Abelian Varieties over ℚ

with Large Endomorphism Algebras

and

Their Simple Components over \overline{ℚ}

Elisabeth Eve Pyle

B.S. (Stanford University) 1988

C.Phil. (University of California at Berkeley) 1992

A dissertation submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophy

in Mathematics

in the Graduate Division

of the

University of California at Berkeley

Committee in Charge: Professor Kenneth A. Ribet, Chair

Professor Andrew P. Ogg Professor Alan Schoenfeld

Contents

Introduction 1 1 Abelian Varieties over ℚ of GL_{2}-type 4 2 The Brauer Class of End(B) 12 3 The Tensor Product of a Module with an Abelian Variety 21 4 Descent from \overline{ℚ} to ℚ 25 5 Fields of Definition for Building Blocks 35 6 An Example of a ℚ-Curve 47 7 Further Questions 53 A A Non-Building Block 55 B Useful Facts Concerning Central Simple Algebras 62 Bibliography 64

Introduction

This dissertation expands upon the results of K. Ribet in [Ri 1] and [Ri 2] con-

cerning abelian varieties “of GL_{2}-type”. An abelian variety A/ℚ is said to be of GL_{2}-type if its algebra of ℚ-endomorphisms ℚ ⊗ End_{ℚ}(A) is a number field E of degree equal to the dimension of A.

The study of abelian varieties of GL_{2}-type grew out of research surrounding the

Taniyama-Shimura Conjecture. Recall that the usual statement of the Taniyama-Shimura Conjecture asserts that every elliptic curve over ℚ is a quotient of a modular curve of the form X_{0}(M). Recent manuscripts circulated by Wiles and Taylor-Wiles prove the Taniyama-Shimura Conjecture for a large class of elliptic curves over ℚ.

It is well known that an equivalent formulation of the Taniyama-Shimura Conjec-

ture requires only that every elliptic curve over ℚ be a quotient of a standard modular curve of the form X_{1}(N). Furthermore, in [Ma], B. Mazur proved the conjecture’s equivalence to yet another seemingly weaker statement: an elliptic curve over ℚ is a quotient over ℚ of some X_{1}(N) if and only if it is a quotient over \overline{ℚ} of some X_{1}(N)_{/\overline{ℚ}}.

Now every map from X_{1}(N) to an elliptic curve sending ∞ to O extends uniquely

to a map from its Jacobian J_{1}(N) to the elliptic curve. Hence, this line of inquiry leads naturally to the (closely related) questions: how may we characterize those simple abelian varieties over ℚ (respectively, over \overline{ℚ}) that appear as quotients of J_{1}(N) (respectively, J_{1}(N)_{/\overline{ℚ}}) for some N ≥ 1? We will loosely use the word “modular” to refer to such abelian varieties.

In [Ri 1], Ribet gives a conjectural answer to one of these questions.

Conjecture. A simple abelian variety A over ℚ is modular if and only if A is of GL_{2}-type.

It is known that every simple quotient of J_{1}(N)/ℚ is of GL_{2}-type. Indeed, the

primary source of examples of abelian varieties of GL_{2}-type is a construction resulting from extensive work by Shimura on the subject [Sh 2, Sh 4]. Let f = ∑ a_{n}q^{n} be a normalized cuspidal eigenform of weight two on Γ_{1}(N). Let E be the number field ℚ(…, a_{n}, …). Then Shimura constructs, as a quotient of J_{1}(N), an abelian variety A_{f} over ℚ with the following properties: there is an action E ⟶^{∼} ℚ ⊗ End_{ℚ}(A_{f}), and the dimension of A_{f} is equal to [E: ℚ]. Moreover, every simple quotient of J_{1}(N)/ℚ is isogenous to an A_{f} constructed in this way.

Thus the conjecture is reduced to: all abelian varieties over ℚ of GL_{2}-type are

isogenous to some A_{f}. In support of this assertion, Ribet has proved [Ri 1, Thm. 4.4] that if one assumes Serre’s conjecture [Se 4, 3.2.4_{?}] about representations of Gal(\overline{ℚ}/ℚ), then it follows that all abelian varieties over ℚ of GL_{2}-type are modular. (See also [Sh 1].)

Now in this dissertation, we look at the complete decomposition over \overline{ℚ} of abelian

varieties of GL_{2}-type. We determine necessary and sufficient conditions for an abelian variety over \overline{ℚ} to be a simple component of an abelian variety over ℚ of GL_{2}-type; such abelian varieties will be called “building blocks”. These results enable us to restate the above conjecture as: a simple abelian variety B over \overline{ℚ} is modular if and only if B is a building block.

We would like to emphasize that the work contained herein deals exclusively with

the properties of abelian varieties of GL_{2}-type, the properties of their component building blocks, and the relationships between these two kinds of abelian varieties, without any recourse to their possible “modular” structure.

In Chapter 1 we define our terminology and review the structure of abelian va-

rieties over ℚ of GL_{2}-type. Chapter 2 contains a discussion of the Brauer class of the endomorphism algebra of a building block; it culminates in a theorem we will apply in Chapter 4. Chapter 3 is a short preparation of a category-theoretic construction that we will also use in Chapter 4.

Chapter 4 contains the bulk of the technical work; it describes the correspondence

between building blocks over \overline{ℚ} and abelian varieties over ℚ of GL_{2}-type. Then Chapter 5 is a collection of further results, mostly connected to fields of definition for these abelian varieties and their endomorphisms. In Chapter 6 we illustrate the theory of Chapters 4 and 5 with explicit examples of a one-dimensional building block and an associated four-dimensional abelian variety of GL_{2}-type. Chapter 7 is a brief consideration of some unanswered questions and avenues for future work.

Finally there are two appendices. In the first, we construct a counterexample to demonstrate the necessity of a certain condition we impose on building blocks. The second is a list of facts we quote from the theory of central simple algebras; we apply these theorems to the endomorphism algebras we encounter.

Notation

Throughout this dissertation we will work in the category of abelian varieties up to

isogeny. That is the category whose objects are abelian varieties A, B, … with morphisms ℚ ⊗ Hom(A, B). In other words, we formally invert all isogenies A → B. In this category, notation is simplified by writing Hom(A, B) for ℚ ⊗ Hom(A, B).

The endomorphism algebra Hom(A, A) will be denoted End(A), and the following

variants of Hom will also apply to End.

If A and B are abelian varieties defined over a field K, then Hom(A, B) will refer

to all morphisms A_{/\overline{K}} → B_{/\overline{K}}. If we wish to refer to only those morphisms defined over K, we will write Hom_{K}(A, B).

In some cases, A and B may be equipped with an action of a ring R; that is, there

are fixed maps R → End(A) and R → End(B). A morphism φ : A → B will be said to “respect R” (or “respect the actions of R”, or “be compatible with R”…) if φ r = r φ for all r ∈ R. The subgroup of morphisms that respect R will be denoted Hom^{R}(A, B).

We will write Z(R) for the center of a ring R.

The sign “∼” will designate the equivalence relation of isogeny.

The symbol G_{ℚ} will denote the absolute Galois group Gal(\overline{ℚ}/ℚ) with its usual

profinite topology.

Acknowledgments

My warmest thanks go to my dissertation advisor, Ken Ribet, for all his helpful

suggestions. Without his patient and continued support, this work would never have been realized.

This work was partially supported by NSF Grant #DMS 93-06898.

Chapter 1

Abelian Varieties over ℚ of

GL_{2}-type

In this first chapter we will recapitulate many of the results from [Ri 1], on which

we rely heavily in the remainder of this dissertation. Our approach differs from Ribet’s in that it focuses not on ℓ-adic representations, but rather on the structure of endomorphism algebras as semi-simple ℚ-algebras. We remind the reader that for an abelian variety A, the notation End(A) will always refer to the ℚ-algebra of endomorphisms of A.

Definition. An abelian variety A/ℚ is of GL_{2}-type if End_{ℚ}(A), the algebra of endomorphisms of A defined over ℚ, is a number field of degree equal to the dimension of A.

Note that this definition applies only to abelian varieties defined over ℚ, although we may not always say so explicitly; i.e., the phrase “A is of GL_{2}-type” will imply that A is defined over ℚ. Also note that abelian varieties of GL_{2}-type are always simple over ℚ, which is a modification of the terminology in [Ri 1].

Why do we say these abelian varieties are “of GL_{2}-type”? If A/ℚ is an abelian

variety such that End_{ℚ}(A) is a number field E with [E: ℚ] = dim A, then it is well known that for any prime ℓ, the Tate module V_{ℓ}(A) is free of rank two over E ⊗_{ℚ} ℚ_{ℓ}. Thus the action of the absolute Galois group Gal(\overline{ℚ}/ℚ) = G_{ℚ} on V_{ℓ}(A) defines a representation G_{ℚ} → GL_{2}(E ⊗_{ℚ} ℚ_{ℓ}).

Abelian varieties of GL_{2}-type with complex multiplication over \overline{ℚ} have a particu-

larly simple structure. Indeed, Shimura has proved the following in [Sh 1, Prop. 1.5]:

CHAPTER 1. ABELIAN VARIETIES OVER ℚ OF GL_{2}-TYPE 5

Proposition 1.1 Let A/ℚ be an abelian variety of GL_{2}-type. Suppose A_{/\overline{ℚ}} contains a simple abelian subvariety of CM-type. Then A_{/\overline{ℚ}} is isogenous to a power of a CM elliptic curve. The theory of elliptic curves with complex multiplication is well understood, and requires different techniques from the non-CM case. For these reasons, we will focus on abelian varieties with no complex multiplication. Henceforth we will consider only abelian varieties of GL_{2}-type with no complex multiplication over \overline{ℚ}.

We begin our analysis of abelian varieties of GL_{2}-type with a review of a familiar

lemma that describes the action of a field on an abelian variety. This result is originally due to Albert, and Mumford thoroughly covers the case of a simple abelian variety in [Mu]. Since we will be concerned mostly with non-simple abelian varieties, we include a proof to remind the reader of the details in that situation. In particular, our proof will help us better understand the special case of abelian varieties of GL_{2}-type.

We will use the following terminology from the theory of central simple algebras.

(See Appendix B for more details.) Let a be a simple ℚ-algebra and let F be the center of a . The Schur index of a is the integer \sqrt{dim_{F} D}, where D is a division algebra (determined up to isomorphism) such that a ≅ M_{n}(D) for some n ≥ 1. A subfield of a is a ℚ-subalgebra E of a such that E is a field. A maximal subfield of a is a subfield that is maximal with respect to inclusion.

Lemma 1.2 Let A be an abelian variety over \overline{ℚ} that has no simple abelian subvariety of CM-type. Let E be a subfield of End(A). Then [E:ℚ] divides dim A.

Proof. We first consider the case where A is isogenous to a power of a simple abelian variety B. Then for some n ≥ 1, we may write A ∼ B^{n}, dim A = n dim B, and End(A) ≅ M_{n}(End(B)). Now B has no complex multiplication by hypothesis. Therefore End(B) is a central division algebra over a totally real field F with Schur index t = 1 or 2; moreover, t[F:ℚ] | dim B, from [Mu, p. 202]. Thus the endomorphism algebra End(A) ≅ M_{n}(End(B)) is seen to be central simple over F with Schur index t. Now E is contained in a maximal subfield of End(A), and because F is a number field, any such maximal subfield has degree nt[F : ℚ], by [Pi, Prop. 13.1]. Thus [E : ℚ] | nt[F : ℚ]. From above, however, we have t[F:ℚ] | dim B. Therefore, [E:ℚ] | n dim B = dim A.

In general, of course, A is isogenous to a (finite) product ∏ B_{i}^{n_{i}}, where each B_{i}

is simple, B_{i} is non-isogenous to B_{j} if i ≠ j, and no B_{i} has complex multiplication. Then End(A) is isomorphic to the direct product of the simple algebras M_{n_{i}}(End(B_{i})). Now because E is a ℚ-subalgebra of End(A), we must have E ↪ M_{n_{i}}(End(B_{i})) for each i (via the ith projection map). But then for each i we apply the above argument to find [E:ℚ] | n_{i} dim B_{i}. Hence [E:ℚ] | ∑ n_{i} dim B_{i} = dim A. QED

Thus we see that another way to characterize abelian varieties of GL_{2}-type is as

those abelian varieties which are simple over ℚ but have a “large” or “maximal” number of endomorphisms defined over ℚ. By applying the lemma in this situation, when [E:ℚ] = dim A, we immediately obtain the following description of the endomorphism algebra of an abelian variety of GL_{2}-type.

Theorem 1.3 Let A/ℚ be an abelian variety of GL_{2}-type with no complex multiplication over \overline{ℚ}. Let E = End_{ℚ}(A). Then the full endomorphism algebra End(A) is a central simple algebra over a totally real field F ⊆ E; the Schur index of End(A) is 1 or 2; and End(A) contains E as a maximal subfield.

Proof. The equality [E:ℚ] = dim A forces equalities everywhere in the proof of the above lemma. In particular, we see that over \overline{ℚ}, we must have A ∼ B^{n} for some simple abelian variety B/\overline{ℚ} with no complex multiplication; otherwise E would act on proper subvarieties of A, and it is too big to do so. This proves that End(A) is a central simple algebra over a totally real field F with Schur index t = 1 or 2. Moreover, we have [E: ℚ] = nt[F : ℚ] = n dim B = dim A. Thus E must be a maximal subfield of End(A), which then implies that E contains the center F of End(A). QED

The Simple Components of A_{/\overline{ℚ}}

Let A be an abelian variety of GL_{2}-type. We now consider the properties of an

abelian variety B/\overline{ℚ} appearing as a simple component of A_{/\overline{ℚ}}. Note that the proof of the above theorem provides us with some information about End(B) as well as End(A). In fact, we have already proved the following proposition:

Proposition 1.4 Let B be a simple component over \overline{ℚ} of an abelian variety of GL_{2}-type. Then the endomorphism algebra End(B) is a central division algebra over a totally real field F; the Schur index t of End(B) is 1 or 2; and t[F:ℚ] = dim B.

Thus B, like A, also has the property that its endomorphism algebra is maximal, if compared with other simple abelian varieties of the same dimension. We remark here that in the case t = 2, the condition t[F : ℚ] = dim B means that End B must be an indefinite quaternion algebra over F. Indeed, the endomorphism algebra of a simple abelian variety is always totally definite or totally indefinite, and the equality t[F : ℚ] = dim B can never hold if End(B) is a definite quaternion algebra [Sh 3, Prop. 15].

It turns out that B has another interesting property. Namely, it is isogenous to

all of its Galois conjugates. More technically, we say that B is a ℚ-abelian variety, which is defined as follows.

Definition. An abelian variety X/\overline{ℚ} is a ℚ-abelian variety if for each σ ∈ Gal(\overline{ℚ}/ℚ), there exists an isogeny μ_{σ} : ^{σ}X → X compatible with the endomorphisms of X; that is, μ_{σ} ^{σ}φ = φ μ_{σ} for all φ ∈ End(X).

It is easy to see that for each σ ∈ G_{ℚ}, the conjugate ^{σ}B must be isogenous to B.

Indeed, ^{σ}B^{n} ∼ ^{σ}A = A ∼ B^{n}, because A is defined over ℚ. Then, by the uniqueness of decomposition of abelian varieties up to isogeny, we must have ^{σ}B ∼ B. It takes a little more work to show that there is actually an isogeny which is compatible with End(B).

Proposition 1.5 Let B be a simple component over \overline{ℚ} of an abelian variety A of GL_{2}- type. Then B is a ℚ-abelian variety.

Proof. Fix an isogeny A_{/\overline{ℚ}} ∼ B^{n}, or equivalently, an isomorphism End(A) ≅ M_{n}(End(B)). For each σ ∈ G_{ℚ}, we form the σ-twist of this isogeny, ^{σ}B^{n} ∼ ^{σ}A. Let φ ∈ End(B), and write φ̃ for the endomorphism of A which is φ in each place. Then, because, A is defined over ℚ, there are commutative diagrams

[figure: ^{σ}B ⟶^{diag.} ^{σ}B^{n} ∼ ^{σ}A = A; ^{σ}B ⟶^{diag.} ^{σ}B^{n} ∼ ^{σ}A = A; ^{σ}φ; ^{^{σ}φ̃}; A ∼ B^{n} ⟶^{π_{j}} B; A ∼ B^{n} ⟶^{π_{j}} B; φ̃; φ]

where π_{j} is projection onto the jth place of B^{n}.

Now End(A) is a central simple algebra, and for each σ ∈ G_{ℚ}, there is an automor-

phism of End(A) given by ψ ↦ ^{σ}ψ. Hence by the Skolem-Noether Theorem, there exists an invertible α(σ) ∈ End(A) such that ^{σ}ψ = α(σ) ψ α(σ)^{−1} for all ψ ∈ End(A). Therefore we may connect the above diagrams with

[figure: A; A; A; A; ^{^{σ}φ̃}; φ̃; α(σ)^{−1}; α(σ)^{−1}]

Then the composite map

σB → ^{σ}B^{n} ∼ ^{σ}A = A ⟶^{α(σ)^{−1}} A ∼ B^{n} ⟶^{π_{j}} B

is a morphism from ^{σ}B to B that is compatible with End(B). For some j this morphism must be non-zero, hence an isogeny because B is simple. Thus, we have found an isogeny μ_{σ} : ^{σ}B → B such that μ_{σ} ^{σ}φ = φ μ_{σ} for all φ ∈ End(B), and so B is a ℚ-abelian variety.

QED

We now make a definition based on these properties associated with simple com-

ponents of abelian varieties of GL_{2}-type.

Definition. A building block is an abelian variety B/\overline{ℚ} such that

(i) the endomorphism algebra End(B) is a central division algebra over a totally real field F with Schur index t = 1 or 2, and t[F:ℚ] = dim B.

(ii) B is a ℚ-abelian variety.

We will show in Proposition 4.5 that every building block appears as a simple component of some abelian variety over ℚ of GL_{2}-type.

The Extension E/F

Let A be an abelian variety of GL_{2}-type. Let E = End_{ℚ}(A), and let F =

Z(End(A)) denote the center of End(A). We now look more closely at the field extension E/F. In the proof of the above proposition, we saw that for each σ ∈ G_{ℚ}, the Skolem-Noether Theorem implies that there exists an invertible α(σ) ∈ End(A) such that σφ = α(σ) φ α(σ)^{−1} for all φ ∈ End(A). Notice that because E = End_{ℚ}(A), the endomorphism α(σ) must be contained in the centralizer of E in End(A). But E is a maximal subfield of End(A), and hence its own centralizer, as F is a number field [Pi, Prop. 13.1]. Therefore α(σ) ∈ E. Furthermore, each α(σ) is determined up to (non-zero) elements of the center F of End(A). We now fix a choice of α(σ) ∈ E for each σ ∈ G_{ℚ}, choosing α(σ′) = α(σ) whenever ^{σ′}φ = ^{σ}φ for all φ ∈ End(A). We may view α as a locally constant function G_{ℚ} → E^{∗} that is a lift of the evident homomorphism G_{ℚ} → E^{∗}/F^{∗}.

Proposition 1.6 The field E is generated over F by the α(σ).

Proof. Let E′ = F (…, α(σ), …). Then E′ ⊆ E, since each α(σ) ∈ E. Now let φ be an endomorphism of A that lies in the centralizer of E′; that is, φ α(σ) = α(σ) φ for all σ ∈ G_{ℚ}. But then ^{σ}φ = α(σ) φ α(σ)^{−1} = φ for all σ, which means that φ is defined over ℚ, and thus lies in E. Therefore the centralizer C(E′) of E′ is also contained in E. Now we use the Double Centralizer Theorem [Pi, Thm. 12.7] to count dimensions, and we find (dim_{F} E′)(dim_{F} (C(E′)) = dim_{F} End(A) = (dim_{F} E)^{2}. Thus we must have E′ = C(E′) = E, and so E is generated over F by the collection {α(σ)}. QED

Now recall that the Rosati involution induced by any polarization of A/ℚ is used

to define an involution e ↦ \overline{e} on E which is independent of the polarization. This “complex conjugation” fixes F —indeed, any totally real subfield of E.

Lemma 1.7 For all σ ∈ G_{ℚ}, we have \overline{α(σ)} α(σ) ∈ F.

Proof. Fix a polarization (over ℚ) of A and let φ ↦ φ′ designate the Rosati involution on End(A) induced by the polarization. Fix σ ∈ G_{ℚ}. Now for all φ ∈ End(A) we have ^{σ}φ = α(σ) φ α(σ)^{−1}. Taking the Rosati involution of both sides, we find ^{σ}φ′ = \overline{α(σ)}^{−1} φ′ \overline{α(σ)} for all φ′ ∈ End(A). Replacing φ′ with φ, we find ^{σ}φ = \overline{α(σ)}^{−1} φ \overline{α(σ)} for all φ ∈ End(A). So we see that \overline{α(σ)}^{−1} acts via conjugation on End(A) exactly as α(σ) does. Therefore \overline{α(σ)} α(σ) lies in the center of End(A), which is F. QED

Proposition 1.8 The field E is an abelian Galois extension of F.

Proof. Recall that E is generated over F by the α(σ), so it suffices to study the extensions F (α(σ)). First we write α(σ)^{2} in a more complicated way:

α(σ)^{2} = (α(σ)/\overline{α(σ)}) · (\overline{α(σ)} α(σ)) .

We have seen that \overline{α(σ)} α(σ) lies in F. Now we consider the first factor α(σ)/\overline{α(σ)}.

Note that by construction, the α(σ) satisfy α(σ) α(τ) α(στ)^{−1} ∈ F for all σ, τ ∈

G_{ℚ}. Therefore

α(σ) · α(τ) · α(στ)^{−1} = \overline{α(σ)} · \overline{α(τ)} · \overline{α(στ)}^{−1},

and rearranging terms, we find

(α(σ)/\overline{α(σ)}) · (α(τ)/\overline{α(τ)}) = α(στ)/\overline{α(στ)}  for all σ, τ ∈ G_{ℚ}.

That is, σ ↦ α(σ)/\overline{α(σ)} is a continuous character on G_{ℚ} with values in E^{∗}. In particular, α(σ)/\overline{α(σ)} is a root of unity.

Now we have expressed α(σ)^{2} as a product of a root of unity and an element of F.

Clearly then, α(σ) is contained in an abelian Galois extension of F obtained by adjoining \sqrt{\overline{α(σ)} α(σ)} and some number of roots of unity. Therefore F (α(σ)) is an abelian Galois extension, and this is true for any σ. Finally, since E is the compositum of the F (α(σ)), we see that the field E is also an abelian extension of F. QED

Some Cohomology

Let A be an abelian variety of GL_{2}-type. Let E = End_{ℚ}(A) and F = Z(End(A)).

As in the previous section, let α : G_{ℚ} → E^{∗} be a locally constant function such that σφ = α(σ) φ α(σ)^{−1} for all φ ∈ End(A). Recall that α is well-defined modulo F^{∗}.

We note that c(σ, τ) =^{def} α(σ)^{−1}α(τ)^{−1}α(στ) is a two-cocycle on G_{ℚ} with values in

F^{∗} (where G_{ℚ} acts trivially on F^{∗}). Moreover, a different choice for α leads to a cocycle which is cohomologous to c(σ, τ). Thus the abelian variety A of GL_{2}-type supplies us with a well-defined cohomology class [c] ∈ H^{2}(G_{ℚ}, F^{∗}).

Lemma 1.9 The class of c(σ, τ) has order dividing two in H^{2}(G_{ℚ}, F^{∗}).

Proof. Since c(σ, τ) lies in F^{∗}, we have c(σ, τ) = \overline{c(σ, τ)}. Hence we may write

c(σ, τ)^{2} = \overline{c(σ, τ)} c(σ, τ)
= (\overline{α(σ)} α(σ))^{−1} (\overline{α(τ)} α(τ))^{−1} (\overline{α(στ)} α(στ)) .

Thus we have expressed c(σ, τ)^{2} as a coboundary, because we have already shown that \overline{α(σ)} α(σ) ∈ F^{∗} for all σ ∈ G_{ℚ}. QED

Now consider the G_{ℚ}-module \overline{ℚ} ⊗_{ℚ} F, where σ ∈ G_{ℚ} acts via σ(r ⊗ f) = σ(r) ⊗ f.

The natural inclusion F ↪ \overline{ℚ} ⊗_{ℚ} F induces a map H^{2}(G_{ℚ}, F^{∗}) → H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) and this latter group is canonically isomorphic to the Brauer group Br(F). (The isomorphism will be discussed in detail in Chapter 2.)

The endomorphism algebra End(A) is a central simple F -algebra, and as such,

determines a Brauer class in Br(F). The class has order dividing two in Br(F), because the Schur index of End(A) is 1 or 2 [Pi, Thm. 18.6]. Furthermore, the field E splits End(A) because E is a maximal subfield of End(A) [Pi, Cor. 13.3], and E also splits c(σ, τ), by definition of the cocycle c. These observations motivate one to ask whether there is a connection between the class of End(A) in Br(F) and the class of c in H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}). This question is answered by the following proposition.

Proposition 1.10 The image of [c] in H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) corresponds (under the natural isomorphism) to the class of End(A) in Br(F).

Proof. [Ri 1, Thm. 5.6].

Alternatively, the results of Chapter 2 and Chapter 4 of this dissertation may be

combined to give a different proof of Proposition 1.10.

Chapter 2

The Brauer Class of End(B)

In Chapter 1, we saw how an abelian variety of GL_{2}-type decomposes over \overline{ℚ} into

a product B × … × B of building blocks B with certain properties. Recall our definitions:

Definition. An abelian variety B/\overline{ℚ} is a ℚ-abelian variety if for each σ ∈ Gal(\overline{ℚ}/ℚ), there exists an isogeny μ_{σ} : ^{σ}B → B compatible with the endomorphisms of B; that is, μ_{σ} ^{σ}φ = φ μ_{σ} for all φ ∈ End(B).

Definition. A building block is an abelian variety B/\overline{ℚ} such that

(i) the endomorphism algebra End(B) is a central division algebra over a totally real field F with Schur index t = 1 or 2, and t[F:ℚ] = dim B.

(ii) B is a ℚ-abelian variety.

Our ultimate goal is to show that every building block B appears as a simple

quotient of some abelian variety A/ℚ of GL_{2}-type. Towards that end, in this chapter we will study the class of the endomorphism algebra End(B) in the Brauer group Br(F).

Let G_{ℚ} be the absolute Galois group Gal(\overline{ℚ}/ℚ) with its usual profinite topology.

CHAPTER 2. THE BRAUER CLASS OF END(B) 13

A building block B/\overline{ℚ} may be constructed from a model over a finite extension K of ℚ; by abusing notation we will again call such a model B. Then ^{σ}B = ^{σ′}B if σ, σ′ ∈ G_{ℚ} restrict to the same map on K. Now because B is a ℚ-abelian variety, we may fix a collection of isogenies { μ_{σ} : ^{σ}B → B }_{σ∈G_{ℚ}} respecting End(B) such that μ_{σ} = μ_{σ′} whenever ^{σ}B = ^{σ′}B. Define c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1} for each pair σ, τ ∈ G_{ℚ}. Then c(σ, τ) is an element of End(B), as we may see from the diagram B ⟶^{μ_{στ}^{−1}} ^{στ}B ⟶^{^{σ}μ_{τ}} ^{σ}B ⟶^{μ_{σ}} B. Also, c(σ, τ) φ = φ c(σ, τ) for all φ ∈ End(B); this follows from the compatibility conditions on the μ_{σ}. Thus c(σ, τ) lies in the center of End(B), and may be thought of as an non-zero element of the field F. Therefore, we may view c as a locally constant two-cocycle on G_{ℚ} with values in F^{∗}, and we easily check that a different choice of isogenies ^{σ}B → B changes c by a coboundary. Thus to the abelian variety B we associate a well-defined cohomology class [c] ∈ H^{2}(G_{ℚ}, F^{∗}), where G_{ℚ} acts trivially on F^{∗}.

Next, consider the G_{ℚ}-module \overline{ℚ} ⊗_{ℚ} F, where the action of σ ∈ G_{ℚ} is defined

by σ(r ⊗ f) = σ(r) ⊗ f. The inclusion F ↪ \overline{ℚ} ⊗_{ℚ} F induces a map H^{2}(G_{ℚ}, F^{∗}) → H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}), and this latter group is canonically isomorphic to the Brauer group of F, as we will see.

Theorem 2.1 The image of [c] in H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) corresponds to the class of End(B) in the Brauer group Br(F).

Theorem 2.1 is reminiscent of Proposition 1.10, which referred to another coho-

mology class [c] ∈ H^{2}(G_{ℚ}, F^{∗}) attached to an abelian variety of GL_{2}-type. In Chapter 4 we will justify the use of this notation when we show that every building block B appears as a quotient of an abelian variety A over ℚ of GL_{2}-type in such a way that the definitions of [c] coincide. Since End(A) and End(B) determine the same class in Br(F) whenever A ∼ B × … × B, it is evident that Theorem 2.1 is a necessary condition for this assertion to be valid.

The proof of the Theorem 2.1 will be postponed until after we have reviewed the

construction of the isomorphism H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) ⟶^{∼} Br(F).

The Identification of the Brauer Group Br(F) with H^{2}(Gal(\overline{F} /F), \overline{F}^{∗})

Let G be a profinite group, and let X, X_{1}, X_{2}, X_{3} be discrete (multiplicative)

groups equipped with a left action of G. We recall the following definitions and facts from the theory of non-abelian cohomology. (See [Se 2, p. 131–134] for more details.)

• H^{0}(G, X) =^{def} { x ∈ X : σ(x) = x for all σ ∈ G }.

• A one-cocycle on G with values in X is a continuous map G → X sending σ ↦ x_{σ} such that x_{στ} = x_{σ} · σ(x_{τ}) for all σ, τ ∈ G.

• The one-cocycles x_{σ} and y_{σ} are cohomologous if there exists an x ∈ X such that y_{σ} = x^{−1} · x_{σ} · σ(x) for all σ ∈ G. A one-cocycle is a coboundary if it is cohomologous to the trivial cocycle σ ↦ 1.

• H^{1}(G, X) =^{def} {one-cocycles}/{coboundaries}.

• If 1 → X_{1} → X_{2} → X_{3} → 1 is a short exact sequence of non-abelian G-modules then there exists an exact sequence of pointed sets

1 → H^{0}(G, X_{1}) → H^{0}(G, X_{2}) → H^{0}(G, X_{3})
→ H^{1}(G, X_{1}) → H^{1}(G, X_{2}) → H^{1}(G, X_{3}).

Moreover, if X_{1} is contained in the center of X_{2}, the sequence may be extended by one more term, with a map to H^{2}(G, X_{1}). (Recall that the “kernel” of a morphism of pointed sets is the inverse image of the distinguished element.)

In particular, let G_{F} = Gal(\overline{F} /F), where F is a number field. Then

1 → \overline{F}^{∗} → GL_{n}(\overline{F}) → PGL_{n}(\overline{F}) → 1

is an exact sequence of G_{F} -modules, and the corresponding cohomology sequence is

⋯ → H^{1}(G_{F} , GL_{n}(\overline{F})) ⟶ H^{1}(G_{F} , PGL_{n}(\overline{F})) ⟶^{δ_{n}} H^{2}(G_{F} , \overline{F}^{∗}).

A generalization of Hilbert’s Theorem 90 states that H^{1}(G_{F} , GL_{n}(\overline{F})) is trivial [Se 2, Prop. X.3], so in fact the map δ_{n} is injective.

Now H^{1}(G_{F} , PGL_{n}(\overline{F})) classifies the forms of M_{n}(\overline{F}). That is, there is a canonical

bijection

H^{1}(G_{F} , PGL_{n}(\overline{F})) ⟶ Br_{n}(F), (2.2)

where Br_{n}(F) denotes the set of Brauer classes of F -algebras B such that \overline{F} ⊗_{F} B is isomorphic to M_{n}(\overline{F}) as an \overline{F} -algebra. This correspondence is described explicitly by associating to each one-cocycle σ ↦ P_{σ} ∈ PGL_{n}(\overline{F}) the Brauer class of the F -algebra

B =^{def} \{ X ∈ M_{n}(\overline{F}) | P_{σ} · σ(X) · P_{σ}^{−1} = X for all σ ∈ G_{F} \} .

Now because Br(F) is the union (over n) of the Br_{n}(F), we may define a map from Br(F) to H^{2}(G_{F} , \overline{F}^{∗}) by using the inverse of (2.2):

[figure: H^{1}(G_{F} , PGL_{n}(\overline{F})); H^{2}(G_{F} , \overline{F}^{∗}); Br(F); Br_{n}(F); δ_{n}; (2.2)]

This is the desired isomorphism. (See [Se 2, X.5] for the details.)

Remark. In the above definition of B we have written σ(X) when σ ∈ G_{F} and X ∈ M_{n}(\overline{F}). Here we are regarding M_{n}(\overline{F}) as the ring of \overline{F} -endomorphisms of the vector space \overline{F}^{n}, with its standard G_{F} -module structure. Then σ(X) is defined to be the endomorphism σ X σ^{−1} on \overline{F}^{n}. It is useful to note, however, that we may also describe σ(X) as the matrix obtained from X ∈ M_{n}(\overline{F}) by applying σ to each entry.

Co-Induced Modules

Let H be a closed subgroup of a profinite group G, and let M be a discrete H-

module (possibly non-abelian). The co-induced G-module Coind_{H}^{G} M is defined to be the set of continuous functions θ : G → M such that θ(ηx) = η(θ(x)) for all x ∈ G and η ∈ H. The group structure on Coind_{H}^{G} M is defined by pointwise multiplication, and the action of σ ∈ G is defined by (^{σ}θ)(x) = θ(xσ).

In the case M = \overline{ℚ}, which we consider below, it will be helpful to put a \overline{ℚ}-vector

space structure on Coind_{H}^{G} \overline{ℚ}. This is achieved by defining (rθ)(x) = x(r) · θ(x) for r ∈ \overline{ℚ}, θ ∈ Coind_{H}^{G} \overline{ℚ}, and x ∈ G_{ℚ}.

Now we return to the situation G = G_{ℚ} to study the G_{ℚ}-module \overline{ℚ}⊗_{ℚ} F. Consider

the finite set of field embeddings \{ι : F ↪ \overline{ℚ} \}. The group G_{ℚ} acts (on the left) on this set in the usual manner; that is to say, if ι is an embedding of F into \overline{ℚ} and σ is an element of G, then σ ι is another such embedding. We now fix an embedding ι_{0}, let H = Gal(\overline{ℚ}/ι_{0}(F)), and view \overline{ℚ} as an H-module.

Proposition 2.3 There is a \overline{ℚ}-linear isomorphism of G_{ℚ}-modules

\overline{ℚ} ⊗_{ℚ} F ⟶^{∼} Coind_{H}^{G} \overline{ℚ}, (2.3)

in which 1 ⊗ f is sent to the function θ_{f} : x ↦ ι_{0}(f).

Proof. Note that θ_{f} ∈ Coind_{H}^{G} \overline{ℚ} because H fixes ι_{0}(F) by definition. Also, one may check that^{σ}(rθ) = σ(r)^{σ}θ, and use this formula to verify that (2.3) is a G_{ℚ}-module homomorphism.

Now we want to show that this homomorphism is injective. Recall that we may

write \overline{ℚ} ⊗_{ℚ} F ⟶^{∼} ∏_{ι} \overline{ℚ} via the map r ⊗ f ↦ (r · ι(f))_{ι}. We use this isomorphism of \overline{ℚ}- algebras to factor (2.3) through the product; the element (r_{ι})_{ι} ∈ ∏_{ι} \overline{ℚ} is then mapped to the function x ↦ x(r_{x^{−1}ι_{0}}) in Coind_{H}^{G} \overline{ℚ}. This correspondence is clearly injective.

Finally we note that a function θ ∈ Coind_{H}^{G} \overline{ℚ} is determined by its values on any

set of right coset representatives of H in G_{ℚ}. Thus the \overline{ℚ}-dimension of Coind_{H}^{G} \overline{ℚ} must be [G_{ℚ}: H] = [F : ℚ], which is the same as the \overline{ℚ}-dimension of \overline{ℚ} ⊗_{ℚ} F. But (2.3) is already injective, hence it must actually be an isomorphism. QED

Proposition 2.4 We obtain the following isomorphisms in an analogous manner.

1.  (\overline{ℚ} ⊗_{ℚ} F)^{∗} ≅ Coind_{H}^{G} \overline{ℚ}^{∗}.
2.  M_{n}(\overline{ℚ} ⊗_{ℚ} F) ≅ Coind_{H}^{G} M_{n}(\overline{ℚ}).
3. GL_{n}(\overline{ℚ} ⊗_{ℚ} F) ≅ Coind_{H}^{G} GL_{n}(\overline{ℚ}).
4. PGL_{n}(\overline{ℚ} ⊗_{ℚ} F) ≅ Coind_{H}^{G} PGL_{n}(\overline{ℚ}).

Proof.

1. Follows directly from above because the G_{ℚ}-action on \overline{ℚ} ⊗_{ℚ} F respects the multiplicative structure as well as the additive structure.

2. We can see from the definition of “Coind_{H}^{G}” that M_{n}(Coind_{H}^{G} \overline{ℚ}) ≅ Coind_{H}^{G} M_{n}(\overline{ℚ}).

3. The action of G_{ℚ} still respects the multiplicative structure of the matrix algebra.

4. Follows from (1) and (3), and the fact that the functor Coind_{H}^{G} commutes with the

formation of quotient modules.

QED

Remark. By using the isomorphism \overline{ℚ} ⊗_{ℚ} F ⟶^{∼} ∏_{ι} \overline{ℚ} described in the proposition, we may similarly write any of the above G_{ℚ}-modules as a product over the set of embeddings {ι}. The action of σ ∈ G_{ℚ} on such a product sends the ι-place to the σ^{−1}ι-place via σ.

Cohomology of Co-Induced Modules

Let G be a profinite group, H a closed subgroup of G, M a discrete H-module, and

Coind_{H}^{G} M the associated co-induced G-module. In the usual case, where M is an abelian group, Shapiro’s Lemma states that the G-cohomology of Coind_{H}^{G} M may be identified with the H-cohomology of M. See for example [Br, Prop. III.6.2]. This fact is also well known for non-abelian modules [Se 1, §5.8b], but for the convenience of the reader we include an explicit proof.

Proposition 2.5 (“Non-Abelian Shapiro’s Lemma”) There is a natural bijection of pointed sets H^{1}(G, Coind_{H}^{G} M) → H^{1}(H, M).

Proof. First note that the projection π : Coind_{H}^{G} M → M defined by θ ↦ θ(1) is compatible with the inclusion H ⊂ G, in the sense that π(^{η}θ) = η(π(θ)) for η ∈ H and θ ∈ Coind_{H}^{G} M. Therefore there exists an induced map H^{1}(G, Coind_{H}^{G} M) → H^{1}(H, M), given by sending the one-cocycle σ ↦ θ_{σ} to the cocycle η ↦ θ_{η}(1).

To show that this map is one-to-one, suppose that η ↦ θ_{η}(1) is a coboundary,

i.e., that there is a fixed m ∈ M such that θ_{η}(1) = m^{−1} · η(m) for all η ∈ H. One defines ψ(x) = m·θ_{x}(1) for x ∈ G, and checks that ψ ∈ Coind_{H}^{G} M. Then θ_{σ} = ψ^{−1} ^{σ}ψ for all σ ∈ G, so θ_{σ} is also a coboundary. Thus we have an injective map H^{1}(G, Coind_{H}^{G} M) → H^{1}(H, M).

In order to show this map is surjective, let η ↦ m_{η} be a one-cocycle on H with

values in M. Fix a set of right coset representatives of H in G, with the representative for H being 1. Now for each x ∈ G, define h_{x} to be the element of H such that h_{x}^{−1}x is the chosen coset representative of Hx. Clearly for η ∈ H we have h_{η} = η, and more generally h_{ηx} = ηh_{x} for all x ∈ G. Define θ_{σ}(x) = m_{h_{x}}^{−1} · m_{h_{xσ}}. One verifies

• θ_{σ} ∈ Coind_{H}^{G} M,

• σ ↦ θ_{σ} is a one-cocycle on G, and

• θ_{η}(1) = m_{η} for η ∈ H

to finish the proof of surjectivity. QED

The Bijection between H^{1}(G_{ℚ}, PGL_{n}(\overline{ℚ} ⊗_{ℚ} F)) and Br_{n}(F)

Again we return to the specific case G = G_{ℚ} and H = Gal(\overline{ℚ}/F), where we have

identified F with ι_{0}(F) ⊂ \overline{ℚ} for some fixed embedding ι_{0}. Shapiro’s Lemma, together with our proof that (\overline{ℚ} ⊗_{ℚ} F)^{∗} ≅ Coind_{H}^{G} \overline{ℚ}^{∗}, has already given us an isomorphism of the group H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) with H^{2}(H, \overline{ℚ}^{∗}) ≅ Br(F). We will need a more precise description of the correspondence, however. We have the following commutative diagram for each n:

[figure: H^{1}(H, PGL_{n}(\overline{ℚ})); H^{2}(H, \overline{ℚ}^{∗}); Br(F); Br_{n}(F); δ_{n}; H^{1}(G_{ℚ}, PGL_{n}(\overline{ℚ} ⊗_{ℚ} F)); H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}); δ_{n}; (2.2); (2.5)]

Here the δ_{n} are the connecting homomorphisms from the cohomology sequences derived from the two short exact sequences:

1 → (\overline{ℚ} ⊗_{ℚ} F)^{∗} → GL_{n}(\overline{ℚ} ⊗_{ℚ} F) → PGL_{n}(\overline{ℚ} ⊗_{ℚ} F) → 1 (co-induced G_{ℚ}-modules) 1 → \overline{ℚ}^{∗} → GL_{n}(\overline{ℚ}) → PGL_{n}(\overline{ℚ}) → 1 (H-modules).

Note that Proposition 2.5 allows us to identify the two corresponding cohomology sequences. In particular, the map (2.5) is a bijection for which we may find an explicit formula. First write PGL_{n}(\overline{ℚ}⊗_{ℚ} F) as the product ∏_{ι} PGL_{n}(\overline{ℚ}). Let σ ↦ (P_{ι,σ})_{ι} represent a cohomology class in H^{1}(G_{ℚ}, PGL_{n}(\overline{ℚ} ⊗_{ℚ} F)). Then the image under (2.5) of this class is the class of η ↦ P_{ι_{0},η} in H^{1}(H, PGL_{n}(\overline{ℚ})).

Now the image of the class of this latter cocycle under bijection (2.2) is the Brauer

class of the algebra

B = \{ X ∈ M_{n}(\overline{ℚ}) | P_{ι_{0},η} · η(X) · P_{ι_{0},η}^{−1} = X for all η ∈ H \} .

But there is an alternate description of this algebra. Consider the parallel construction

B′ = \{ (X_{ι})_{ι} ∈ M_{n}(\overline{ℚ} ⊗_{ℚ} F) | (P_{ι,σ})_{ι} · σ(X_{ι})_{ι} · (P_{ι,σ})_{ι}^{−1} = (X_{ι})_{ι} for all σ ∈ G_{ℚ} \} . (2.6)

It is clear that B′ is also an F -algebra, and that there is an F -algebra homomorphism B′ → B given by (X_{ι})_{ι} ↦ X_{ι_{0}}. In fact this map is an isomorphism. Indeed, if we fix left coset representatives {g_{ι}} of H in G, then X ↦ (g_{ι}(X))_{ι} is the inverse isomorphism (assuming that (P_{ι,σ})_{ι} is constructed from P_{ι_{0},η} as in Proposition 2.5). The verification, however, is tedious and we leave it to the reader.

Now we see that B and B′ determine the same class in Br_{n}(F). Observe, however,

that the construction of B′ avoids the choice of the embedding ι_{0} : F ↪ \overline{ℚ}. Therefore, even though bijections (2.5) and (2.2) depend on ι_{0}, their composition is independent of this choice, and thus the isomorphism H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) ≅ Br(F) is completely canonical.

The Proof of Theorem 2.1

Let B be a building block, and let c(σ, τ) be the associated two-cocycle on G_{ℚ} with

values in F^{∗}. Now that we have established the isomorphism H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) ≅ Br(F), we aim to show that the image in H^{2}(G_{ℚ}, (\overline{ℚ}⊗_{ℚ} F)^{∗}) of the cohomology class [c] corresponds to the class of End(B) in Br(F). Note that the Brauer class of End(B) lies in Br_{t}(F) where t = 1 or 2, since by hypothesis, End(B) is a central division algebra over F with Schur index t = 1 or 2. Thus from the commutative diagram of the previous section we see it will suffice to find a one-cocycle with values in PGL_{t}(\overline{ℚ} ⊗_{ℚ} F) whose class maps to both the image of [c] in H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) and the Brauer class of End(B) in Br_{t}(F).

First, we consider the Lie algebra Lie B. Recall that the space of tangent vectors

Lie B is a \overline{ℚ}-vector space canonically attached to the abelian variety B, whose dimension over \overline{ℚ} is equal to the dimension of B. (See e.g. [Mu, §11].) Also, Lie B has the property that Lie^{σ}B ≅ ^{σ}(Lie B) =^{def} \overline{ℚ} ⊗_{\overline{ℚ}} Lie B, where the map from the base \overline{ℚ} to the base extension \overline{ℚ} is given by σ. Furthermore, the action of F on B induces an F -vector space structure on Lie B, and it is well known that since F is totally real, Lie B is free of rank t over \overline{ℚ} ⊗_{ℚ} F. (Here is where we use the condition that the dimension of B is exactly t[F:ℚ].)

Now each μ_{σ} : ^{σ}B → B induces an isomorphism of \overline{ℚ}-vector spaces λ_{σ} : Lie^{σ}B →

Lie B, and the relation μ_{σ} ^{σ}μ_{τ} = c(σ, τ) μ_{στ} implies that λ_{σ} ^{σ}λ_{τ} = c(σ, τ) λ_{στ} by functoriality. Here ^{σ}λ_{τ} is the map 1 ⊗ λ_{τ} : ^{σ}(Lie^{τ}B) → ^{σ}(Lie B). Fix a basis {e_{i}}_{i=1}^{t} for Lie B over \overline{ℚ} ⊗_{ℚ} F. Then for each σ ∈ G_{ℚ}, \{1 ⊗ e_{i}\}_{i=1}^{t} is a basis for the σ-twist ^{σ}(Lie B) ≅ Lie^{σ}B. We may use this collection of bases to express the maps λ_{σ} as matrices Λ_{σ} ∈ GL_{t}(\overline{ℚ} ⊗_{ℚ} F), and the identity λ_{σ} ^{σ}λ_{τ} = c(σ, τ) λ_{στ} translates into Λ_{σ} ·^{σ}Λ_{τ} = c(σ, τ) Λ_{στ}. At this point one easily verifies that the matrix ^{σ}Λ_{τ} describing the map ^{σ}λ_{τ} is the matrix Λ_{τ} with its entries twisted by σ, i.e., exactly σ(Λ_{τ}).

We are now provided with a one-cocycle with values in PGL_{t}(\overline{ℚ} ⊗_{ℚ} F), namely

σ ↦ Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗}. Proposition 2.7 The class of Λ_{σ} mod (\overline{ℚ}⊗_{ℚ} F)^{∗} in H^{1}(G_{ℚ}, PGL_{t}(\overline{ℚ}⊗_{ℚ} F)) is independent of the choice of basis for Lie B.

Proof. A different choice of basis for Lie B is described by a change-of-basis matrix P ∈ GL_{t}(\overline{ℚ} ⊗_{ℚ} F). The corresponding change of basis on Lie^{σ}B ≅ ^{σ}(Lie B) will be given by ^{σ}P. Thus the new matrix for the map λ_{σ} : Lie^{σ}B → Lie B will be P · Λ_{σ} ·^{σ}P^{−1}, and the cocycle σ ↦ P · Λ_{σ} ·^{σ}P^{−1} is cohomologous to σ ↦ Λ_{σ}. QED

It remains to show that the class of the cocycle σ ↦ Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗} corresponds to both the image of [c] and the Brauer class of End(B), as desired.

Proposition 2.8 The class of Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗} in H^{1}(G_{ℚ}, PGL_{t}(\overline{ℚ} ⊗_{ℚ} F)) maps to the image of [c] in H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}).

Proof. The connecting homomorphism which supplies the inclusion

H^{1}(G_{ℚ}, PGL_{t}(\overline{ℚ} ⊗_{ℚ} F)) ↪ H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗})

is defined by the following procedure. First, choose a one-cocycle to represent the given cohomology class. In our case we take Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗}. Then lift the chosen cocycle to a map G_{ℚ} → GL_{t}(\overline{ℚ} ⊗_{ℚ} F); we choose σ ↦ Λ_{σ} for our application. Then Λ_{σ} · σ(Λ_{τ}) · Λ_{στ}^{−1} is a two-cocycle with values in (\overline{ℚ}⊗_{ℚ} F)^{∗}, and the class of this two-cocycle in H^{2}(G_{ℚ}, (\overline{ℚ}⊗_{ℚ} F)^{∗}) is independent of all the choices we made. In our situation, Λ_{σ} · σ(Λ_{τ}) · Λ_{στ}^{−1} = c(σ, τ), which is exactly what we wished to show. QED

Proposition 2.9 The class of Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗} in H^{1}(G_{ℚ}, \overline{ℚ} ⊗_{ℚ} F) corresponds to the class of End(B) in Br_{t}(F).

Proof. By applying Equation 2.6, we find the Brauer class of F -algebras corresponding to the cocycle Λ_{σ} mod (\overline{ℚ} ⊗_{ℚ} F)^{∗} is represented by the algebra

B =^{def} \{ X ∈ M_{t}(\overline{ℚ} ⊗_{ℚ} F) | Λ_{σ} · σ(X) · Λ_{σ}^{−1} = X for all σ ∈ G_{ℚ} \} .

On the other hand, by the functoriality of “Lie”, there is a ring homomorphism

End(B) → End(Lie B). In fact this map is an injection, because End(B) is a division algebra. Then by using our fixed basis of Lie B, we have

End(B) ↪ End(Lie B) ≅ M_{t}(\overline{ℚ} ⊗_{ℚ} F).

We claim that under these maps, the image of End(B) lands in B. Indeed, for

φ ∈ End(B), let Φ ∈ M_{t}(\overline{ℚ} ⊗_{ℚ} F) be the corresponding matrix. Recall that the original μ_{σ} satisfied φ μ_{σ} = μ_{σ} ^{σ}φ. Hence by functoriality again, we have Φ · Λ_{σ} = Λ_{σ} · σ(Φ) for all σ. Thus Λ_{σ} · σ(Φ) · Λ_{σ}^{−1} = Φ for all σ ∈ G_{ℚ}, which means Φ ∈ B.

Finally we see that both End(B) and B are t^{2}-dimensional over F, so the inclusion

must in fact be an isomorphism End(B) ⟶^{∼} B. QED

This proves Theorem 2.1.

Chapter 3

The Tensor Product of a Module

with an Abelian Variety

Here we prepare some “abstract nonsense” we will need for our construction of the

correspondence between building blocks and abelian varieties of GL_{2}-type. The results in this chapter, however, are valid for arbitrary abelian varieties.

Let A be an abelian variety up to isogeny, let R be a ring with a map R → End(A),

and let M be a finitely generated right R-module. By abusing notation, we will identify R with its image in End(A), and say that A is an abelian variety with an action of R.

Theorem 3.1 Let ℭ be the category of abelian varieties up to isogeny, and let 𝔄 be the category of abelian groups. Let T : ℭ → 𝔄 be the covariant functor defined by T (X) = Hom_{R}(M, Hom(A, X)). Then T is representable.

Proof. [Mit, Thm. VI.3.1].

We will denote the object representing the functor T as M ⊗_{R} A, and devote this chapter to summarizing some properties of this “tensor product”. First, we may reformulate the above theorem as an assertion of the existence of an abelian variety M ⊗_{R} A characterized by the following universal property:

M ⊗_{R} A is an abelian variety together with a homomorphism of right R-modules Δ : M → Hom(A, M ⊗_{R} A), which, for an abelian variety X, induces an isomorphism of abelian groups Hom(M ⊗_{R} A, X) → Hom_{R}(M, Hom(A, X)) via ψ ↦ (m ↦ ψ Δ_{m}).

(3.2)

Next, standard procedures produce the following facts. In the list below, it is

understood that each tensor product is equipped with the evident homomorphism Δ, and that the symbol “ ≅ ” designates a natural equivalence that respects these homomorphisms.

Elementary Properties of M ⊗_{R} A:

(i) If A is an abelian variety with an action of R, then R ⊗_{R} A ≅ A. (ii) If M is a right R-module, N is an R-S-bimodule, and A is an abelian variety with an

action of S, then M ⊗_{R} (N ⊗_{S} A) ≅ (M ⊗_{R} N) ⊗_{S} A. (iii) If {M_{i}} is a finite collection of right R-modules, and A is an abelian variety with an

action of R, then (⊕M_{i}) ⊗_{R} A ≅ ∏(M_{i} ⊗_{R} A). (iv) If M is a right R-module and {A_{i}} is a finite collection of abelian varieties, each with

a map R → End(A_{i}), then M ⊗_{R} ∏ A_{i} ≅ ∏(M ⊗_{R} A_{i}).

(v) M ⊗_{R} A is functorial in each variable separately. More explicitly, let M_{1} and M_{2} be right R-modules and f ∈ Hom_{R}(M_{1}, M_{2}). Let A_{1} and A_{2} be abelian varieties with actions of R and suppose φ : A_{1} → A_{2} is a morphism of abelian varieties that respects R; i.e., r φ = φ r for all r ∈ R. We write φ ∈ Hom^{R}(A_{1}, A_{2}). Then m ↦ Δ_{f(m)} φ is a homomorphism of right R-modules M_{1} → Hom(A_{1}, M_{2} ⊗_{R} A_{2}). Hence by (3.2), there exists a unique morphism f ⊗ φ ∈ Hom(M_{1} ⊗_{R} A_{1}, M_{2} ⊗_{R} A_{2}) such that (f ⊗ φ) Δ_{m} = Δ_{f(m)} φ for all m ∈ M. Moreover, it is easy to check that if g : M_{2} → M_{3} is another R-module homomorphism and ψ : A_{2} → A_{3} another morphism of abelian varieties respecting R, then (g ⊗ ψ) (f ⊗ φ) = (g f) ⊗ (ψ φ).

If we apply property (v) in the particular case A_{1} = A_{2} = A and M_{1} = M_{2} = M,

we have ring homomorphisms

\begin{matrix}End_{R} M & → & End(M ⊗_{R} A) \\ f & ↦ & f ⊗ 1 \end{matrix}  \begin{matrix}End^{R}(A) & → & End(M ⊗_{R} A) \\ φ & ↦ & 1 ⊗ φ \end{matrix}

whose images commute. Note that the center Z(R) ⊆ End^{R}(A), and that we may also map Z(R) → End_{R} M via r ↦ ρ_{r} = right multiplication by r on M. (The condition r ∈ Z(R) makes ρ_{r} a homomorphism of right R-modules.)

Proposition 3.3 The two maps Z(R) → End(M ⊗_{R} A) coincide. Specifically, 1 ⊗ r = ρ_{r} ⊗ 1 ∈ End(M ⊗_{R} A).

Proof. By (3.2), it suffices to show that (ρ_{r}⊗1) Δ_{m} = (1⊗r) Δ_{m} for all m ∈ M. But this plainly follows from the definitions: (ρ_{r} ⊗ 1) Δ_{m} = Δ_{ρ_{r}(m)} = Δ_{mr} = Δ_{m} r = (1 ⊗ r) Δ_{m}.

QED

Thus we obtain a homomorphism of Z(R)-algebras

(3.4) End_{R} M ⊗_{Z(R)} End^{R}(A) ⟶ End^{Z(R)}(M ⊗_{R} A).

Now suppose M is an S-R-bimodule. Then we may map S → End_{R} M via s ↦ λ_{s}

= left multiplication by s on M. By composing this map with End_{R} M → End(M ⊗_{R} A) from above, we find that we have an action of S on M ⊗_{R} A. Again, we will usually identify S with its image in End(M ⊗_{R}A). We now reconsider the defining property (3.2) of M ⊗_{R}A.

Proposition 3.5 Let M be an S-R-bimodule, and let A be an abelian variety with an action of R. Suppose X is an abelian variety with an action of S. Then Δ induces an isomorphism Hom^{S}(M ⊗_{R} A, X) → Hom_{S-R}(M, Hom(A, X)).

Proof. It is clear that if ψ : M ⊗_{R} A → X respects S, then m ↦ ψ Δ_{m} is a homomorphism of S-R-bimodules. On the other hand, suppose m ↦ φ_{m} is an element of Hom_{S-R}(M, Hom(A, X)). Then there exists a morphism of abelian varieties φ̃ ∈ Hom(M ⊗_{R} A, X) such that φ_{m} = φ̃ Δ_{m} for all m ∈ M. We have to show that φ̃ respects S, i.e., that s φ̃ = φ̃ s for all s ∈ S. As usual, by (3.2) it suffices to show that s φ̃ Δ_{m} = φ̃ s Δ_{m} for all s ∈ S and m ∈ M, and this follows directly from the relevant definitions: s φ̃ Δ_{m} = s φ_{m} = φ_{sm} = φ̃ Δ_{sm} = φ̃ s Δ_{m}. QED

Finally, if M = S is a ring, then it is an S-R-bimodule and the above results

hold. Also, we have a ring homomorphism R → S via r ↦ 1·r, so an abelian variety X with an action of S also has an action of R. With the extra structure in this case, we find an isomorphism Hom_{S-R}(S, Hom(A, X)) ⟶^{∼} Hom^{R}(A, X), by taking the element s ↦ φ_{s} of Hom_{S-R}(S, Hom(A, X)) to φ_{1}. The reason that φ_{1} : A → X respects R is because R commutes with 1 ∈ S. Thus we may now rephrase the defining property of S ⊗_{R} A in the category of abelian varieties up to isogeny:

Let A be an abelian variety with an action of R, and let S be a ring which is also a right R-module. Then S⊗_{R}A is an abelian variety with an action of S, together with a morphism Δ_{1} : A → S ⊗_{R} A respecting R, which satisfies the following universal property: given an abelian variety X with an action of S, and a morphism φ : A → X respecting R, there exists a unique morphism φ̃ : S ⊗_{R} A → X respecting S such that φ = φ̃ Δ_{1}.

Galois Action

Now suppose that A is defined over \overline{ℚ}; let us consider the effect of twisting by

σ ∈ Gal(\overline{ℚ}/ℚ) = G_{ℚ} on the tensor product construction. Note that if A is an abelian variety with an action of R, then so is ^{σ}A, for each σ ∈ G_{ℚ}; we may map R → End(A) → End(^{σ}A) via the usual “twist-by-σ”.

Lemma 3.6 Let A, A_{1}, A_{2} be abelian varieties over \overline{ℚ} with actions of R, and let M, M_{1}, M_{2} be right R-modules.

1. For each σ ∈ G_{ℚ}, we have ^{σ}(M ⊗_{R} A) ∼ M ⊗_{R} ^{σ}A.

2. Let f ∈ Hom_{R}(M_{1}, M_{2}), let φ ∈ Hom^{R}(A_{1}, A_{2}), and let f ⊗ φ be the induced map M_{1} ⊗_{R} A_{1} → M_{2} ⊗_{R} A_{2} defined by property (v) above. Then for each σ ∈ G_{ℚ}, we have σ(f ⊗ φ) = f ⊗^{σ}φ.

Proof.

1. Fix σ ∈ G_{ℚ}. We will demonstrate that ^{σ}(M ⊗_{R} A) satisfies the defining property (3.2) of M ⊗_{R} ^{σ}A. First construct M ⊗_{R} A and Δ : M → Hom(A, M ⊗_{R} A). Then for an abelian variety X/\overline{ℚ}, the map Δ induces an isomorphism Hom(M ⊗_{R} A, X) → Hom_{R}(M, Hom(A, X)). Now if one defines ^{σ}Δ : M → Hom_{R}(^{σ}A, ^{σ}(M ⊗_{R} A)) as (^{σ}Δ)_{m} =^{def} ^{σ}(Δ_{m}), then ^{σ}Δ induces an isomorphism Hom(^{σ}(M ⊗_{R} A), ^{σ}X) → Hom_{R}(M, Hom(^{σ}A,^{σ}X)). But of course any abelian variety may be written as ^{σ}X for some X, so ^{σ}(M ⊗_{R} A) and ^{σ}Δ do satisfy the characteristic property of M ⊗_{R} ^{σ}A.

2. The map f ⊗ φ is defined to be the unique morphism M_{1} ⊗_{R} A_{1} → M_{2} ⊗_{R} A_{2} such that (f ⊗ φ) Δ_{m} = Δ_{f(m)} φ for all m ∈ M_{1}. Applying σ to both sides, we find σ(f ⊗ φ) ^{σ}Δ_{m} = ^{σ}Δ_{f(m)} ^{σ}φ = (f ⊗ ^{σ}φ) ^{σ}Δ_{m} for all m ∈ M_{1}, which implies that σ(f ⊗ φ) = f ⊗^{σ}φ. QED

Chapter 4

Descent from \overline{ℚ} to ℚ

In this chapter we will show how to construct, from a given building block B, an

abelian variety A/ℚ of GL_{2}-type with B as a simple quotient over \overline{ℚ}. We will then describe a precise correspondence between these two types of abelian varieties. Recall the following definitions.

Definition. An abelian variety B/\overline{ℚ} is a ℚ-abelian variety if for each σ ∈ Gal(\overline{ℚ}/ℚ), there exists an isogeny μ_{σ} : ^{σ}B → B compatible with the endomorphisms of B; that is, μ_{σ} ^{σ}φ = φ μ_{σ} for all φ ∈ End(B).

Definition. A building block is an abelian variety B/\overline{ℚ} such that

(i) the endomorphism algebra End(B) is a central division algebra over a totally real field F with Schur index t = 1 or 2, and t[F:ℚ] = dim B.

(ii) B is a ℚ-abelian variety.

Definition. An abelian variety A/ℚ is of GL_{2}-type if End_{ℚ}(A), the algebra of endomorphisms of A defined over ℚ, is a number field of degree equal to the dimension of A.

Construction of an Abelian Variety of GL_{2}-type

We write G_{ℚ} for the absolute Galois group Gal(\overline{ℚ}/ℚ) with its usual profinite

CHAPTER 4. DESCENT FROM \overline{ℚ} TO ℚ 26

topology. Let B be a building block. Then, by definition, B is a ℚ-abelian variety. We have seen in Chapter 2 how to fix for each σ ∈ G_{ℚ} a choice of μ_{σ} : ^{σ}B → B respecting End(B) such that c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1} is a locally constant two-cocycle on G_{ℚ} with values in F^{∗} (a G_{ℚ}-module with trivial action). The class [c] ∈ H^{2}(G_{ℚ}, F^{∗}) is independent of our choices for μ_{σ}.

Now consider the cohomology group H^{2}(G_{ℚ}, \overline{F}^{∗}), where G_{ℚ} acts trivially on \overline{F}^{∗}.

By a theorem of Tate [Se 3, Thm. 4], this group is trivial. Therefore the image of [c] in H^{2}(G_{ℚ}, \overline{F}^{∗}) is a coboundary; that is, there exists a locally constant function β : G_{ℚ} → \overline{F}^{∗} such that c(σ, τ) = β(σ) · β(τ) · β(στ)^{−1} for all σ, τ ∈ G_{ℚ}. Fix such a function β. Let E be the field extension of F obtained by adjoining all the values β(σ). Then E is a finite extension of F (as β is locally constant); set d = [E:F].

Consider the abelian variety E ⊗_{F} B. From Equation 3.4 there is a map

(4.1) End_{F} E ⊗_{F} End(B) → End(E ⊗_{F} B).

We claim this map is an isomorphism of F -algebras. Indeed, fix a basis of E over F. Then we see that End_{F} E is isomorphic to the matrix algebra M_{d}(F), and so the image of End_{F} E ⊗_{F} End(B) in End(E ⊗_{F} B) is isomorphic to M_{d}(End(B)). On the other hand, the same choice of basis for E/F gives us E ⊗_{F} B ∼ (⨁^{d} F)⊗_{F} B ∼ ∏^{d} B, so the endomorphism algebra End(E ⊗_{F} B) must itself be isomorphic to M_{d}(End(B)). From now on we will frequently identify End(E ⊗_{F} B) with End_{F} E ⊗_{F} End(B).

Now we claim that we may descend the abelian variety E ⊗_{F} B to ℚ.

For each σ ∈ G_{ℚ}, define ν_{σ} =^{def} β(σ)^{−1} ⊗ μ_{σ} ∈ Hom(E ⊗_{F} ^{σ}B, E ⊗_{F} B), using our

fixed β(σ) and μ_{σ}. Recall that by Lemma 3.6, we have E ⊗_{F} ^{σ}B ∼ ^{σ}(E ⊗_{F} B); therefore ν_{σ} may be viewed as an isogeny ^{σ}(E ⊗_{F} B) → E ⊗_{F} B.

Proposition 4.2 There exists an abelian variety A_{0} over ℚ and an isogeny (over \overline{ℚ}) κ : E ⊗_{F} B → A_{0} such that κ^{−1} ^{σ}κ = ν_{σ} for all σ ∈ G_{ℚ}.

Proof. First, we see that ν_{σ} ^{σ}ν_{τ} = ν_{στ} for all σ, τ ∈ G_{ℚ}. Indeed, by using the definitions of c(σ, τ) and β(σ), and the properties of the tensor product from Chapter 3, we have

ν_{σ} ^{σ}ν_{τ} = β(σ)^{−1}β(τ)^{−1} ⊗ μ_{σ} ^{σ}μ_{τ}
= β(στ)^{−1}c(σ, τ)^{−1} ⊗ c(σ, τ)μ_{στ}
= β(στ)^{−1} ⊗ μ_{στ}
= ν_{στ}.

Therefore, we may apply the method of [Ri 1, Thm. 8.2] to construct A_{0}. More

explicitly, let K be a finite Galois extension of ℚ such that E ⊗_{F} B is defined over K and all ν_{σ} are defined over K. Let X be the abelian variety Res_{K/ℚ}(E ⊗_{F} B), where “Res” denotes Weil’s “restriction of scalars” functor. (See [We, §1.3].) Then X is defined over ℚ and X_{/K} ≅ ∏_{g} ^{g}(E ⊗_{F} B), where the product runs over g ∈ Gal(K/ℚ). We now define κ = (ν_{g}^{−1})_{g} : E ⊗_{F} B → X_{/K}. Lastly, write A_{0} for the image of κ (or technically, the image of a suitable multiple of κ).

A direct (if tiresome) computation yields the formula κ^{−1} ^{σ}κ = ν_{σ}. QED

Next we ask which endomorphisms of A_{0} are defined over ℚ? Or more generally,

how does σ ∈ G_{ℚ} act on End(A_{0})?

Proposition 4.3 For each σ ∈ G_{ℚ}, the automorphism φ ↦ ^{σ}φ of End(A_{0}) is equal to conjugation by β(σ)^{−1} ∈ E. (Here the action of E on A_{0} is induced by κ.)

Proof. The isogeny κ induces an isomorphism End(A_{0}) ⟶^{∼} End(E ⊗_{F} B) via φ ↦ κ^{−1} φ κ. Thus, the automorphism φ ↦ ^{σ}φ on End(A_{0}) translates into the automorphism κ^{−1} φ κ ↦ κ^{−1} ^{σ}φ κ on End(E ⊗_{F} B). Now we may write

κ^{−1} ^{σ}φ κ = κ^{−1} (^{σ}κ ^{σ}κ^{−1}) ^{σ}φ (^{σ}κ ^{σ}κ^{−1}) κ
= ν_{σ} ^{σ}(κ^{−1} φ κ) ν_{σ}^{−1}  because κ^{−1} ^{σ}κ = ν_{σ};
= (β(σ)^{−1} ⊗ 1) (1 ⊗ μ_{σ}) ^{σ}(κ^{−1} φ κ) (1 ⊗ μ_{σ}^{−1}) (β(σ) ⊗ 1) by def. of ν_{σ};
= (β(σ)^{−1} ⊗ 1) (κ^{−1} φ κ) (β(σ) ⊗ 1).

The last equality follows because for all ψ ∈ End(E⊗_{F} B), we have (1⊗μ_{σ}) ^{σ}ψ = ψ (1⊗μ_{σ}). Indeed, write ψ as an element of the tensor product End_{F} E ⊗_{F} End(B) and use the fact that each μ_{σ} was originally chosen to be compatible with End(B).

Finally, translating back to End(A_{0}), we see ^{σ}φ = β(σ)^{−1} φ β(σ). QED

Proposition 4.4 Under the isomorphism End(A_{0}) ⟶^{∼} End_{F} E ⊗_{F} End(B) induced by the isogeny κ, we have End_{ℚ}(A_{0}) ≅ E ⊗_{F} End(B).

Proof. By applying the above proposition we find that the endomorphisms of A_{0} which are fixed by all σ ∈ G_{ℚ} are exactly those which commute with all β(σ). Thus End_{ℚ}(A_{0}) is an F -subalgebra of End(A_{0}); in fact, End_{ℚ}(A_{0}) is the centralizer C(E) of E in End(A_{0}), since E was defined to be the field generated over F by the β(σ).

Now it is clear that C(E) is contained in E ⊗_{F} End(B), so it suffices to show

that these subalgebras have the same dimension over F. Recall that we have written d = [E:F] and t = 1 or 2 = Schur index of End(B). Then dim_{F} E · dim_{F} (E ⊗_{F} End(B)) = d(dt^{2}) = dim_{F} End(A_{0}). Therefore, by the Double Centralizer Theorem [Pi, Thm. 12.7], dim_{F} C(E) = dim_{F} (E ⊗_{F} End(B)). QED

Proposition 4.5 There exists an abelian variety A/ℚ of GL_{2}-type containing B as a simple component.

Proof. We have seen by Theorem 2.1 that if we send the class of the cocycle c(σ, τ),

[c] ∈ H^{2}(G_{ℚ}, F^{∗}) → H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}) ⟶^{∼} Br(F) via the natural maps, then its image in the Brauer group is the class of End(B). Now consider what happens upon extending to the field E. There is a commutative diagram:

[figure: H^{2}(G_{ℚ}, F^{∗}); H^{2}(G_{ℚ}, E^{∗}); H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} F)^{∗}); H^{2}(G_{ℚ}, (\overline{ℚ} ⊗_{ℚ} E)^{∗}); Br(F); Br(E); ≅; ≅]

On the one hand, [c] ∈ H^{2}(G_{ℚ}, F^{∗}) maps to the class of End(B) in Br(F), which

maps to the class of E ⊗_{F} End(B) in Br(E). But on the other hand, c splits when taking values in E; in fact, this is how E was originally defined. In other words, the image of [c] is trivial in H^{2}(G_{ℚ}, E^{∗}), and hence maps to the trivial class in Br(E). Therefore E ⊗_{F} End(B) must be isomorphic to the matrix algebra M_{t}(E), where t = 1 or 2 is the Schur index of End(B).

Fix an isomorphism E ⊗_{F} End(B) ⟶^{∼} M_{t}(E). Then by the previous proposition,

we have End_{ℚ}(A_{0}) ⟶^{∼} M_{t}(E). This isomorphism implies that A_{0} must be isogenous over ℚ to A^{t}, where A is an abelian variety over ℚ with End_{ℚ}(A) = E. (For instance, we may define A to be the image of the endomorphism of A_{0} corresponding to the matrix that has a 1 in the upper left corner and zeroes elsewhere.)

Now remember that E ⊗_{F} B is isogenous to a product of d copies of B, where

d = [E : F]. It follows that A_{/\overline{ℚ}}^{t} ∼ B^{d}, and therefore A_{/\overline{ℚ}} ∼ B^{d/t}. Thus B is a simple component of A. We have already seen that End_{ℚ}(A) = E, so to complete the proof we need only calculate the dimension of A in order to show that A is of GL_{2}-type. Recall that because B is a building block, we have dim B = t[F : ℚ]. Thus the dimension of A is \frac{d}{t} dim B = d[F:ℚ] = [E:ℚ], and we see that A is of GL_{2}-type, as desired. QED

A More Precise Correspondence

Notice that it was necessary to make several choices in our construction of an

abelian variety A/ℚ of GL_{2}-type from a building block B. We now ask if there is any way to make the correspondence between these two types of abelian varieties more precise. If we start with an A/ℚ of GL_{2}-type and decompose it into building blocks, what information must we record in order to recover the “same” A/ℚ via the above procedure? The main idea is to keep track of the “descent data”—the ν_{σ}—which provide the isogeny from B × … × B (defined over \overline{ℚ}) to A (defined over ℚ).

First, however, we must say what it means for two abelian varieties of GL_{2}-type

to be the “same”.

Definition. A GL_{2}-type pair is a pair (A, i) where A is an abelian variety over ℚ of GL_{2}-type and i is an injection End_{ℚ}(A) ↪ \overline{ℚ}.

Definition. Two GL_{2}-type pairs (A, i) and (A′, i′) are equivalent if there exists an isogeny a : A → A′ that is defined over ℚ and satisfies i′(a φ a^{−1}) = i(φ) for all φ ∈ End_{ℚ}(A).

We make similar definitions for building blocks.

Definition. A building block triple is a triple (B, j, {ν_{σ}}_{σ∈G_{ℚ}}) where • B is a building block, and we write B = End(B);

• j is an injection of the center Z(End(B)) ↪ \overline{ℚ}, and we write F for the image of j;

• for each σ ∈ G_{ℚ}, ν_{σ} is a non-zero element of \overline{ℚ} ⊗_{F} Hom^{B}(^{σ}B, B) such that the family {ν_{σ}} is locally constant and satisfies ν_{σ} ^{σ}ν_{τ} = ν_{στ} for all σ, τ ∈ G_{ℚ}.

Definition. Two building block triples (B, j, {ν_{σ}}) and (B′, j′, {ν′_{σ}}) are equivalent if there exists an isogeny b : B → B′ such that j′(b φ b^{−1}) = j(φ) for all φ ∈ Z(End(B)), and ν′_{σ} = b ν_{σ} ^{σ}b^{−1} for all σ ∈ G_{ℚ}.

Suppose we are given a building block triple (B, j, {ν_{σ}}). Note that for each σ,

Hom^{B}(^{σ}B, B) is a one-dimensional F -vector space; the choice of an isogeny μ_{σ} : ^{σ}B → B respecting B then determines β(σ) ∈ \overline{ℚ}^{∗} such that ν_{σ} may be written as β(σ)^{−1} ⊗ μ_{σ}. Let E ⊂ \overline{ℚ} be the field that is generated over F by the values β(σ); then because each β(σ) is well-defined modulo F^{∗}, we see that E depends only on the ν_{σ}, and not on our choices for μ_{σ}. The field E is a finite extension of F because the collection {ν_{σ}} is locally constant. Now we may view each ν_{σ} as an element of Hom(E ⊗_{F} ^{σ}B, E ⊗_{F} B) and construct an abelian variety of GL_{2}-type using the procedure described in the previous section.

We claim there is the following relationship between building block triples and

GL_{2}-type pairs.

Theorem 4.6 The above construction, which associates a GL_{2}-type pair (A, i) to a building block triple (B, j, {ν_{σ}}), induces a bijection on equivalence classes.

We break the proof into several steps.

Step 1 The correspondence (B, j, {ν_{σ}}) ⇝ (A, i) is well-defined on equivalence classes.

Suppose the triple (B, j, {ν_{σ}}) is equivalent to (B′, j′, {ν′_{σ}}). Fix an isogeny b :

B → B′ satisfying the stated conditions for equivalence. Our first observation is that the image of j′ is equal to the image of j; we will call their common image F and identify Z(B) and Z(B′) with F. Also, the isogeny b induces an isomorphism End(B) ≅ End(B′), and in particular, we note that End(B) and End(B′) have the same Schur index t = 1 or 2 as central simple F -algebras.

Next, for each σ ∈ G_{ℚ}, pick μ_{σ} ∈ Hom^{B}(^{σ}B, B) and write ν_{σ} = β(σ)^{−1} ⊗ μ_{σ}.

Then the fact that ν′_{σ} = b ν_{σ} ^{σ}b^{−1} implies that we may write ν′_{σ} = β(σ)^{−1} ⊗ (b μ_{σ} ^{σ}b^{−1}). Thus the subfield of \overline{ℚ} generated over F by the coefficients of the ν′_{σ} is the same as the one generated by the coefficients of the ν_{σ}. Call this field E. Now our construction yields isogenies κ : E ⊗_{F} B → A^{t} and κ′ : E ⊗_{F} B′ → A′^{t} where A and A′ are abelian varieties over ℚ of GL_{2}-type with E ⟶^{∼} End_{ℚ}(A) and E ⟶^{∼} End_{ℚ}(A′). These actions of E on A and A′ define the pairs (A, i) and (A′, i′), respectively.

Now a = κ′ (1 ⊗ b) κ^{−1} is an isogeny A^{t} → A′^{t}. We claim it is an isogeny defined

over ℚ. Indeed, for any σ ∈ G_{ℚ} we have:

σ(κ′ (1 ⊗ b) κ^{−1}) = (κ′ κ′^{−1}) ^{σ}κ′ (1 ⊗^{σ}b) ^{σ}κ^{−1} (κ κ^{−1})
= κ′ ν′_{σ} (1 ⊗^{σ}b) ν_{σ}^{−1} κ^{−1}
= κ′ (1 ⊗ b) ν_{σ} ν_{σ}^{−1} κ^{−1}
= κ′ (1 ⊗ b) κ^{−1}.

Moreover, κ, 1 ⊗ b and κ′ all respect the action of E ⊗_{F} B. (Here we are using

the isomorphism B′ ≅ B induced by b to define the action of E ⊗_{F} B on E ⊗_{F} B′ and A′^{t}.)

Therefore, a respects the action of E ⊗_{F} B ≅ M_{t}(E). But it is easy to see that the natural map Hom^{E}(A, A′) → Hom^{M_{t}(E)}(A^{t}, A′^{t}) is an isomorphism, so in fact we may regard a as a morphism A → A′. Then a is an isogeny defined over ℚ that respects E; that is, a defines an equivalence of pairs (A, i) and (A′, i′), as desired.

Step 2 Define the inverse correspondence from pairs (A, i) to triples (B, j, {ν_{σ}}).

Let (A, i) be a GL_{2}-type pair. Write a for the full endomorphism algebra End(A),

E for the image of i, and F = i(Z(a )). By our usual abuse of notation, we will use i to identify E and F with the corresponding subalgebras of a . Recall that a is a central simple algebra over F, by Theorem 1.3. Thus a has a unique (up to isomorphism) simple right a -module. Fix such a simple module M.

Let B = M ⊗a A. We claim that B is a building block. By Propositions 1.4

and 1.5, it suffices to show that B is a simple component of A. Now because a is a simple algebra, for some n we have a ≅ ⨁^{n} M as right a -modules. Fix such an isomorphism. Then

A = a ⊗a A ∼ (⨁^{n} M) ⊗a A ∼ ∏^{n}(M ⊗a A) ∼ ∏^{n} B.

Thus B is a subvariety of A. It remains to show that B is simple.

By [Pi, Cor. 3.5a], a is isomorphic to M_{n}(B), where B =^{def} Enda M is a central

division algebra over F. But the above isogeny A ∼ ∏^{n} B implies that a ≅ M_{n}(End(B)). Therefore we must have that End(B) ≅ B is a division algebra, and so B is simple.

We remark that in fact we have an explicit isomorphism B → End(B), namely the

one arising from the general map (3.4):

Enda M ⊗_{F}Enda(A) ⟶ End(B)
B ⊗_{F}F  ⟶^{∼} End(B).

(4.7)

This map induces an isomorphism F ⟶^{∼} Z(End(B)), whose inverse provides us with the definition of j in the building block triple we will associate with B.

For future reference, we write t for the common Schur index of End(A) and End(B)

(remember t = 1 or 2), and note that [E: F] = nt, because E is a maximal subfield of a by Theorem 1.3.

Now we need to define ν_{σ} for each σ ∈ G_{ℚ}. We will do so by constructing an

isogeny κ : E ⊗_{F} B → A^{t}. Because B = M ⊗a A, we will view E ⊗_{F} B as (E ⊗_{F} M) ⊗a A. Note that E ⊗_{F} M is an (E ⊗_{F} B)-a -bimodule. Furthermore, we know that E splits B (because E is a maximal subfield of a ), so we may fix an isomorphism

E ⊗_{F} B ⟶^{∼} M_{t}(E) ≅ End_{ℚ}(A^{t}).

In this way we obtain an action of E ⊗_{F} B on A^{t}, and so, a left action of E ⊗_{F} B on Hom(A, A^{t}). Now choose an isomorphism of (E ⊗_{F} B)-a -bimodules E ⊗_{F} M → Hom(A, A^{t}). (They are abstractly isomorphic because they have the same dimension—n^{2}t^{3}—over F [Pi, Cor. 3.3b].) Then by the defining property of the tensor product (3.2), there is a morphism κ : (E ⊗_{F} M) ⊗a A → A^{t} that respects the actions of E ⊗_{F} B (Proposition 3.5).

We claim this morphism is an isogeny. Indeed, the image of κ is a subvariety of A^{t}

with an action of E ⊗_{F} B ≅ M_{t}(E). But the only subvarieties of A^{t} mapped into themselves by M_{t}(E) are the zero subvariety and all of A^{t}. Now κ cannot be the zero morphism, because the homomorphism of modules used to define κ was non-zero. Therefore the image of κ must be A^{t}. Now we note that as right a -modules, E ⊗_{F} M ≅ ⨁^{nt} M ≅ ⨁^{t} a , so the dimension of (E ⊗_{F} M) ⊗a A is equal to the dimension of A^{t}. Thus κ is an isogeny.

We are now able to define the ν_{σ}. Consider κ^{−1} ^{σ}κ : E ⊗_{F} ^{σ}B → E ⊗_{F} B, which is

an isogeny respecting E ⊗_{F} B. The fact that

Hom^{E⊗B}(E ⊗_{F} ^{σ}B, E ⊗_{F} B) ≅ E ⊗_{F} Hom^{B}(^{σ}B, B)

follows from End^{E⊗B}(E ⊗_{F} B) ≅ E (which we know from the proof of Proposition 4.4). Hence we may write ν_{σ} = κ^{−1} ^{σ}κ ∈ E ⊗_{F} Hom^{B}(^{σ}B, B). Plainly, ν_{σ} ^{σ}ν_{τ} = ν_{στ} as desired.

Lastly we must check that the ν_{σ} do not depend on the various module isomor-

phisms chosen above. Suppose we make different choices and construct κ′ : E ⊗_{F} B → A^{t} respecting E ⊗_{F} B. Then κ′ κ^{−1} : A^{t} → A^{t} is an isogeny respecting E ⊗_{F} B ≅ M_{t}(E). But the only endomorphisms of A^{t} that respect M_{t}(E) lie in E, and are therefore defined over ℚ. Hence ^{σ}(κ′ κ^{−1}) = κ′ κ^{−1} for all σ ∈ G_{ℚ}, and we see ν′_{σ} = κ′^{−1} ^{σ}κ′ = κ^{−1} ^{σ}κ = ν_{σ} is independent of our choices.

Thus to the GL_{2}-type pair (A, i) we associate the triple (B, j, {ν_{σ}}).

Step 3 The correspondence defined in Step 2 is well-defined on equivalence classes.

Suppose (A′, i′) is a pair that is equivalent to (A, i). Write a = End(A) and

a′ = End(A′). Let a : A → A′ be an isogeny which satisfies the stated conditions for the equivalence of pairs. Then a induces an isomorphism a ⟶^{∼} a′ via φ ↦ a φ a^{−1}.

First notice that the image of i′ is the same as the image of i; we will call their common image E. Also i′(Z(a′)) = i(Z(a )), which we will call F. Henceforth we will identify E and F with these subalgebras of the endomorphism algebras.

Now let M be a simple right a -module, and M′ a simple right a′-module. Form

B = M ⊗a A and B′ = M′ ⊗a′ A′, and the accompanying data j, j′, ν_{σ}, and ν′_{σ} as above. Now we use the isomorphism a ⟶^{∼} a′ to view A′ as an abelian variety with an action of a , and M′ as an a -module. Then we may write B′ = M′ ⊗a A′.

Because a has a unique simple right module, there must be an isomorphism of a -

modules f : M → M′. Define b : B → B′ to be the morphism f ⊗ a : M ⊗a A → M′ ⊗a A′. Then b is an isogeny because both f and a are invertible, and b respects the action of F, because F commutes with everything in sight.

It remains to show that ν′_{σ} = b ν_{σ} ^{σ}b^{−1} for each σ ∈ G_{ℚ}. Remember that ν_{σ} is

defined as κ^{−1} ^{σ}κ where κ is an isogeny E ⊗_{F} B → A^{t} respecting E ⊗_{F} B, and similarly ν′_{σ} is defined in terms of κ′ : E ⊗_{F} B′ → A′^{t}. Now notice that a^{−1} κ′ (1 ⊗ b) κ^{−1} is an endomorphism of A^{t} respecting E ⊗_{F} B ≅ M_{t}(E). But the only such endomorphisms are elements of E, which are endomorphisms defined over ℚ. Therefore κ′ (1 ⊗ b) κ^{−1} is defined over ℚ, so ^{σ}(κ′ (1 ⊗ b) κ^{−1}) = κ′ (1 ⊗ b) κ^{−1} for all σ ∈ G_{ℚ}, and it follows that ν′_{σ} = b ν_{σ} ^{σ}b^{−1}.

Step 4 The above correspondences are inverse to each other.

Suppose we start with a GL_{2}-type pair (A, i), and use it to construct a building

block triple (B, j, {ν_{σ}}). Then from the map (4.7) we have Z(End(A)) ⟶^{∼} Z(End(B)), so the field F is the same from both viewpoints. Next, we need to show the field E = image of i is the same as the field E generated over F by the coefficients of the ν_{σ}. Write ν_{σ} = β(σ)^{−1} ⊗ μ_{σ}, and remember these ν_{σ} are defined by an isogeny κ : E ⊗_{F} B → A^{t}, where t = 1 or 2. Then by Proposition 4.3 we find that for φ ∈ End(A^{t}) and σ ∈ G_{ℚ}, we have ^{σ}φ = β(σ)^{−1} φ β(σ). Hence the same relationship holds for φ ∈ End(A). In other words, these β(σ)^{−1} are the same (up to F^{∗}) as the α(σ) appearing in Chapter 1, and we have seen there (Proposition 1.6) that these elements do indeed generate End_{ℚ}(A) over F.

Now clearly when we construct E ⊗_{F} B and descend it to ℚ via the map κ defined

by the ν_{σ}, we are just recovering the map κ : E ⊗_{F} (M ⊗a A) → A^{t} we used to define the ν_{σ} to begin with. Therefore the map from pairs to triples and back to pairs is the identity on equivalence classes.

On the other hand, what happens when we start with a triple (B, j, {ν_{σ}}) and construct a pair (A, i)? From (4.1) we have Z(End(B)) ⟶^{∼} Z(End(A^{t})) ≅ Z(End(A)), and so again we may write F with no confusion. Now we want to show that the building block triple constructed from (A, i) is equivalent to the triple (B, j, {ν_{σ}}).

Let M be a simple right a -module; we may view it as an F -a -bimodule. Choose

an isomorphism of F -a -bimodules M → Hom(A, B); they are isomorphic because they have the same F -dimension. Then by the defining property of the tensor product (3.2), there is a morphism M ⊗a A → B respecting F. This morphism is an isogeny because it is non-zero by construction and because M ⊗a A and B are simple abelian varieties. Call the inverse of this map b.

Finally, we have to show that the ν′_{σ} defined by (A, i) above are equal to b ν_{σ} ^{σ}b^{−1}

for the original ν_{σ} associated with B. Let κ : E ⊗_{F} B → A^{t} be defined by the original ν_{σ}. Then κ′ = κ (1⊗b)^{−1} is an isogeny E⊗_{F} (M ⊗aA) → A^{t} respecting E⊗_{F} End(B), where the action of End(B) on M ⊗a A is induced by b. We have already seen that any such isogeny is valid for defining the ν′_{σ}, and so ν′_{σ} = κ′^{−1} ^{σ}κ′ = (1 ⊗ b) κ^{−1} ^{σ}κ (1 ⊗^{σ}b^{−1}) = b ν_{σ} ^{σ}b^{−1}.

This proves Theorem 4.6.

Summary

Let (B, j, {ν_{σ}}) be a building block triple; we will write ν_{σ} = β(σ)^{−1} ⊗ μ_{σ} with the

understanding that the following formulas are independent of the transcriptions chosen. Let (A, i) be a GL_{2}-type pair corresponding to (B, j, {ν_{σ}}) under Theorem 4.6. The chart below summarizes how to define, from either perspective, the most frequently used constructs associated to this situation.

GL_{2}-type pair  Building block triple
Construct  (A, i)  (B, j, {ν_{σ}}_{σ∈G_{ℚ}})
number field  i(Z(End(A)))  j(Z(End(B)))
F
number field i(End_{ℚ}(A)) extension of F
E  generated by { β(σ) : σ ∈ G_{ℚ} }
α : G_{ℚ} → E^{∗} ^{σ}φ = α(σ) φ α(σ)^{−1} β^{−1}
(well-defined mod F^{∗})  for all φ ∈ End(A)
[c] ∈ H^{2}(G_{ℚ}, F^{∗}) class of class of
c(σ, τ) = α(στ) α(σ)^{−1}α(τ)^{−1}  c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1}

Chapter 5

Fields of Definition for Building

Blocks

In this chapter we generalize results of Shimura and Ribet concerning various

properties of building blocks, in particular, regarding their fields of definition. Recall that a (finite) compositum of quadratic extensions of a field is said to be a (2, …, 2)-extension, and that such an extension is Galois with group (ℤ/2ℤ)^{r} for some r.

Theorem 5.1 Let B/\overline{ℚ} be a building block. Then B is isogenous over \overline{ℚ} to an abelian variety defined over a (2, …, 2)-extension of ℚ.

Elkies first proved this in the case when B is an elliptic curve [El]. Ribet generalized

the result to the case of a building block with End(B) a totally real field [Ri 2, Thm. 1.2]. In fact, similar arguments apply to the general case, and we follow his approach as we outline the proof below.

Remember that to a building block B we have associated a well-defined cohomology

class [c] ∈ H^{2}(G_{ℚ}, F^{∗}) (where G_{ℚ} acts trivially on F^{∗}) by choosing isogenies μ_{σ} : ^{σ}B → B which respect End(B) and defining a two-cocycle c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1}. We will see that the class [c] provides certain information about a field of definition for B.

Definition. Let K be a number field. A building block over K is an abelian variety B over K such that B_{/\overline{ℚ}} is a building block and all endomorphisms of B are defined over K.

Proposition 5.2 Let K be a number field. Let B be a building block and [c] its associated cohomology class. Then B is isogenous over \overline{ℚ} to a building block B_{0} over K if and only if [c] lies in the kernel of the restriction map H^{2}(G_{ℚ}, F^{∗}) → H^{2}(Gal(\overline{ℚ}/K), F^{∗}).

Proof. Suppose B_{0} is an abelian variety over K that is isogenous to B. Fix an isogeny κ : B → B_{0}. Then for each σ ∈ Gal(\overline{ℚ}/K), define μ_{σ} = κ^{−1} ^{σ}κ : ^{σ}B → B; these isogenies respect End(B) because by hypothesis all endomorphisms of B_{0} are defined over K. Clearly now c(σ, τ) = 1 for all σ, τ ∈ Gal(\overline{ℚ}/K) , and so [c] = 1 ∈ H^{2}(Gal(\overline{ℚ}/K), F^{∗}).

Conversely, suppose [c] is trivial in H^{2}(Gal(\overline{ℚ}/K), F^{∗}). Choose a collection \{μ_{σ}\}

as usual to define the two-cocycle c(σ, τ). Then for σ, τ ∈ Gal(\overline{ℚ}/K), we may write c(σ, τ) as a coboundary; i.e., c(σ, τ) = β(σ) · β(τ) · β(στ)^{−1} where β : Gal(\overline{ℚ}/K) → F^{∗} is a locally constant function. Now replace each μ_{σ} with β(σ)^{−1} μ_{σ}, which is again an isogeny ^{σ}B → B respecting End(B). The new family \{μ_{σ}\} satisfies μ_{σ} ^{σ}μ_{τ} = μ_{στ} for all σ, τ ∈ Gal(\overline{ℚ}/K). Therefore we may apply [Ri 1, Thm. 8.2] to construct B_{0}/K and an isogeny κ : B → B_{0}_{/\overline{ℚ}}.

Finally, we need to check that all endomorphisms of B_{0} are defined over K,

but this follows easily from the facts that the μ_{σ} are compatible with End(B) and that μ_{σ} = κ^{−1} ^{σ}κ by construction of κ. Indeed, for φ ∈ End(B_{0}), we have κ^{−1} φ κ ∈ End(B), so μ_{σ} ^{σ}(κ^{−1} φ κ) = (κ^{−1} φ κ) μ_{σ} for all σ ∈ Gal(\overline{ℚ}/K). We may rewrite this as κ^{−1} ^{σ}φ ^{σ}κ = κ^{−1} φ ^{σ}κ, and therefore ^{σ}φ = φ. Since this holds for all σ ∈ Gal(\overline{ℚ}/K), we see that φ ∈ End(B_{0}) is defined over K. QED

Note that [c] has order dividing two in H^{2}(G_{ℚ}, F^{∗}). Indeed, we know from Chap-

ter 4 that there exists an abelian variety A of GL_{2}-type containing the building block B as a simple component, and that the class of the cocycle c(σ, τ) associated to B is exactly that of the cocycle we associated to A in Chapter 1. We saw there (Lemma 1.9) that the class of this cocycle has order one or two.

Write H^{2}(G_{ℚ}, F^{∗})[2] for the group of elements of the cohomology group H^{2}(G_{ℚ}, F^{∗})

of order dividing two.

Proposition 5.3 Let F be a totally real field and define P = F^{∗}/{±1}. Then H^{2}(G_{ℚ}, F^{∗})[2] is abstractly isomorphic to H^{2}(G_{ℚ}, {±1}) × Hom(G_{ℚ}, P/P^{2}).

Proof. (Ribet) We follow the procedure described in [Ri 2, Thm. 3.3].

Consider the exact sequence of trivial G_{ℚ}-modules

1 → {±1} → F^{∗} → P → 1.

This sequence is split because P is a free group [Ri 2, Lemma 3.5]. Thus we may write F^{∗} ≅ {±1} × P. We note that this isomorphism is in general non-canonical (although when [F:ℚ] is odd, we may embed P ↪ F^{∗} as the group of elements of F^{∗} of positive norm). In any case, there is an isomorphism

H^{2}(G_{ℚ}, F^{∗})[2] ≅ H^{2}(G_{ℚ}, {±1})[2] × H^{2}(G_{ℚ}, P)[2].

Clearly all elements of H^{2}(G_{ℚ}, {±1}) are of order dividing two, so H^{2}(G_{ℚ}, {±1})[2] = H^{2}(G_{ℚ}, {±1}).

We now claim that there is an isomorphism Hom(G_{ℚ}, P/P^{2}) ⟶^{∼} H^{2}(G_{ℚ}, P)[2].

Indeed, note that the squaring map x ↦ x^{2} is an injective map P → P, because P is torsion-free. Now look at the exact sequence

1 ⟶ P ⟶^{sq.} P ⟶ P/P^{2} ⟶ 1.

The corresponding long exact cohomology sequence is

⋯ ⟶ H^{1}(G_{ℚ}, P) ⟶ H^{1}(G_{ℚ}, P/P^{2}) ⟶ H^{2}(G_{ℚ}, P) ⟶^{sq.} H^{2}(G_{ℚ}, P).

Now G_{ℚ} is acting trivially, so we may regard each H^{1} as a Hom. Furthermore, P is torsionfree, so H^{1}(G_{ℚ}, P) = Hom(G_{ℚ}, P) = 1. Also, the image of H^{1}(G_{ℚ}, P/P^{2}) in H^{2}(G_{ℚ}, P) is the kernel of the squaring map H^{2}(G_{ℚ}, P) → H^{2}(G_{ℚ}, P), which consists exactly of the subgroup of elements of order dividing two. Therefore we may rewrite the cohomology sequence as

1 → Hom(G_{ℚ}, P/P^{2}) \overrightarrow{→} H^{2}(G_{ℚ}, P)[2].

Hence Hom(G_{ℚ}, P/P^{2}) is isomorphic to H^{2}(G_{ℚ}, P)[2], proving the proposition. QED

We are now ready to prove Theorem 5.1.

Proof. By the previous propositions, it suffices to show that the images of the cohomology class [c] in H^{2}(G_{ℚ}, {±1}) and Hom(G_{ℚ}, P/P^{2}) under the isomorphism of Proposition 5.3 are trivialized by restriction to Gal(\overline{ℚ}/K) where K is a (2, …, 2)-extension of ℚ.

Recall that H^{2}(G_{ℚ}, {±1}) may be identified with Br(ℚ)[2] = {Brauer classes

of quaternion algebras over ℚ}. A quaternion algebra over ℚ may always be split by a quadratic extension; therefore the image of [c] in H^{2}(G_{ℚ}, {±1}) is trivial under restriction to some quadratic extension K_{±} of ℚ.

The other “piece” of [c] lies in Hom(G_{ℚ}, P/P^{2}). Since the image of any continuous

homomorphism G_{ℚ} → P/P^{2} is a finite elementary 2-group, the kernel of the homomorphism corresponding to the image of [c] is Gal(\overline{ℚ}/K_{P}), where K_{P} is a (2, …, 2)-extension of ℚ.

Now let K be the composite of the fields K_{±} and K_{P}. Then K is a (2, …, 2)-

extension such that [c] is in the kernel of the restriction H^{2}(G_{ℚ}, F^{∗}) → H^{2}(Gal(\overline{ℚ}/K), F^{∗}).

QED

Elkies’s proof of the theorem for elliptic curves shows that the field K_{P} already

trivializes [c] in that case. Ribet’s method demonstrates this interesting fact to be true whenever [F : ℚ] is odd [Ri 2, Cor. 4.5]; it is unknown, however, whether it is true in general. In any case, the field K_{P} is a significant part of the (2, …, 2)-extension K, and in order to determine K_{P}, we will now discuss an explicit description of the image of [c] in Hom(G_{ℚ}, P/P^{2}).

The “Degree” of μ_{σ}

Let B_{1} and B_{2} be isogenous building blocks, and fix polarizations θ_{1} : B_{1} → B̂_{1} and

θ_{2} : B_{2} → B̂_{2}. We will say an isogeny μ : B_{1} → B_{2} is Rosati compatible if the isomorphism End(B_{1}) ≅ End(B_{2}) induced by μ is compatible with the Rosati involutions defined by θ_{1} and θ_{2}. For such an isogeny, we define the “degree” δ(μ) = μ θ_{1}^{−1} μ̂ θ_{2} ∈ Z(End(B_{2})). Note that δ(μ) depends on the chosen polarizations θ_{1}, θ_{2} (much as the Rosati involution of an endomorphism algebra depends on a choice of polarization).

The following properties of δ are elementary. If φ : B_{2} → B_{2}, then δ(φ) = φ φ′,

where φ′ is the Rosati involution of φ with respect to the polarization θ_{2}. Also, δ is multiplicative in the following sense: if λ : B_{2} → B_{3} is another isogeny that is Rosati compatible, then δ(λ μ) = (λ δ(μ) λ^{−1}) δ(λ) (assuming we use a fixed set of polarizations to calculate the degrees). In particular, if each building block is equipped with an action F ⟶^{∼} Z(End(B_{i})), and λ and μ respect F, then we may write δ(λ μ) = δ(λ) · δ(μ) as elements of F. Proposition 5.4 Let B_{1}, B_{2} be building blocks with polarizations θ_{1}, θ_{2}, respectively, and let μ : B_{1} → B_{2} be an isogeny that is Rosati compatible. Then the degree δ(μ) is a totally positive element of the totally real field F. Proof. We will regard the isogeny μ : B_{1} → B_{2} as an element of End(B_{1}×B_{2}), and use the properties of the Rosati involution on this endomorphism algebra to prove the proposition.

First define an isomorphism

M_{2}(End(B_{2})) ⟶ End(B_{1} × B_{2})
( \begin{matrix}φ_{11} & φ_{12} \\ φ_{21} & φ_{22} \end{matrix} ) ⟼ ( \begin{matrix}μ^{−1} φ_{11} μ & μ^{−1} φ_{12} \\ φ_{21} μ & φ_{22} \end{matrix} ) .

Now the polarizations θ_{1}, θ_{2} define a polarization θ_{1} × θ_{2} on End(B_{1} × B_{2}) which, in turn, induces a Rosati involution on End(B_{1} ×B_{2}). Pulling back this involution to M_{2}(End(B_{2})), we find it is given by the formula

Φ = ( \begin{matrix}φ_{11} & φ_{12} \\ φ_{21} & φ_{22} \end{matrix} ) ⟼ ( \begin{matrix}φ′_{11} & δ(μ) φ′_{21} \\ φ′_{12} δ(μ)^{−1} & φ′_{22} \end{matrix} ) = Φ′.

where φ ↦ φ′ is the Rosati involution on End(B_{2}) coming from θ_{2}.

Now the Rosati involution is a positive involution; that is, for a non-zero endo-

morphism Φ, the trace Tr(ΦΦ′) ∈ ℚ is positive [Mu, §21 Thm. 1]. In particular, consider Φ ∈ End(B_{1} × B_{2}) of the form \binom{0 0}{φ 0}. Then

Φ′ = ( \begin{matrix}0 & δ(μ) φ′ \\ 0 & 0 \end{matrix} ) and ΦΦ′ = ( \begin{matrix}0 & 0 \\ 0 & δ(μ) φ φ′ \end{matrix} ) .

Therefore Tr(ΦΦ′) = Tr(δ(μ) φ φ′) > 0 for all φ ∈ End(B_{2}).

We now claim the fact that this inequality holds for all φ implies that δ(μ) must

be totally positive. Indeed, let I be the finite set of embeddings {ι : F ↪ ℝ}. Suppose some embedding, call it ι_{0}, is such that ι_{0}(δ(μ)) < 0. Now by the Strong Approximation Theorem, we may choose φ ∈ F such that |ι_{0}(φ))|^{2} is large, say greater than #I/|ι_{0}(δ(μ))|, but |ι(φ)|^{2} is small, say less than 1/ max_{ι≠ι_{0}}{|ι(δ(μ))|} for all ι ≠ ι_{0}.

Now by [Mu, p. 182], Tr(δ(μ) φ φ′) is a positive multiple of Tr_{F/ℚ}(δ(μ) · φ^{2}),

which we may calculate as

Tr_{F/ℚ}(δ(μ) · φ^{2}) = ∑_{ι∈I} ι(δ(μ)) · ι(φ)^{2}
= ι_{0}(δ(μ)) · ι_{0}(φ)^{2} + ∑_{ι≠ι_{0}} ι(δ(μ)) · ι(φ)^{2}
< −#I  + ∑_{ι≠ι_{0}} 1
< 0

because we chose φ carefully.

This contradicts the fact that Tr(δ(μ) φ φ′) is positive for all φ. Therefore ι(δ(μ))

must be positive for all embeddings {ι : F ↪ ℝ}. QED

Next, we look at the relationship between δ(μ) and the standard degree deg μ ∈ ℚ.

Proposition 5.5 Let δ(μ) be the “degree” of an isogeny of building blocks μ : B_{1} → B_{2} defined in terms of polarizations θ_{1} and θ_{2} with deg θ_{1} = deg θ_{2}. Then deg μ = N_{B/ℚ}(δ(μ)), where N_{B/ℚ} is the canonical reduced norm of B = End(B_{2}) over ℚ.

Proof. Write t = \sqrt{dim_{F} B} (= 1 or 2) as usual. First, because B_{2} is simple, we know deg δ(μ) = (N_{B/ℚ}(δ(μ)))^{n} for some n ≥ 1 [Mu, p. 182]. But “deg” is a polynomial function of degree 2 dim B_{2}, and N_{B/ℚ} is a polynomial function of degree t[F:ℚ] = dim B_{2} (since B_{2} is a building block). Thus n = 2 and deg δ(μ) = (N_{B/ℚ}(δ(μ)))^{2}.

On the other hand, δ(μ) = μ θ_{1}^{−1} μ̂ θ_{2} means that

deg δ(μ) = (deg μ)(deg θ_{1})^{−1}(deg μ̂)(deg θ_{2}) = (deg μ)^{2},

because deg μ̂ = deg μ, and deg θ_{1} = deg θ_{2} by hypothesis.

Therefore, (deg μ)^{2} = (N_{B/ℚ}(δ(μ))^{2}, so deg μ = ± N_{B/ℚ}(δ(μ)) = ±(N_{F/ℚ}(δ(μ)))^{t},

as δ(μ) lies in F. Now we have seen that δ(μ) is totally positive; thus N_{F/ℚ}(δ(μ)) > 0. Of course, deg μ is also positive, and so we must have deg μ = + N_{B/ℚ}(δ(μ)). QED

Now let B be a building block and fix a collection {μ_{σ} : ^{σ}B → B} respecting

End(B), as usual. Let θ : B → B̂ be a polarization of B. Then for each σ ∈ G_{ℚ}, the isogeny ^{σ}θ : ^{σ}B → ^{σ}B̂ is a polarization of ^{σ}B and we define, as above, the degree δ(μ_{σ}) = μ_{σ} ^{σ}θ^{−1} μ̂_{σ} θ. It is easy to check that the isogenies {μ_{σ}} are Rosati compatible, because each μ_{σ} was chosen to be compatible with End(B). Note, however, that by requiring the polarization of ^{σ}B to be ^{σ}θ, now δ(μ_{σ}) ∈ F is independent of the chosen polarization θ.

Moreover, the μ_{σ} are well-defined modulo F^{∗}, and therefore δ(μ_{σ}) mod F^{∗}^{2} de-

pends only on the building block B and not on the choices for μ_{σ} or the polarization defining δ. Also, μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1} ∈ F^{∗} implies that δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1} ∈ F^{∗}^{2}, and hence σ ↦ δ(μ_{σ}) mod F^{∗}^{2} is a locally constant homomorphism G_{ℚ} → F^{∗}/F^{∗}^{2}.

Recall from the previous section the two-cocycle c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1} associ-

ated to the building block B. We have seen there is an isomorphism H^{2}(G_{ℚ}, F^{∗})[2] ≅ H^{2}(G_{ℚ}, {±1}) × Hom(G_{ℚ}, P/P^{2}), where P is defined to be F^{∗}/{±1}, and we are interested in the image of [c] in Hom(G_{ℚ}, P/P^{2}).

Proposition 5.6 The image of [c] in Hom(G_{ℚ}, P/P^{2}) is exactly the homomorphism σ ↦ δ(μ_{σ}) mod {±1}F^{∗}^{2}.

Proof. The projection map H^{2}(G_{ℚ}, F^{∗})[2] → Hom(G_{ℚ}, P/P^{2}) is broken into two parts: first, the map H^{2}(G_{ℚ}, F^{∗})[2] → H^{2}(G_{ℚ}, P)[2] induced by F^{∗} → F^{∗}/{±1} = P, followed by the inverse of the isomorphism Hom(G_{ℚ}, P/P^{2}) ⟶^{∼} H^{2}(G_{ℚ}, P)[2] coming from the exact sequence 1 → P ⟶^{sq.} P ⟶ P/P^{2} → 1. Thus we have to show that the image of the homomorphism σ ↦ δ(μ_{σ}) mod {±1}F^{∗}^{2} in H^{2}(G_{ℚ}, P)[2] is the same as the image of [c] in this group.

The isomorphism Hom(G_{ℚ}, P/P^{2}) ⟶^{∼} H^{2}(G_{ℚ}, P)[2] maps the homomorphism

σ ↦ δ(μ_{σ}) mod {±1}F^{∗}^{2} to a cohomology class as follows: Lift the homomorphism to a map G_{ℚ} → P, e.g., σ ↦ δ(μ_{σ}) mod {±1}. Then δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1} mod {±1} is a square in P, and σ, τ ↦ \sqrt{δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1}} mod \{±1\} is the two-cocycle that defines the image in H^{2}(G_{ℚ}, P)[2].

On the other hand, c(σ, τ) = μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1}, and taking the degree δ of both sides

we find c(σ, τ)^{2} = δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1}. Hence c(σ, τ) = ±\sqrt{δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1}}, and therefore c(σ, τ) mod {±1} gives the same class in H^{2}(G_{ℚ}, P)[2] that the homomorphism σ ↦ δ(μ_{σ}) mod {±1}F^{∗}^{2} does. QED

Some Characters

Let B be a building block. As we have seen in Chapter 4, we form a building

block triple (B, j, \{ν_{σ}\}) by attaching some extra data to B: namely, j : Z(End(B)) ↪ \overline{ℚ}, whose image we call F, and for all σ ∈ G_{ℚ}, a non-zero ν_{σ} ∈ \overline{ℚ} ⊗_{F} Hom^{B}(^{σ}B, B) such that ν_{σ} ^{σ}ν_{τ} = ν_{στ}. Because Hom^{B}(^{σ}B, B) is a one-dimensional F -vector space, we can, and do, write ν_{σ} = β(σ)^{−1} ⊗ μ_{σ} for some μ_{σ} : ^{σ}B → B respecting End(B) and some β(σ) ∈ \overline{ℚ}^{∗}. Let E be the field extension of F generated by the β(σ).

Now let (A, i) be a pair of GL_{2}-type corresponding to (B, j, {ν_{σ}}) under Theo-

rem 4.6. Then i is an isomorphism of End_{ℚ}(A) onto the field E, and i(Z(End(A)) = F. Recall that E is a Galois extension of F. In [Ri 1], Ribet associates a character ε : G_{ℚ} → E^{∗} to the abelian variety A. Our goal in this section is to recover ε directly from the triple (B, j, {ν_{σ}}).

The condition ν_{σ} ^{σ}ν_{τ} = ν_{στ} tells us that

μ_{σ} ^{σ}μ_{τ} μ_{στ}^{−1} = c(σ, τ) = β(σ) · β(τ) · β(στ)^{−1} ∈ F^{∗},

and therefore, if δ is the degree map defined in the previous section, δ(μ_{σ}) · δ(μ_{τ}) · δ(μ_{στ})^{−1} = δ(c(σ, τ)) = c(σ, τ)^{2} = β(σ)^{2} · β(τ)^{2} · β(στ)^{−2}.

Hence we see that σ ↦ δ(μ_{σ})/β(σ)^{2} is a continuous character G_{ℚ} → E^{∗}. Note that this character depends only on the triple (B, j, {ν_{σ}}) and not on our choices for μ_{σ}; we claim that it is in fact equal to the character ε.

In order to define ε and to prove our claim, we need to study the “λ-adic” repre-

sentations of G_{ℚ} provided by the abelian variety A of GL_{2}-type.

Let ℓ be a prime number and let V_{ℓ} = V_{ℓ}(A) be the Tate module associated to

A. Because A is of GL_{2}-type, V_{ℓ}(A) is free of rank two over E ⊗_{ℚ} ℚ_{ℓ}. Let {λ} be the set of primes of E lying over ℓ, and for each λ, write E_{λ} for the completion of E at λ; then E ⊗_{ℚ} ℚ_{ℓ} ≅ ⨁_{λ|ℓ} E_{λ}. Now define V_{λ} = E_{λ} ⊗_{E⊗ℚ_{ℓ}} V_{ℓ}. We see immediately that each V_{λ} is a two-dimensional E_{λ}-vector space, and that V_{ℓ} ≅ ⨁_{λ|ℓ} V_{λ}. Each V_{λ} is still also a G_{ℚ}-module, since the action of G_{ℚ} on V_{ℓ} commutes with E ⊗_{ℚ} ℚ_{ℓ}.

Our starting point for studying the V_{λ} will be the following result of Faltings.

Theorem 5.7 Let K be a finite Galois extension of ℚ such that all endomorphisms of A are defined over K, and let H = Gal(\overline{ℚ}/K). Then

End(A) ⊗_{ℚ} ℚ_{ℓ} → End_{ℚ_{ℓ}[H]} V_{ℓ}

is an isomorphism of ℚ_{ℓ}[G_{ℚ}]-modules.

The G_{ℚ}-module structure on V_{ℓ} induces an action of G_{ℚ} on End_{ℚ_{ℓ}} V_{ℓ} via ^{σ}f = σ f σ^{−1}. Now End_{ℚ_{ℓ}[H]} V_{ℓ} is the submodule of End_{ℚ_{ℓ}} V_{ℓ} fixed by H; it is still a G_{ℚ}-submodule because H is a normal subgroup. This defines the action of G_{ℚ} on End_{ℚ_{ℓ}[H]} V_{ℓ} in the theorem. Of course, the action of G_{ℚ} on End(A) ⊗_{ℚ} ℚ_{ℓ} comes from the natural action of G_{ℚ} on End(A).

Corollary 5.8 Let H ⊂ G_{ℚ} be a normal subgroup of finite index with the properties described in the above theorem. Then End_{E_{λ}[H]} V_{λ} ≅ E_{λ} for all λ | ℓ.

Proof. Recall that E is its own centralizer in End(A). Therefore the centralizer of E ⊗_{ℚ} ℚ_{ℓ} in End(A) ⊗_{ℚ} ℚ_{ℓ} is E ⊗_{ℚ} ℚ_{ℓ}. Applying the isomorphism in the theorem, we find that the centralizer of E ⊗_{ℚ} ℚ_{ℓ} in End_{ℚ_{ℓ}[H]} V_{ℓ} is again E ⊗_{ℚ} ℚ_{ℓ}. Therefore,

End_{(E⊗ℚ)[H]} V_{ℓ} ≅ E ⊗_{ℚ} ℚ_{ℓ}
⨁_{λ|ℓ} End_{E_{λ}[H]} V_{λ} ≅ ⨁_{λ|ℓ} E_{λ}
End_{E_{λ}[H]} V_{λ} ≅ E_{λ}. QED

Corollary 5.9 Let ℓ be a prime that splits completely in E. Then Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{λ′}) is non-zero if and only if λ′ = γ(λ) for some γ ∈ Gal(E/F).

Proof. Regard the isomorphisms

End(A) ⊗_{ℚ} ℚ_{ℓ} ≅ End_{ℚ_{ℓ}[H]} V_{ℓ} ≅ ⨁_{λ,λ′} Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{λ′})

as isomorphisms of (F ⊗_{ℚ} ℚ_{ℓ})-modules. Because F ⊗_{ℚ} ℚ_{ℓ} is the center of End(A) ⊗_{ℚ} ℚ_{ℓ}, every non-zero f ∈ Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{λ′}) must be an (F ⊗_{ℚ}ℚ_{ℓ})-isomorphism. But it is clear that V_{λ} ≅ V_{λ′} as (F ⊗_{ℚ} ℚ_{ℓ})-modules if and only if E_{λ} ≅ E_{λ′} as F -modules, which is equivalent to the condition λ′ = γ(λ) for some γ ∈ Gal(E/F). In other words, if λ′ ≠ γ(λ), then Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{λ′}) = 0.

Now because ℓ splits completely in E, there are [E : ℚ] primes λ lying over ℓ,

and we have E_{λ} ≅ ℚ_{ℓ} for each λ. By the previous corollary, End_{ℚ_{ℓ}[H]} V_{λ} ≅ ℚ_{ℓ}. Therefore, the ℚ_{ℓ}-dimension of Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{γ(λ)}) must be zero or one. Hence the ℚ_{ℓ}-dimension of ⨁_{γ} ⨁_{λ} Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{γ(λ)}) is less than or equal to [E:F][E:ℚ], with equality holding if and only if each Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{γ(λ)}) is one-dimensional. On the other hand, the ℚ_{ℓ}-dimension of End(A) ⊗_{ℚ} ℚ_{ℓ} is [E:F]^{2} · [F:ℚ], which forces equality above. QED

Now consider the action of G_{ℚ} on End(A) ⊗_{ℚ} ℚ_{ℓ}. Recall that for each σ ∈ G_{ℚ} we

have defined α(σ) ∈ E^{∗} to be an element of End(A) such that ^{σ}φ = α(σ) φ α(σ)^{−1}. That is to say, σ ∈ G_{ℚ} acts on End(A) via conjugation by α(σ), where α(σ) ∈ E^{∗} is well-defined modulo F^{∗}. We fix a choice of α(σ) for each σ; these α(σ) satisfy α(σ)·α(τ)·α(στ)^{−1} ∈ F^{∗}, as always. Therefore, for each γ ∈ Gal(E/F), we may define a character χ_{γ} : G_{ℚ} → E^{∗} by χ_{γ}(σ) = γ(α(σ))/α(σ). Note that these characters depend only on the abelian variety A.

Notation: Let ℓ be a prime which splits completely in E, and let λ be a prime of E dividing ℓ. Then E_{λ} will denote ℚ_{ℓ} with the map E ↪ ℚ_{ℓ} induced by λ, and the image in ℚ_{ℓ} of an element e ∈ E will be written e_{λ}.

Lemma 5.10 Let ℓ be a prime which splits completely in E. Let γ ∈ Gal(E/F). Then the ℚ_{ℓ}[G_{ℚ}]-module V_{λ} is isomorphic to V_{γ(λ)} ⊗ χ_{γ,λ}^{−1}.

Proof. We know that V_{λ} and V_{γ(λ)} are isomorphic ℚ_{ℓ}[H]-modules for some normal subgroup H of G_{ℚ} of finite index. Let f ∈ Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{γ(λ)}) be an isomorphism. Now End(A) ⊗_{ℚ} ℚ_{ℓ} is G_{ℚ}-isomorphic to End_{ℚ_{ℓ}[H]} V_{ℓ}, and we know σ acts on End(A) ⊗_{ℚ} ℚ_{ℓ} via conjugation by α(σ). Also, α(σ) ∈ E acts on V_{λ} as α(σ)_{λ} for each λ. Thus we may write σf = α(σ)_{γ(λ)} · f · α(σ)_{λ}^{−1} in Hom_{ℚ_{ℓ}[H]}(V_{λ}, V_{γ(λ)}). Now because E_{λ} ≅ ℚ_{ℓ}, we have

σf = \frac{α(σ)_{γ(λ)}}{α(σ)_{λ}} · f = (\frac{γ(α(σ))}{α(σ)})_{λ} · f = χ_{γ,λ}(σ) · f.

Hence for all σ ∈ G_{ℚ} and v ∈ V_{λ}, we have σ(f(v)) = χ_{γ,λ}(σ) · f(σ(v)), and we see that f is an isomorphism of ℚ_{ℓ}[G_{ℚ}]-modules V_{λ} ⟶^{∼} V_{γ(λ)} ⊗ χ_{γ,λ}^{−1}. QED

Now what is the character ε? The action of G_{ℚ} on V_{λ} defines a representation

ρ_{λ} : G_{ℚ} → Aut V_{λ}. By [Ri 1, Lemma 3.1] there exists a character of finite order ε : G_{ℚ} → E^{∗} such that for each prime λ | ℓ of E, we have det ρ_{λ} = ε · χ_{ℓ}, where χ_{ℓ} : G_{ℚ} → ℚ_{ℓ}^{∗} is the ℓ-adic cyclotomic character. As we will soon see, ε is equal to the character χ_{conj.}^{−1}, where “conj.” ∈ Gal(E/F) is the canonical complex conjugation e ↦ \overline{e} on E.

Lemma 5.11 Let ℓ be a prime that splits completely in E. Then the G_{ℚ}-module V_{λ} is isomorphic to V\overline{λ} ⊗ ε_{λ}.

Proof. (Ribet) Let < , >: V_{ℓ} × V_{ℓ} → ℚ_{ℓ}(1) be the Weil pairing on V_{ℓ}(A) induced by a fixed polarization of A/ℚ. Here ℚ_{ℓ}(1) denotes the ℓ-adic Tate module ℚ_{ℓ} ⊗_{ℤ_{ℓ}} \underleftarrow{lim} lu_{ℓ^{n}} of ℓ-power roots of unity in \overline{ℚ}. Recall that the Weil pairing is non-degenerate and Galois invariant; that is, for σ ∈ G_{ℚ} and v, w ∈ V_{ℓ} we have ^{σ}< v, w > = < σ(v), σ(w) >. Thus we obtain an isomorphism of G_{ℚ}-modules

V_{ℓ} ≅ Hom_{ℚ_{ℓ}}(V_{ℓ}, ℚ_{ℓ}(1)).

By writing V_{ℓ} as the direct sum of the V_{λ}, we then have

(∗)  ⨁_{λ|ℓ} V_{λ} ≅ ⨁_{λ|ℓ} Hom_{ℚ_{ℓ}}(V_{λ}, ℚ_{ℓ}(1)).

We claim that under this isomorphism, the \overline{λ}-component on the left-hand side corresponds to the λ-component on the right-hand side. Indeed, recall that if φ ↦ φ′ is the Rosati involution on End(A) coming from our fixed polarization, then φ and φ′ are adjoint with respect to the Weil pairing; in particular, < ev, w > = < v, \overline{e}w > for e ∈ E and v, w ∈ V_{ℓ}. Now we may extend the automorphism “conj.” to E ⊗_{ℚ} ℚ_{ℓ}; it is simply the map which permutes the components of ⨁_{λ|ℓ} E_{λ} by taking the λ-component to the \overline{λ}-component. The new “conj.” still has the same property with respect to < , >.

Now let {π_{λ}} be a set of orthogonal idempotents for E ⊗_{ℚ} ℚ_{ℓ}, so that V_{λ} = π_{λ}V_{ℓ}.

Note that \overline{π_{λ}} = π_{\overline{λ}}. By using <π_{λ}v, w > = <v, \overline{π_{λ}}w >, it is easy to verify that if v = π_{λ}v ∈ V_{λ}, then under (∗) the corresponding homomorphism f_{v} is f_{v} π_{\overline{λ}} ∈ Hom_{ℚ_{ℓ}}(V\overline{λ}, ℚ_{ℓ}(1)). This proves the claim, and we have

V\overline{λ} ≅ Hom_{ℚ_{ℓ}}(V_{λ}, ℚ_{ℓ}(1)).

On the other hand, V_{λ} is a two-dimensional E_{λ}-vector space. Therefore the deter-

minant gives a pairing V_{λ} × V_{λ} → E_{λ} which, by definition of ε, defines an isomorphism of ℚ_{ℓ}[G_{ℚ}]-modules

V_{λ} ≅ Hom_{ℚ_{ℓ}}(V_{λ}, ℚ_{ℓ}(1) ⊗ ε_{λ}) ≅ Hom_{ℚ_{ℓ}}(V_{λ}, ℚ_{ℓ}(1)) ⊗ ε_{λ}.

Combining this with the previous isomorphism, we see V_{λ} ≅ V\overline{λ} ⊗ ε_{λ}. QED

Theorem 5.12 Let (B, j, {ν_{σ}}) be a building block triple, and let (A, i) be a pair of GL_{2}- type corresponding to the triple under Theorem 4.6. Then the character σ ↦ δ(μ_{σ})/β(σ)^{2} defined by (B, j, {ν_{σ}}) is equal to the character ε defined by (A, i).

Proof. Let ℓ be a prime that splits completely in E. Together, the previous two lemmas imply that ε_{λ} = χ_{conj.,λ}^{−1} for all λ dividing ℓ, and so ε = χ_{conj.}^{−1} : G_{ℚ} → E^{∗}. That is, we may write ε(σ) = α(σ)/\overline{α(σ)} for the usual collection \{α(σ)\}_{σ∈G_{ℚ}} defined by A.

On the other hand, we know that under the correspondence (B, j, {ν_{σ}}) ⇝ (A, i),

we may take α(σ) = β(σ)^{−1}. Thus ε(σ) = \overline{β(σ)}/β(σ).

Now we compare our two characters by taking their quotient; we find a new char-

acter σ ↦ ε(σ)^{−1} · δ(μ_{σ})/β(σ)^{2} = δ(μ_{σ})/β(σ)\overline{β(σ)}, which takes values in F^{∗}. Therefore δ(μ_{σ})/β(σ)\overline{β(σ)} ∈ \{±1\} for all σ ∈ G_{ℚ}, because F is a totally real field. But δ(μ_{σ}) and β(σ)\overline{β(σ)} are both totally positive elements of F^{∗}, so their quotient must be +1. Hence we have δ(μ_{σ}) = β(σ)\overline{β(σ)} for all σ, and ε(σ) = δ(μ_{σ})/β(σ)^{2}. QED

This theorem has some interesting consequences. First, recall our discussion of

fields of definition for building blocks. We found that a building block B is always isogenous to a building block B_{0} over a (2, …, 2)-extension K of ℚ. Furthermore, that extension may frequently be taken to be the fixed field K_{P} of the subgroup { σ : δ(μ_{σ}) ∈ {±1}F^{∗}^{2} } ⊂ G_{ℚ}, and in all cases, a quadratic extension of K_{P} suffices. Now if A is an abelian variety of GL_{2}-type corresponding to B, this information translates into the knowledge of a field K where A breaks up into its simple components, i.e., A_{/K} ∼ B_{0} × … × B_{0}. The theorem lets us compute the field K_{P} from data associated with A only.

Corollary 5.13 Let lu_{E} denote the group of roots of unity in E^{∗}. Then the field K_{P} is the fixed field of { σ : α(σ) ∈ lu_{E}F^{∗} } ⊂ G_{ℚ}.

Proof. Fix σ ∈ G_{ℚ}. It suffices to show that δ(μ_{σ}) ∈ {±1}F^{∗}^{2} if and only if α(σ) ∈ lu_{E}F^{∗}. Suppose that δ(μ_{σ}) ∈ {±1}F^{∗}^{2}. Then by the theorem, α(σ)^{2} = ε(σ)/δ(μ_{σ}) ∈ lu_{E}F^{∗}^{2}, which implies that α(σ) ∈ lu_{E}F^{∗}. Conversely, suppose that α(σ) ∈ lu_{E}F^{∗}. Then δ(μ_{σ}) = ε(σ)/α(σ)^{2} ∈ (lu_{E} ∩ F^{∗})F^{∗}^{2} = {±1}F^{∗}^{2}. QED

Finally, we may formulate a criterion to determine when, given a building block,

we may construct from it an abelian variety A of GL_{2}-type with ε = 1. This question is motivated by the discussion in the Introduction of a possible “modular” structure on A. Recall that A is “modular” if it appears as a quotient of the Jacobian J_{1}(N) for some N. In this event, A is isogenous to an abelian variety A_{f}, where f is a normalized cuspidal eigenform of weight two on Γ_{1}(N). (Conjecturally, all abelian varieties of GL_{2}-type may be constructed in this way.) Now when A is modular, the character ε is exactly the character of the modular form f on Γ_{0}(N) [Ri 1, Lemma 4.3]. Thus the case ε = 1 is especially interesting because the morphism J_{1}(N) → A then factors through J_{0}(N).

The previous theorem tells us that ε depends on our choice of ν_{σ} = β(σ)^{−1} ⊗ μ_{σ}.

Corollary 5.14 Let B be a building block, and [c] its associated cohomology class. Then the following are equivalent:

(i) There exists a choice of {ν_{σ}} such that the character ε is trivial. (ii) The class [c] lies in the kernel of the natural map

H^{2}(G_{ℚ}, F^{∗}) → H^{2}(G_{ℚ}, F (…, \sqrt{δ(μ_{σ})}, …)).

Proof. First note that the field extension F_{1} =^{def} F (…, \sqrt{δ(μ_{σ})}, …) is independent of our choices for μ_{σ}, because δ(μ_{σ}) is well-defined modulo F^{∗}^{2}. Furthermore, since all δ(μ) are totally positive elements of F^{∗}, we see that the field F_{1} is totally real.

Now suppose ε = 1. Then by the theorem, β(σ)^{2} = δ(μ_{σ}) for all σ ∈ G_{ℚ}. Thus

β(σ) ∈ F (\sqrt{δ(μ_{σ})}), and so c(σ, τ) splits in F_{1}.

On the other hand, if c splits in F_{1}, then we may choose all β(σ) in this totally

real field. Therefore \overline{β(σ)} = β(σ) for all σ, and so ε = 1. QED

Chapter 6

An Example of a ℚ-Curve

In this chapter we give a concrete example of a building block of dimension one,

and we calculate many of the associated functions and characters that were discussed in the previous chapter.

We see from its definition that a building block of dimension one is simply an

elliptic curve over \overline{ℚ} without complex multiplication that is isogenous to all of its Galois conjugates. The following example of such a ℚ-curve was suggested by N. Elkies of Harvard University.

Let B be the elliptic curve over ℚ(\sqrt{−3}) defined by the equation

B : y^{2} = x^{3} + 2x^{2} + bx, where b ∈ ℚ(\sqrt{−3}) is an algebraic integer of trace 1.

Proposition 6.1 The curve B does not have complex multiplication.

Proof. Let ω be a primitive cube root of unity in ℚ(\sqrt{−3}), and let 𝔭 = (1 − ω) be the unique prime of ℚ(\sqrt{−3}) lying over 3. We claim that b ≡ −1 mod 𝔭, because b has trace 1. Indeed, write b = r + sω with r, s ∈ ℤ. Then Tr(b) = 1 implies that s = 2r + 1, and so b + 1 = (1 − ω)(1 + rω) ∈ 𝔭.

Now the discriminant of B is 64b^{2}(1 − b) ≢ 0 mod 𝔭, so 𝔭 is a prime of good

CHAPTER 6. AN EXAMPLE OF A ℚ-CURVE 48

reduction for B. Let B_{𝔭} be the reduction of B modulo 𝔭; the equation for B_{𝔭} over the field of three elements 𝔽_{3} is then y^{2} = x^{3} + 2x^{2} − x. Recall that End(B) is a subfield of End(B_{𝔭}), and that it is possible to determine End(B_{𝔭}) by calculating the characteristic polynomial of the Frobenius endomorphism at 𝔭. We can write down this polynomial after counting the number of 𝔽_{3}-rational points on B_{𝔭}. It is easy to see there are only two such points: ∞ and (0, 0). Therefore the trace of Frobenius is 1 + 3 − 2 = 2, and the characteristic polynomial is X^{2} − 2X + 3, by [Si, Chap. V]. Hence End(B_{𝔭}) ≅ ℚ(\sqrt{−2}), which is a field with class number one.

Now End(B) ↪ ℚ(\sqrt{−2}), but B cannot have complex multiplication by this field.

Indeed, an elliptic curve over \overline{ℚ} with complex multiplication by a quadratic imaginary field K of class number one must have a rational j-invariant [Si, Thm. C.11.2], and the j-invariant of B is (4 − 3b)^{3}/b^{2}(1 − b) ∉ ℚ. Thus End(B) ≅ ℚ. QED

Let τ be the non-trivial automorphism of ℚ(\sqrt{−3}) over ℚ. Then τ(b) = 1 − b,

because b has trace 1, and we may define an isogeny μ : ^{τ}B → B of degree two by

\begin{matrix}^{τ}B : y^{2} = x^{3} + 2x^{2} + (1 − b)x & \underrightarrow{μ} & B : y^{2} = x^{3} + 2x^{2} + bx \\ (x, y) & ⟼ & (−\frac{y^{2}}{2x^{2}} , \frac{\sqrt{−2}y(1−b−x^{2})}{4x^{2}}) . \end{matrix}

This isogeny may be factored into two steps: first twist^{τ}B by \sqrt{−2}, then use the two-isogeny provided by [Si, Ex. III.4.5]. Similarly, we find a formula for the dual μ̂ : B → ^{τ}B.

(−\frac{y^{2}}{2x^{2}} , −\frac{\sqrt{−2}y(b−x^{2})}{4x^{2}}) μ̂ ; ⟵⊣ (x, y)

Thus B/ℚ(\sqrt{−3}) is a ℚ-curve, but we note that the isogeny μ is not defined over ℚ(\sqrt{−3}). In fact, we may make the following stronger statement.

Proposition 6.2 There is no twist of B/ℚ(\sqrt{−3}) such that the isogeny corresponding to μ is defined over ℚ(\sqrt{−3}).

Proof. Any twist of B/ℚ(\sqrt{−3}) is a twist by \sqrt{a} for some a ∈ ℚ(\sqrt{−3}); the resulting curve is B_{a} : ay^{2} = x^{3} + 2x^{2} + bx. Then the isogeny corresponding to μ is

\begin{matrix}^{τ}(B_{a}) & ⟶ & B_{a} \\ (x, y) & ⟼ & (−\frac{τ(a)y^{2}}{2x^{2}} , \sqrt{\frac{−2τ(a)}{a}} \frac{y(1−b−x^{2})}{4x^{2}}) , \end{matrix}

which is not defined over ℚ(\sqrt{−3}), unless −2τ(a)/a is a square in ℚ(\sqrt{−3}). But 2 is a prime in this field, and the valuation at 2 of −2τ(a)/a must be 1; thus −2τ(a)/a cannot be a square. QED

Now we wish to investigate the two-cocycle c(g, h) = μ_{g} ^{g}μ_{h} μ_{gh}^{−1}. In order to do

so, we must extend our base field to K = ℚ(\sqrt{−3}, \sqrt{−2}). By abusing notation, we now write τ for the non-trivial element of Gal(K/ℚ) fixing ℚ(\sqrt{−2}), so that τ restricts to our previous τ on ℚ(\sqrt{−3}). Also, we introduce σ ∈ Gal(K/ℚ) as the non-trivial automorphism fixing ℚ(\sqrt{−3}). Thus we have ^{σ}B = B and ^{στ}B = ^{τ}B, and isogenies μ_{σ} = μ_{1} = id : B → B, and μ_{στ} = μ_{τ} = μ : ^{τ}B → B. We remark that by Proposition 5.5, the “degree” δ(μ_{g}) is exactly the usual degree of an isogeny; that is, δ(μ_{σ}) = δ(μ_{1}) = 1 and δ(μ_{στ}) = δ(μ_{τ}) = 2.

Note that the following identities are immediate from the formulas for μ and μ̂:

σμ = −μ,  ^{τ}μ = −μ̂,  ^{στ}μ = μ̂,  and, of course, μμ̂ = 2.

We can now calculate the values of c(g, h):

h
1  σ  τ στ
1  1  1  1 1
g σ  1  1 −1 −1
τ  1  1 −2 −2
στ  1  1  2 2
c(g, h)

We would like to split this cocycle as

c(g, h) = \frac{β(g) · β(h)}{β(gh)}

for some (locally constant) function β : G_{ℚ} → \overline{ℚ}^{∗}. Notice that in order to define such an β, we will again have to extend our base field. There is no such β : G_{ℚ} → Gal(K/ℚ) → \overline{ℚ}^{∗} because, for example, στ = τσ, but c(σ, τ) ≠ c(τ, σ). Therefore, we will have to find an extension of K which is non-abelian over ℚ.

Let L be the splitting field over ℚ of the polynomial X^{4} − 4X^{2} + 6. Figure 6.3

diagrams the subfields of L; in particular, L is a quadratic extension of K.

Again we redefine our notation: σ is now an element of Gal(L/ℚ) generating the

subgroup fixing ℚ(\sqrt{−3}), and τ is now the (non-trivial) element of Gal(L/ℚ) fixing one of the (∗) fields, so that σ and τ restrict to our previous σ, τ on K. The Galois group Gal(L/ℚ) is of course the dihedral group D_{4} = <σ, τ : σ^{4} = τ^{2} = 1, τσ = σ^{3}τ >.

We are now able to split c(g, h) by defining β(g) as follows:

\begin{matrix}g & 1 & σ & σ^{2} & σ^{3} & τ & στ & σ^{2}τ & σ^{3}τ \\ β(g) & 1 & \sqrt{−1} & −1 & −\sqrt{−1} & \sqrt{−2} & \sqrt{2} & −\sqrt{−2} & −\sqrt{2} \end{matrix}

[figure: ℚ; ℚ(\sqrt{−2}); ℚ(\sqrt{−3}); ℚ(\sqrt{6}); K; L; ∗; ∗; •; •]

Figure 6.3: The subfields of L

Let E be the field generated by the values of β, i.e., E = ℚ(\sqrt{−1}, \sqrt{−2}). Then E

is a Galois extension of ℚ, and we write the elements of Gal(E/ℚ) as follows.

[figure: ℚ; ℚ(\sqrt{−2}); ℚ(\sqrt{−1}); ℚ(\sqrt{2}); E; γ_{−2}; γ_{−1}; γ_{2}]

Now we know from Proposition 4.5 that there is an abelian variety A/ℚ of GL_{2}-

type with End_{ℚ}(A) ≅ E, such that A_{/\overline{ℚ}} is isogenous to B×B×B×B. To compute a specific such A, we will use Weil’s “restriction of scalars” functor. We first remind the reader of the salient features of this construction [We, §1.3].

Let K/k be a finite Galois extension with group G, and let V be an abelian variety

over K. The restriction of scalars Res_{K/k} V from K to k is an abelian variety W over k, together with a morphism p : W_{/K} → V, which represents the functor that takes an abelian variety X over k to the abelian group Hom_{K}(X_{/K}, V). The construction of W shows that W_{/K} is isomorphic to ∏_{g∈G} ^{g}V. Thus, applying the universal property with X = W, we find

End_{k}(W) ≅ Hom_{K}(W_{/K}, V) ≅ Hom_{K}(∏_{g∈G} ^{g}V, V) ≅ ⨁_{g∈G} Hom_{K}(^{g}V, V).

Now we return to the specific ℚ-curve B, which we view as a curve over the

field L. We claim that we may construct A as a quotient of the restriction of scalars R = Res_{L/ℚ} B. Indeed, Ribet [Ri 1, Lemma 6.4] has shown that R = End_{ℚ}(R) is the “twisted group algebra” ℚ[Gal(L/ℚ)]^{c}, with multiplication defined by [g][h] = c(g, h)[gh] for all g, h ∈ Gal(L/ℚ). The algebra R maps onto E via [g] ↦ β(g). Moreover, it is easy to see that the kernel of this homomorphism is the ideal (1 + [σ^{2}]). Now note that \frac{1}{2}(1 + [σ^{2}]) is the central idempotent of the semisimple algebra R generating this ideal. Therefore, \frac{1}{2}(1 − [σ^{2}]) is the complementary central idempotent which projects R onto E. Thus we may take the abelian variety A ⊂ R to be the subvariety that is the image of \frac{1}{2}(1 − [σ^{2}]) ∈ End_{ℚ}(R).

In fact we may describe A more explicitly, by further analysis of the abelian variety

R and its endomorphisms over ℚ. We break down R as Res_{K/ℚ}(Res_{L/K} B), and first consider Res_{L/K} B, which is an abelian variety over K.

Let B′ be the twist of B_{/K} by the quadratic extension L/K. We will continue to

write Gal(L/K) = {1, σ^{2}} as a subgroup of Gal(L/ℚ).

Lemma 6.4 The restriction of scalars Res_{L/K} B is isomorphic over K to B × B′.

Proof. Recall that by definition, B′ is an abelian variety over K which is isomorphic over L (but not over K) to B. Let λ : B → B′ be such an isomorphism over L; then λ^{−1} ^{σ^{2}}λ = −1. Now define p : B × B′ → B by p = (\frac{1}{2}, \frac{1}{2}λ^{−1}). We will show that the pair (B × B′, p) satisfies the universal property of Res_{L/K} B.

Let X be an abelian variety over K, and f : X → B a morphism over L. We

must demonstrate the existence of a unique morphism g : X → B × B′ over K such that p g = f. Note that if g is to be defined over K, we must also have ^{σ^{2}}p g = ^{σ^{2}}f. Now using the definition of p, we are able to produce a unique solution to these two equations: g = (f +^{σ^{2}}f, λ f − λ ^{σ^{2}}f). Thus B × B′ must be isomorphic to Res_{L/K} B. QED

Now on the one hand, End_{K}(B × B′) is plainly equal to End_{K}(B) × End_{K}(B′) ≅

ℚ×ℚ, while on the other hand, we know End_{K}(Res_{L/K} B) is the group algebra ℚ[Gal(L/K)]. Using the isomorphism (p,^{σ^{2}}p) : B × B′ → B ×^{σ^{2}}B from the above proof, we find that the induced isomorphism End_{K}(B)×End_{K}(B′) ⟶^{∼} ℚ[Gal(L/K)] is given by (1, 0) ↦ \frac{1}{2}(1+[σ^{2}]) and (0, 1) ↦ \frac{1}{2}(1 − [σ^{2}]). In other words, \frac{1}{2}(1 + [σ^{2}]) is the projection Res_{L/K} B → B, and \frac{1}{2}(1 − [σ^{2}]) is the projection Res_{L/K} B → B′.

Proposition 6.5 The abelian variety A/ℚ of GL_{2}-type is Res_{K/ℚ} B′, where B′ is the twist of B_{/K} by the quadratic extension L/K.

Proof. By the universal property of Res_{K/ℚ}, there is a map

End_{K}(Res_{L/K} B) → End_{ℚ}(Res_{K/ℚ}(Res_{L/K} B)).

This is, of course, the natural map ℚ[Gal(L/K)] → ℚ[Gal(L/ℚ)]^{c}. Now as we have seen, \frac{1}{2}(1 − [σ^{2}]) is the projection Res_{L/K} B → B′; therefore as an endomorphism of Res_{L/ℚ} B, this element must project Res_{K/ℚ}(Res_{L/K} B) onto Res_{K/ℚ} B′. But we have defined A to be the image of \frac{1}{2}(1 − [σ^{2}]) in Res_{L/ℚ} B. Thus A = Res_{K/ℚ} B′. QED

Remark. It is important to note that the other factor Res_{K/ℚ} B of R is not isogenous over ℚ to A. Indeed, an analysis similar to above shows that over ℚ,

Res_{K/ℚ} B ≅ Res_{ℚ(\sqrt{−3})/ℚ} B × Res_{ℚ(\sqrt{−3})/ℚ} B_{−2}

where B_{−2} is the twist of B/ℚ(\sqrt{−3}) by \sqrt{−2}. Furthermore, recall from our original construction of μ : ^{τ}B → B that B is isogenous over ℚ(\sqrt{−3}) to ^{τ}B_{−2}. Therefore Res_{ℚ(\sqrt{−3})/ℚ} B is isogenous over ℚ to Res_{ℚ(\sqrt{−3})/ℚ} ^{τ}B_{−2} = Res_{ℚ(\sqrt{−3})/ℚ} B_{−2}. Finally, since we know that τB is not isogenous to B over ℚ(\sqrt{−3}), we have End_{ℚ}(Res_{ℚ(\sqrt{−3})/ℚ} B) = ℚ. Thus the endomorphism algebra End_{ℚ}(Res_{K/ℚ} B) is isomorphic to M_{2}(ℚ).

Lastly, we calculate some of the functions we have associated to the abelian variety

A of GL_{2}-type. For example, for each g ∈ G_{ℚ}, the automorphism φ ↦ ^{g}φ on End(A) is given by conjugation by α(g) = β(g)^{−1}. We now use these α(g) to calculate the characters χ_{γ} : G_{ℚ} → E^{∗} defined by χ_{γ}(g) = γ(α(g))/α(g) for γ ∈ Gal(E/ℚ). Then writing σ, τ ∈ Gal(L/ℚ) as above, we find

\begin{aligned}χ_{γ_{−2}}(σ) & = & −1 \\ χ_{γ_{−2}}(τ) & = & 1 \end{aligned}  \begin{aligned}χ_{γ_{−1}}(σ) & = & 1 \\ χ_{γ_{−1}}(τ) & = & −1 \end{aligned}  \begin{aligned}χ_{γ_{2}}(σ) & = & −1 \\ χ_{γ_{2}}(τ) & = & −1 \end{aligned}

Also, we remind the reader that the character ε : G_{ℚ} → E^{∗} is equal to the inverse of χ_{γ_{2}}, because γ_{2} acts as complex conjugation on E. Thus ε(σ) = −1 and ε(τ) = −1.

Chapter 7

Further Questions

In conclusion, we would like to discuss a few unanswered questions and suggest

some avenues of further exploration regarding the topic of abelian varieties of GL_{2}-type and their simple components.

The best choice for β?

One question concerns the construction of abelian varieties of GL_{2}-type from build-

ing blocks. Given a building block B, we form the two-cocycle c(σ, τ) on G_{ℚ} with values in F^{∗}, which we then split as c(σ, τ) = β(σ) · β(τ) · β(στ)^{−1}, where β : G_{ℚ} → E^{∗} ⊂ \overline{F}^{∗} is a locally constant function. The question is: what is the “best” choice for β? And what do we mean by “best”? One possible interpretation is to try to minimize the degree [E : F], which would then minimize the dimension of the abelian variety A of GL_{2}-type that we construct from B. Another “best” choice might be to minimize the order of the character ε attached to A. A related question is: what is the connection between abelian varieties of GL_{2}-type constructed from the same building block but with different choices of β?

Generalizing the base field to K?

We would like to consider the possibility of extending our results from abelian

varieties defined over ℚ to abelian varieties defined over an arbitrary field K of characteristic zero. The only trouble seems to be in making the “right” definitions; most of the actual work appears to generalize easily. One complication that arises over K but not over ℚ is the possibility of quaternionic multiplication over K. It is not hard to show from our results that if B is a building block over ℚ, then End_{ℚ}(B) = End(B) cannot be a quaternion algebra. (There is a more elementary proof of this fact as well, which follows from looking at the action of End_{ℚ}(B) on the Lie algebra Lie(B/ℚ).) If K is an arbitrary field, however, then End_{K}(B) may be a quaternion algebra.

Now suppose that we try to generalize the definition of building block in the

obvious way: a building block is an abelian variety B/\overline{K} such that (i) the endomorphism algebra End(B) is a central division algebra over a totally real field F with Schur index t = 1 or 2, and t[F : ℚ] = dim B; (ii) B is a K-abelian variety; i.e., for each σ ∈ Gal(\overline{K}/K) there exists an isogeny μ_{σ} : ^{σ}B → B that is compatible with End(B).

Suppose K is a field where a building block and all of its endomorphisms are

defined; for instance, consider the case when K is algebraically closed. Then in order to preserve the correspondence between building blocks and abelian varieties of GL_{2}-type, it is necessary to broaden our definition of GL_{2}-type. We need: an abelian variety A/K is of GL_{2}-type if End_{K}(A), the algebra of endomorphisms of A defined over K, is a central division algebra over a number field E with Schur index t = 1 or 2, and t[E:ℚ] = dim A.

These new definitions certainly demonstrate a symmetry that was lacking in the

case K = ℚ. On the other hand, the terminology “of GL_{2}-type” does not make much sense in the case 2[E : ℚ] = dim A. Therefore we might want to preserve the original definition of GL_{2}-type—requiring End_{K}(A) = E and [E : ℚ] = dim A—and change the definition of building block accordingly, which would probably require another restriction on the structure of End(B).

Modularity?

The ultimate goal is to prove that all abelian varieties of GL_{2}-type are modular,

that is, appear as quotients of Jacobians J_{1}(N). Modularity would provide us with a lot more information about the structure of our abelian varieties of GL_{2}-type and their building blocks. As mentioned in the introduction, Ribet has shown that this result is implied by Serre’s conjecture of [Se 4, (3.2.4_{?})], but an in-depth discussion of this issue is beyond the scope of this paper.

Appendix A

A Non-Building Block

Recall that in our definition of a building block B, we have required that B be a

ℚ-abelian variety, which we have defined as follows:

Definition. An abelian variety B/\overline{ℚ} is a ℚ-abelian variety if for each σ ∈ Gal(\overline{ℚ}/ℚ), there exists an isogeny μ_{σ} : ^{σ}B → B compatible with the endomorphisms of B; that is, μ_{σ} ^{σ}φ = φ μ_{σ} for all φ ∈ End(B).

It is natural to ask whether the compatibility conditions on the μ_{σ} are necessary. Given an abelian variety B over \overline{ℚ} that is isogenous to all of its Galois conjugates, is it perhaps always possible to find a collection of isogenies {μ_{σ}} that respect End(B)? We answer this question in the negative.

We will construct a two-dimensional abelian variety A over ℚ such that the full

endomorphism algebra End(A) is isomorphic to ℚ(\sqrt{5}), but not all of the endomorphisms are defined over ℚ. Thus, if σ ∈ G_{ℚ} acts non-trivially on End(A), there is no isogeny μ_{σ} : A = ^{σ}A → A such that φ μ_{σ} = μ_{σ} ^{σ}φ for all φ ∈ End(A).

The abelian variety A will be the Jacobian of a certain hyperelliptic curve of genus

two. A theorem due to Mestre describes how to construct the necessary curve.

Theorem A.1 Let d be a positive integer. Let E be an elliptic curve over \overline{ℚ} equipped with a point S of order 2d + 1 and a point Q such that 2Q does not belong to < S >; let x_{E} be a function of degree two on E with a double pole at O_{E}. Define C to be the hyperelliptic curve of genus d described by the equation

y^{2} = ∏_{i=0}^{2d}(x − x_{E}(Q + iS)).

Then the Jacobian of C admits an endomorphism whose characteristic polynomial is

∏_{m=1}^{d} (X − 2 cos \frac{2mπ}{2d + 1}) .

Proof. [Me, Thm. 1].

We remark that we will apply the theorem in the case d = 2, when in fact it

reduces (via a theorem of Poncelet) to a classical result of Humbert.

Note: All the calculations that follow were done on a SparcStation with the GP/PARI Calculator, version 1.37.

Construction of the Curve C

In order for the equation of the curve C to be defined over ℚ, the set of points

{Q + iS}_{i=0}^{4} must be stable under the action of G_{ℚ}. We thus look for an elliptic curve over ℚ with a rational point Q and a subgroup <S > of order five that is G_{ℚ}-stable. Take E to be the elliptic curve J_{0}(11), given by the equation

E : y^{2} + y = x^{3} − x^{2} − 10x − 20.

The point Q = (16, 60) is easily seen to be a rational point on E; in fact, it has order five, and it generates all the ℚ-rational points on E. Now we need to find the rest of the five-torsion subgroup, E[5]. According to [Si, III.2.3d], the “duplication formula” for points P on E is

x_{E}(2P) = \frac{x4_{P} + 20x2_{P} + 158x_{P} + 21}{4x3_{P} − 4x2_{P} − 40x_{P} − 79} where x_{P} = x_{E}(P).

We use this formula to solve for the points P on E satisfying x_{E}(4P) = x_{E}(−P) = x_{E}(P). We find that the x-coordinates of such points are exactly the roots of the following irreducible polynomials:

x − 16
x − 5
(∗) 5x^{2} + 5x − 29
x^{4} + x^{3} + 11x^{2} + 41x + 101
x^{4} + 15x^{3} + 120x^{2} + 200x + 155
3x^{4} − 4x^{3} − 60x^{2} − 237x − 21.

The last polynomial clearly corresponds to the points of order three on E, and we may safely forget about it; the others correspond to G_{ℚ}-stable sets of points of order five; e.g., (x − 16) corresponds to {±Q}, and (x − 5) to {±2Q}.

Let S be a point of E such that x_{E}(S) is a root of the quadratic polynomial

(∗); hence x_{E}(S) ∈ ℚ(\sqrt{5}). Note that x_{E}(2S) must also lie in ℚ(\sqrt{5}) by the duplication formula, and so x_{E}(2S) must be the other root of (∗). Therefore the points of order five corresponding to (∗) are {±S, ±2S}, and furthermore, these points must be stable under G_{ℚ}. In other words, the subgroup < S > is G_{ℚ}-stable, as desired. (In fact, < S > is the Shimura subgroup of J_{0}(11).)

It is now clear that the monic quartic polynomials above correspond to the sets

{Q + iS}_{i=1}^{4} and {2Q + iS}_{i=1}^{4}, but we need to determine which is which. There are several ways to do this; we choose brute force, and use a computer.

First we claim that S (and hence E[5], which is generated by Q and S) is defined

over ℚ(ζ), where ζ is a primitive fifth root of unity. Indeed, suppose that σ ∈ G_{ℚ} fixes ℚ(ζ). Then, because < S > is stable under G_{ℚ}, we may write ^{σ}S = aS for some integer a (mod 5). Now let e_{5} be the Weil pairing E[5] × E[5] → lu_{5} ⊂ ℚ(ζ). Note that because Q and S generate E[5], e_{5}(Q, S) ≠ 1. Now

e_{5}(Q, S) = ^{σ}e_{5}(Q, S) because σ fixes ℚ(ζ).
= e_{5}(^{σ}Q,^{σ}S) because the Weil pairing is Galois invariant.
= e_{5}(Q, aS) by definition of a.
= e_{5}(Q, S)^{a}.

Thus a must be congruent to 1 (mod 5); i.e., σ must fix S, so S is defined over ℚ(ζ).

Now we may write down explicit coordinates in ℚ(ζ) for S. Take S to be the point

(α, β), where

α = −\frac{11}{5}(ζ + ζ^{−1}) − \frac{8}{5},
β = \frac{11}{5}(2ζ^{3} + ζ^{2} + 3ζ + \frac{14}{11}).

A quick computer calculation supplies us with

∏_{i=0}^{4}(x − x_{E}(Q + iS)) = (x − 16)(x^{4} + x^{3} + 11x^{2} + 41x + 101).

We now define C to be the hyperelliptic curve of genus two described by the equation

C : y^{2} = (x − 16)(x^{4} + x^{3} + 11x^{2} + 41x + 101),

and write A for the Jacobian of C. Then A is a two-dimensional abelian variety over ℚ, and by Theorem A.1, there is an endomorphism of A with characteristic polynomial X^{2} +X −1; thus End(A) ⊇ ℚ(\sqrt{5}).

The Structure of End(A)

The author would like to thank F. Hazama of Tokyo-Denki University for suggest-

ing the approach used in this section.

Let A be the abelian variety constructed in the previous section. It is well known

that if p is a prime of good reduction for A, then the natural map End(A) → End(A_{p}) is injective, where A_{p} is the reduction of A mod p. Indeed, let ℓ be a prime not equal to p; write T_{ℓ}(A) for the Tate module and V_{ℓ}(A) for T_{ℓ}(A) ⊗ ℚ. Then use the facts that the reduction map is injective on ℓ-power torsion [Si, Prop. VII.3.1], that End(A) ↪ End V_{ℓ}(A) [W-M, Thm. 3], and that End(A_{p}) ≅ End V_{ℓ}(A_{p}), because A_{p} is defined over a finite field [W-M, Thm. 6]. The result then follows from the commutative diagram:

[figure: End V_{ℓ}(A); End(A); End(A_{p}); End V_{ℓ}(A_{p}); ≅]

Therefore, in order to study End(A), we will look at End(A_{p}) for various choices of p.

To ease calculation, we search for primes p of good reduction with the properties:

(i) End_{𝔽_{p}}(A_{p}) ≅ ℚ(π_{p}), where π_{p} is the Frobenius endomorphism. (ii) All endomorphisms of A_{p} are defined over 𝔽_{p}.

We will use [W-M, Thm. 8]:

Theorem A.2 Let X be a simple abelian variety over the finite field with q elements. Let π_{q} be the Frobenius endomorphism of X, and let f be the characteristic polynomial of π_{q} acting on T_{ℓ}(X), where ℓ is a prime not dividing q. Then

1. f = g^{e}, where g is a monic irreducible polynomial with integer coefficients.

2. End_{𝔽_{q}}(X) is a division algebra whose center is ℚ(π_{q}).

3. dim_{ℚ} End_{𝔽_{q}}(X) = e^{2}[ℚ(π_{q}):ℚ], and 2 dim X = e[ℚ(π_{q}):ℚ].

Now, in order to prove that a prime p satisfies conditions (i) and (ii), we claim

that it is enough to show that the characteristic polynomial of each power of π_{p} is irreducible of degree four over ℚ. Indeed, the full endomorphism algebra End(A_{p}) is the union ⋃_{r} End_{𝔽_{p^{r}}} (A_{p}), and the Frobenius endomorphism π_{p^{r}} is simply π_{p}^{r}. Therefore, if the characteristic polynomial of π_{p}^{r} is irreducible of degree 4 = 2 dim A_{p} for all r, then by (3) we have End_{𝔽_{p^{r}}} (A_{p}) ≅ ℚ(π_{p}^{r}) for all r. Lastly, we see that ℚ(π_{p}^{r}) = ℚ(π_{p}) for all r, because ℚ(π_{p}^{r}) is certainly contained in ℚ(π_{p}), and by hypothesis both these fields have degree four over ℚ.

To calculate the characteristic polynomial of π_{p}, we recall the Frobenius endo-

morphism π_{C_{p}} of the curve C_{p}, which induces the Frobenius endomorphism π_{p} on A_{p}. (We remind the reader that for any variety over 𝔽_{q} with a projective embedding, the “Frobenius” morphism is simply the map on points (x_{0}: …:x_{n}) ↦ (x_{0}^{q}: …:x_{n}^{q}).) The map π_{C_{p}} has a characteristic polynomial obtained by considering the action of π_{C_{p}} on the étale cohomology group H_{et}^{1}(C_{p}, ℤ_{ℓ}), where ℓ is prime to p. But by [Mil, Cor. 9.6],

H_{et}^{1}(C_{p}, ℤ_{ℓ}) ≅ H_{et}^{1}(A_{p}, ℤ_{ℓ}) ≅ T_{ℓ}(A_{p})^{∨},

where ^{∨} denotes the ℤ_{ℓ}-dual. Hence the characteristic polynomial of π_{C_{p}} is in fact the same as the characteristic polynomial of π_{p} acting on T_{ℓ}(A_{p}), which is what we want. Therefore we will use the curve C_{p} to calculate this polynomial.

Here we must note that the equation defining C is singular at infinity, so as yet

it does not make sense to speak of primes of good reduction for C. However, C has a non-singular model C̃ ⊂ ℙ^{4}. The map C̃ → C is an isomorphism over all affine points of C, and has a unique point over the singularity of C [Ca, §2]. Furthermore, this “point at infinity” of C̃ is a smooth point mod p for each prime p. Hence to study the reduction of C̃ at p, it suffices to consider the reduction mod p of our affine equation for C.

First, we calculate the discriminant of the fifth degree polynomial appearing in

the equation for C; it is 5^{5} · 11^{14}. Thus the primes of bad reduction for this equation are exactly p = 5 or p = 11, and these we will avoid. Now we need [M-S, Lemma 3]:

Lemma A.3 Let C be a curve of genus two over ℚ with good reduction at the prime p. Then the characteristic polynomial of the Frobenius element at p is given by

X^{4} − tX^{3} + sX^{2} − ptX + p^{2}, (A.3)

where \begin{aligned}t & = & p + 1 − N_{1}, \\ s & = & \frac{1}{2}(N_{1}^{2} + N_{2}) + p − N_{1} − pN_{1}, \end{aligned} \begin{aligned}N_{1} & = & #C_{p}(𝔽_{p}), \\ N_{2} & = & #C_{p}(𝔽_{p^{2}}). \end{aligned}

Abusing notation, we write π_{p} for a root of (A.3). Recall that we want all powers

of π_{p} to generate a field of degree four over ℚ. In particular, we must have t ≠ 0 for this situation to occur (since otherwise π_{p}^{2} is quadratic over ℚ). A computer search now finds that p = 19 gives N_{1} = 16 and N_{2} = 390, so (A.3) becomes

X^{4} − 4X^{3} + 22X^{2} − 76X + 361, (A.4)

which is irreducible over ℚ. Note that over ℚ(\sqrt{5}) the polynomial (A.4) factors as

(X^{2} + 4εX + 19)(X^{2} − (4 + 4ε)X + 19) where ε = (−1 + \sqrt{5})/2.

To fix ideas, let π_{19} be a root of the first factor and let K = ℚ(π_{19}). The field K contains the complex conjugate \overline{π}_{19}, but is not Galois over ℚ; it is straightforward to check that the other quadratic polynomial defines a different extension of ℚ(\sqrt{5}). As a consequence, we see that K has only one non-trivial subfield, namely ℚ(\sqrt{5}). (Indeed, if K contained another quadratic subfield, it would be biquadratic over ℚ, and hence Galois.)

We now claim that all powers of π_{19} have degree four over ℚ. Suppose this were

not the case. Then π_{19}^{n} would lie in ℚ(\sqrt{5}) for some n, and we would have π_{19}^{n} = \overline{π}_{19}^{n}. But then π_{19}/\overline{π}_{19} would be a root of unity in K, and clearly the only roots of unity in K are ±1. But \overline{π}_{19} cannot equal ±π_{19}, since by definition of π_{19} above, π_{19}\overline{π}_{19} = 19 and π_{19} + \overline{π}_{19} = −4ε. Therefore π_{19}^{n} generates ℚ(π_{19}) over ℚ.

We have now shown that End(A_{19}) = End_{𝔽_{19}}(A_{19}) ≅ ℚ(π_{19}).

The next prime we investigate is p = 59. In this case N_{1} = 76 and N_{2} = 3550, so

the characteristic polynomial of Frobenius is

X^{4} + 16X^{3} + 162X^{2} + 944X + 59^{2}. (A.5)

The polynomial (A.5) is irreducible over ℚ and factors over ℚ(\sqrt{5}) as

(X^{2} + (6 − 4ε)X + 59)(X^{2} + (10 − 4ε)X + 59).

We let π_{59} be a root of the first factor and then exactly the same analysis as the case p = 19 demonstrates that End(A_{59}) = End_{𝔽_{59}}(A_{59}) ≅ ℚ(π_{59}), which is again a non-Galois extension of ℚ of degree four whose only non-trivial subfield is ℚ(\sqrt{5}).

Finally, recall that End(A) injects into both End(A_{19}) and End(A_{59}), and that

End(A) ⊇ ℚ(\sqrt{5}). Hence to show End(A) ≅ ℚ(\sqrt{5}), it suffices to show that ℚ(π_{19}) and ℚ(π_{59}) are distinct number fields. But this follows immediately from the calculation of their discriminants over ℚ, which are 5^{2} · 149 and 2^{6} · 5^{2} · 281, respectively.

To complete our analysis of the structure of End(A), it remains to show that not

all endomorphisms of A are defined over ℚ. Now as above, for any prime p of good reduction we have End_{ℚ}(A) ↪ End_{𝔽_{p}}(A_{p}). But we are now interested in primes p such that not all endomorphisms of A_{p} are defined over 𝔽_{p}. For instance, we take p = 3 and calculate that N_{1} = 4 and N_{2} = 10. Therefore the characteristic polynomial of the Frobenius element at 3 is X^{4} + 9, which is irreducible over ℚ, and so End_{𝔽_{3}}(A_{3}) ≅ ℚ(π_{3}). But ℚ(π_{3}) is the field ℚ(\sqrt{−1}, \sqrt{6}), which clearly does not contain ℚ(\sqrt{5}) as a subfield. Hence End_{ℚ}(A) cannot be isomorphic to ℚ(\sqrt{5}), and we must have End_{ℚ}(A) ≅ ℚ.

Now we see that A/ℚ is an abelian variety that has all of the properties of a

building block, except the compatibility conditions on the μ_{σ}. Of course, a building block over ℚ is exactly an abelian variety of GL_{2}-type, and we can also see now that A is not of GL_{2}-type, since [End_{ℚ}(A) : ℚ] = 1 ≠ 2 = dim A. This demonstrates the importance of requiring in our definition of building block that all of the isogenies μ_{σ} : ^{σ}B → B respect End(B).

Appendix B

Useful Facts Concerning Central

Simple Algebras

The endomorphism algebra of an abelian variety is a finite-dimensional semi-simple

algebra over ℚ. In this dissertation, we are concerned mostly with abelian varieties whose endomorphism algebras are in fact simple ℚ-algebras. We list here a collection of definitions and facts describing the structure of these algebras. Notation

Z(a ) = {x ∈ a : xa = ax ∀a ∈ a } will denote the center of an algebra a .

Ca(B) = {x ∈ a : xb = bx ∀b ∈ B} will denote the centralizer of a subalgebra B ⊆ a .

Let a be a finite-dimensional simple ℚ-algebra.

• The center Z(a ) is a number field F [Pi, Prop. 12.1]. The algebra a is then a central simple algebra over F.

• a has a unique (up to isomorphism) simple right module M [Pi, Prop. 3.3b], and as right a -modules, a ≅ ⨁^{n} M for some n ≥ 1.

• (Wedderburn Structure Theorem.) As F -algebras, a ≅ M_{n}(D), where D = Enda M is a central division algebra over F [Pi, Cor. 3.5a].

• The dimension dim_{F} a is a square [Pi, Prop. 13.1].

• The Schur index of a is \sqrt{dim_{F} D}.

Note that the Schur index of a depends only on the class of a in Br(F).

• (Hasse-Brauer-Noether Theorem.) The Schur index of a is equal to the order of the class of a in the Brauer group Br(F) [Pi, Thm. 18.6]. (Here it is necessary that F be a number field.)

• (Skolem-Noether Theorem.) Let B be a simple F -algebra. If f, g : B → a are homomorphisms, then there is an inner automorphism α : a → a such that α f = g. In particular, any automorphism of a is inner [Pi, Thm. 12.6].

• (Double Centralizer Theorem.) If B is a simple subalgebra of a , then [Pi, Thm. 12.7] (i) Ca(B) is simple. (ii) (dim_{F} B)(dim_{F} Ca(B)) = dim_{F} a . (iii) Ca(Ca(B)) = B.

• A subfield of a is a subalgebra E of a such that E is a field.

A maximal subfield of a is a subfield which is maximal with respect to inclusion.

• The following are equivalent for a subfield E ⊆ a [Pi, Prop. 13.1].

(Here it is necessary that F be a number field.)

(i) E is a maximal subfield of a .

(ii) [E:F] = \sqrt{dim_{F} a} .

(iii) Ca(E) = E.

• A maximal subfield E ⊆ a splits a ; i.e., E ⊗_{F} a ≅ M_{nt}(E) [Pi, Cor. 13.3].

The following diagram summarizes some of the main points.

[figure: ℚ; Z(a ) = F; maximal subfield E; D = division algebra; a ≅ M_{n}(D); nt; t^{2}, where t = Schur index of a; nt; n^{2}]

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