k containing F (E, Φ) and contained in C (recall that F (E, Φ) is the reflex field of (E, Φ)), we say that (A, i) is a CM abelian variety of type (E, Φ) if 2 dim(A) = dim_Q(E), i : E → End0_{(A) is an embedding and T gt}

0(A) ∼=Q kϕ∈Φ as E ⊗Q k module, with e ∈ E acts on

T gt0(A) via e(a)ϕ = (ϕ(e)a)ϕ.

Let (A, i), (A0, i0) be two CM abelian varieties of type (E, Φ) over k, then an isogeny f : A → A0 is called an E-isogeny if f commutes with the action of E: that is, if we

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consider f ⊗ 1 ∈ Hom(A, A0) ⊗_Z_{Q, then (f ⊗ 1) ◦ e = e ◦ (f ⊗ 1) for every e ∈ E (thus} the induced map of f from T gt0(A) to T gt0(A0) commutes with the action of E). And we

say that A is E-isogenous to A0 if there exists an E-isogeny from A to A0. When k is a field containing F (E, Φ) and contained in C, we use Q

ϕ∈Φkϕ (product

of k indexed by Φ) to denote the E-module with the action of E given by e(kϕ)ϕ∈Φ =

(ϕ(e)kϕ)ϕ∈Φ for every e ∈ E and (kϕ)ϕ∈Φ ∈

Q

ϕ∈Φkϕ.

Let OE be the maximal order in E, that is, the product of rings of integers of all factor

fields. Let CΦ be a direct sum of copies of C indexed by Φ. We use ηΦ to denote the homomorphism OE → CΦ given by a 7→ (ϕ(a))ϕ∈Φ. Note that

OE ⊗ZR∼= OE ⊗ZQ ⊗QR∼= E ⊗QR∼= C Φ_,

where the final isomorphism is given by

e ⊗ r 7→ (ϕ(e) · r)ϕ∈Φ.

Thus ηΦ(OE) is a lattice in CΦ. Define iΦ(e)(aϕ)ϕ∈Φ by e(aϕ)ϕ∈Φ = (ϕ(e)aϕ)ϕ∈Φ for e ∈ E

and (aϕ)ϕ∈Φ ∈ T gt0(AΦ) = CΦ.

We use AΦ to denote the complex torus CΦ/ηΦ(OE), then we have the following

proposition:

Proposition 2.2.2. We have that AΦ is an abelian variety.

Proof. Without loss of generality, we may suppose that E is a field (By decomposing CΦ/ηΦ(OE) into product of complex tori).

In the following, we will construct a Riemann form on ΛΦ. Write ηΦ(OE) for ΛΦ and

choose an embedding E → C. Then ΛΦQ is isomorphic to E as E-module. Consider the

bilinear form R : Λ_ΦQ× Λ_ΦQ _{→ Q defined by (x, y) 7→ T r}_E/Q(ζxy) (with the conjugation induced from E, and ζ is an element of E).

We may choose ζ ∈ K such that R gives a rational Riemann form on Λ_ΦQ (see the proof of Lemma 2.2.1 of this thesis), then a suitable multiplication by m would give a Riemann form mR|ΛΦ×ΛΦ on ΛΦ. Thus AΦ is an abelian variety.

The embedding iΦ : E → End0(AΦ) defined by iΦ(e)((aϕ)ϕ∈Φ) = (ϕ(e)aϕ)ϕ∈Φ for

e ∈ E, (aϕ)ϕ∈Φ ∈ T gt0(AΦ) = CΦ makes (AΦ, iΦ) a CM abelian variety of type (E, Φ).

Let A be a CM abelian variety of type (E, Φ) defined over C. Then we may write A = CΦ_/ηΦ_{(R) such that R is a lattice in E (pp.29-30 of [Mil20]).}

Proposition 2.2.3. For an arbitrary CM abelian variety (A, i) defined over C of type (E, Φ), we have that A is E-isogenous to AΦ

Remark 2.2.2. This proposition tells us that all CM abelian varieties over C of the same CM type are isogenous.

Canonical models of Shimura varieties of PEL type

MSc Mathematics

Master Thesis