Contemporary Mathematics Volume 00, 1997

Fields of definition of abelian varieties with real multiplication

Kenneth A. Ribet

1. Introduction

Let K be a field, and let \overline{K} be a separable closure of K. Let C be an elliptic curve over \overline{K}. For each g in the Galois group G := Gal(\overline{K}/K), let ^{g}C be the elliptic curve obtained by conjugating C by g. One says that C is an elliptic K-curve if all the elliptic curves ^{g}C are \overline{K}-isogenous to C.

Recall that a subfield L of \overline{K} is said to be a (2, …, 2)-extension of K if L is a compositum of a finite number of quadratic extensions of K in \overline{K}. The extension L/K is then Galois, and Gal(L/K) is an elementary abelian 2-group. Recently, N. Elkies proved:

(1.1) Theorem (Elkies, [2]). Let C be an elliptic K-curve over \overline{K} with no complex multiplication. Then C is \overline{K}-isogenous to an elliptic curve defined over a (2, …, 2)-extension of K.

In this article, we present an approach to (1.1) which seems different from that of Elkies. At the same time, we generalize (1.1) to include higher-dimensional analogues of elliptic K-curves with no complex multiplication. These are abelian varieties A over \overline{K} whose endomorphism algebras are totally real fields of dimension dim(A).

For lack of a better term, we borrow the phrase “Hilbert-Blumenthal abelian varieties” to refer to abelian varieties whose endomorphism algebras are totally real fields of maximal dimension. Our use of this expression is a bit unusual. Indeed, in standard parlance, a Hilbert-Blumenthal abelian variety relative to a totally real number field F is an abelian variety A over \overline{K} which is furnished with an action of the ring of integers O of F. One requires that the Lie algebra Lie(A/\overline{K}) be free of rank one over O ⊗ \overline{K}, which acts on Lie(A/\overline{K}) by functoriality; in particular, this requirement forces the dimension of A and the degree of F to be equal. In this article, we insist that End(A) ⊗ Q be equal to (and not bigger than) a totally real

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 AMS Subject Classification: 11F11, 11G10 (primary); 14K10, 14K15. This manuscript is in final form. It was written primarily during the program on L-functions at the Isaac Newton Institute for Mathematical Sciences. It is a pleasure to thank the INIMS for its hospitality. The author wishes to thank N. Elkies as well as W. Raskind and J.-L. Colliot-Thélène for helpful e-mail correspondence. This work was partially supported by NSF Grants DMS 88-06815 and DMS 93-06898.

field of dimension dim(A). On the other hand, we do not require the full ring of integers of this field to act on A.

Suppose that A is a Hilbert-Blumenthal abelian variety in our sense, and let F be the totally real number field End(A) ⊗ Q. We say that A is a K-Hilbert-Blumenthal abelian variety (or “K-HBAV”) if ^{g}A is F -equivariantly isogenous to A for all g ∈ G. The equivariance refers to the evident isomorphism φ ↦ ^{g}φ between the endomorphism algebras of A and of ^{g}A: we demand that there be for each g ∈ Gal(\overline{K}/K) an isogeny μ_{g}: ^{g}A → A which satisfies φ^{∘}μ_{g} = ^{g}φ^{∘}μ_{g} for all φ ∈ F.

(1.2) Theorem. Suppose that A is a K-HBAV. Then A is F -equivariantly isogenous to a Hilbert-Blumenthal abelian variety over a finite (2, …, 2) extension of K.

One motivation for proving Theorem 1.2 is the study of Jacobians of modular curves. Indeed, let f be a weight-two newform on the group Γ_{1}(N), and let X_{f} be the abelian variety associated to f by Shimura’s construction [7, Th. 7.14]. Thus X_{f} is a Q-simple factor of the abelian variety J_{1}(N). If f is a cusp form with complex multiplication, then X_{f} becomes isogenous to a power of a CM elliptic curve over \overline{Q}. In the opposite case, Propositions 2.1–2.2 below show that the \overline{Q}-simple factors of X_{f} are then either Q-HBAVs or quaterionic analogues of Q-HBAVs. (It should be possible to prove a version of Theorem 1.2 in the quaternionic case as well.) The absolute decomposition of X_{f} is controlled by coincidences between the Galois conjugates of f and twists of f by Dirichlet characters (see [4]). From the point of view of [4], one sees that this absolute decomposition is achieved over the abelian extension of Q cut out by the set of Dirichlet characters which intervene. This extension is a (2, …, 2)-extension of Q if f has trivial Nebentypus character, but not in general. Nevertheless, Theorem 1.2 tells us that the absolute “building blocks” of X_{f} are defined over a (2, …, 2)-extension of Q in all cases.

We turn now to a discussion of the proof of Theorem 1.2 and some associated results. First of all, let us indicate how Theorem 1.2 follows immediately from a series of results in §3. As we will see, if A is a K-HBAV, then A defines a class γ in the cohomology group H^{2}(G, F^{∗}) made from locally constant cocycles G×G ⟶ F^{∗} and the trivial action of G on F^{∗}. Proposition 3.1 shows that the class γ represents the obstruction to finding a Hilbert-Blumenthal abelian variety over K which is isogenous over \overline{K} to the given one. Therefore, to prove (1.2) is to show that γ becomes trivial under the cohomological restriction map corresponding to a base extension K ⇝ K′, where K′ is a (2, …, 2)-extension of K.

Now, by Proposition 3.2, γ lies in the subgroup H^{2}(G, F^{∗})[2] of H^{2}(G, F^{∗}) consisting of classes of order at most two. On the other hand, Theorem 3.3 asserts that each element of H^{2}(G, F^{∗})[2] becomes trivial under the restriction map corresponding to a base extension K ⇝ K′ of the desired type. This completes our discussion of (1.2).

Secondly, we wish to highlight a technical point that arises in applying Theorem 3.3 to γ. Namely, let P be the quotient F^{∗}/{±1} so that we have an exact sequence of abelian groups

0 → {±1} → F^{∗} → P → 0.

This sequence is split, since the abelian group P is free (Lemma 3.5). Consequently, H^{2}(G, F^{∗})[2] is a split extension of H^{2}(G, P)[2] by H^{2}(G, {±1}). As will be seen in §3, there is an elementary isomorphism Hom(G, P/P^{2}) →^{∼} H^{2}(G, P)[2]. Hence we have a split exact sequence

0 → H^{2}(G, {±1}) → H^{2}(G, F^{∗})[2] → Hom(G, P/P^{2}) → 0.

Call \overline{γ} the image of γ in Hom(G, P/P^{2}). Then \overline{γ} cuts out a (2, …, 2)-extension of K. This is the extension K_{P} of K in \overline{K} such that Gal(\overline{K}/K_{P}) is the kernel of \overline{γ}. Certainly, any extension of K in \overline{K} which trivializes γ must contain K_{P}.

In the case of elliptic curves (i.e., the case F = Q), Elkies shows that γ is trivialized by the field K_{P}. This point, although admittedly technical, seems very striking to us. The present article may be viewed as an attempt to find a generalization of this phenomenon to the Hilbert-Blumenthal case.

Such a generalization is presented in §4, where we introduce the presumably superfluous requirement that K has characteristic zero. (This hypothesis does not intervene in [2].) We show (in Corollary 4.5 below) that γ is trivialized by K_{P} whenever [F: Q] is odd, and more generally whenever there is an embedding F ↪ \overline{K} for which the degree [F K: K] is odd. It would be interesting to determine whether this hypothesis is necessary.

In the case where [F: Q] is odd, F^{∗} is canonically a product P ×{±1}. Indeed, P may be identified with the subgroup of F^{∗} consisting of elements with positive norm to Q^{∗}. Hence we have canonically

H^{2}(G, F^{∗})[2] = H^{2}(G, {±1}) × Hom(G, P/P^{2}).

This suggests a study of the image γ_{±} of γ in the first factor H^{2}(G, {±1}). One may be tempted to think that γ_{±} is trivial, which would certainly explain the vanishing of γ over K_{P}. Although γ_{±} is trivial for the Q-elliptic curves constructed by Shimura in [8], there seem to be examples where γ_{±} can be non-trivial. One such example was communicated to the author by E. Pyle of Berkeley, California; here, K = Q and A is a Q-elliptic curve.

2. K-Hilbert-Blumenthal abelian varieties

Let K, \overline{K} and G be as above, and let F be a totally real number field.

We consider pairs (A, ι), where A is an abelian variety of dimension [F: Q] over \overline{K}, and where ι is an isomorphism F →^{∼} Q ⊗ End(A); such a pair will be called a Hilbert-Blumenthal abelian variety. As mentioned above, it would be more standard to allow ι to be an injection F ↪ Q ⊗ End(A); however, the more restrictive definition seems to be convenient in what follows. It should be stressed that our Hilbert-Blumenthal abelian varieties are, in particular, non-CM abelian varieties.

Abusing notation, we will generally write A for the pair (A, ι). Moreover, we will frequently view A as an object in the category of abelian varieties up to isogeny over \overline{K}. In this category, isogenies of abelian varieties become isomorphisms, and the endomorphism algebra usually denoted Q⊗End(A) can be written more simply as End(A).

Suppose that g is an element of G. Then ^{g}A admits a natural multiplication ^{g}ι by F, so that ^{g}A is again a Hilbert-Blumenthal abelian variety. As mentioned in §1, we say that A is a K-HBAV if there is an F -equivariant isomorphism μ_{g}: ^{g}A →^{∼} A of abelian varieties up to isogeny for each g ∈ G. Notice that a K-HBAV of dimension one is an elliptic K-curve with no complex multiplication.

To motivate the study of K-HBAVs, we record some facts concerning \overline{Q}-simple factors of abelian varieties over Q with many endomorphisms. We begin with some terminology: an abelian variety A over \overline{Q} is said to be a “fake” Hilbert-Blumenthal abelian variety if its endomorphism algebra is a quaternion division algebra over a totally real field F and if dim(A) = 2 · [F: Q]. Also, an abelian variety C over Q is said to be of GL_{2}-type if Q ⊗ End_{Q}(C) is a number field of degree dim(C). One knows that the Q-simple factors of the Jacobian of a modular curve X_{1}(N) are of GL_{2}-type. Moreover, it is reasonable to conjecture that all abelian varieties of GL_{2}-type over Q are Q-simple factors of the Jacobian of some X_{1}(N), cf. [5, 4.4]. (Such a conjecture may be viewed as a higher-dimensional analogue of the conjecture of Taniyama and Shimura to the effect that all elliptic curves over Q are modular.)

(2.1) Proposition. Suppose that C is an abelian variety over Q of GL_{2}-type. Then C_{/\overline{Q}} is “isotypical”: it is isogenous to a product A × ⋯ × A, where A is a simple abelian variety over \overline{Q}. Further, A must be one of the following: (i) an elliptic curve with complex multiplication; (ii) a Hilbert-Blumenthal abelian variety (for some totally real field F); (iii) a “fake” Hilbert-Blumenthal abelian variety.

Proof. Proposition 2.1 is implicit in the discussion of §5 of [5] (which is based principally on results of G. Shimura). For completeness, we shall deduce the proposition from some results included in [5].

First of all, if C_{/\overline{Q}} contains any non-zero abelian subvariety with complex multiplication, then a result of Shimura (Proposition 1.5 of [8]) implies that C_{/\overline{Q}} is a power of an elliptic curve with complex multiplication; thus, we are in case (i). Assume instead that C_{/\overline{Q}} has no non-zero abelian subvariety with complex multiplication. The number field E := Q ⊗ End_{Q}(C) is then its own commutant in X, the algebra of all endomorphisms of C over \overline{Q}. This implies that the center of X is contained in E: it is therefore a subfield F of E. One shows that F is in fact a totally real number field [5, 5.4]. Since the center of X is a single field (as opposed to a product of several fields), C is isotypical, as we claimed. In fact, if we write X as M(n, D), where D is a division algebra with center F, then C_{/\overline{Q}} is isogenous to the nth power of a simple abelian variety A over \overline{Q} whose endomorphism algebra is D.

Let t be such that t^{2} is the rank of D over F. Then a short calculation, based on the fact that E is a maximal commutative semisimple subalgebra of M(n, D), establishes the formula dim(A) = t·[F: Q]. An argument exploiting the action of D on H^{1}(A(C), Q) shows that t is at most 2. (See the proof of Proposition 5.2 of [5] for these facts.) If t = 1, we are in case (ii), while if t = 2 we are in case (iii). ■

(2.2) Proposition. Suppose that C is an abelian variety over Q of GL_{2}-type. Let A be a simple Q-quotient of C whose endomorphism algebra is a totally real field. Then A is a Q-HBAV.

Proof. From what we have seen, the abelian variety C_{/\overline{Q}} is isogenous to a product A×⋯×A (with, say, n factors), and we are in case (ii). Using the analysis of [5], §5 again, we see that the center of the endomorphism algebra X of C_{/\overline{Q}} is a subalgebra of the algebra of Q-endomorphisms of C. If this center is F, then X is isomorphic to M(n, F), and the endomorphism algebra of A is F. After fixing an isomorphism X ≈ M(n, F), we may view A as the image of the matrix whose upper left-hand corner entry is 1 and whose other entries are 0. In this model, there is an obvious F -equivariant isogeny λ: C\overline{Q} → A^{n}, given by the n different matrices with a single 1 in the first column and 0’s elsewhere.

Let g be an element of G and take d ∈ F. Then we have a commutative diagram

A^{n} ←^{λ} C = ^{g}C →^{^{g}λ} ^{g}A^{n}
↓ d ↓ d ↓ ^{g}d ↓ ^{g}d
A^{n} ←^{λ} C = ^{g}C →^{^{g}λ} ^{g}A^{n}

in which the central square expresses the fact that d is defined over Q. Contracting it, we get a diagram

\begin{aligned}A^{n} & →^{∼} & ^{g}A^{n} \\ ↓ d & & ↓ ^{g}d \\ A^{n} & →^{∼} & ^{g}A^{n} \end{aligned}

in which the isomorphism A^{n} →^{∼} ^{g}A^{n} is ^{g}λ^{∘}λ^{−1}.

For each pair of integers (i, j) with 1 ≤ i, j ≤ n, we get a map A → ^{g}A by composing the following maps: the inclusion A ↪ A^{n} which uses the ith coordinate, the isomorphism A^{n} ≈ ^{g}A^{n}, and the projection ^{g}A^{n} → ^{g}A which uses the jth coordinate. For some pair (i, j), this map is non-zero, hence an isogeny. (Recall that A is simple.) Let κ be this isogeny. Then we have

\begin{matrix}A & →^{κ} & ^{g}A \\ ↓ d & & ↓ ^{g}d \\ A & →^{κ} & ^{g}A \end{matrix}

as desired. ■

3. The class γ

Let A be a K-HBAV. For each g, let μ_{g} be an F -equivariant isomorphism up to isogeny ^{g}A → A. We can, and do, assume that the collection (μ_{g}) has been constructed from a model A_{o} of A over a finite extension L of K in such a way that μ_{g} and μ_{g′} are the same map ^{g}A → A whenever g and g′ coincide on L. Thus the association g ↦ μ_{g} is in an obvious sense locally constant. For each pair σ, τ ∈ G, we note that μ_{σ}^{∘}^{σ}μ_{τ}^{∘}μ_{στ}^{−1} is an automorphism of A up to isogeny, and consequently of the form ι(c(σ, τ)) with c(σ, τ) ∈ F^{∗}. The map (σ, τ) ↦ c(σ, τ) is a continuous two-cocycle on G with values in F^{∗}, with F^{∗} being regarded as a trivial G-module. The image γ of c in H^{2}(G, F^{∗}) is independent of the choices of the μ_{σ}.

The following proposition is a mild generalization of [5, Th. 8.2].

(3.1) Proposition. Let E be an extension of K in \overline{K}; let H = Gal(\overline{K}/E) be the corresponding closed subgroup of G. Then the Hilbert-Blumenthal abelian variety A is F -equivariantly isogenous to a Hilbert-Blumenthal abelian variety over E if and only if the class γ lies in the kernel of the restriction map H^{2}(G, F^{∗}) → H^{2}(H, F^{∗}).

Proof. It suffices to prove the proposition in the case E = K, in which case the assertion to be proved is that γ = 1 if and only if A is (F -equivariantly) isogenous to a Hilbert-Blumenthal abelian variety over K.

Suppose first that there is a Hilbert-Blumenthal abelian variety B over K, together with an isomorphism λ: A →^{∼} B_{/\overline{K}} of Hilbert-Blumenthal abelian varieties up to isogeny over \overline{K}. For each g ∈ G, we obtain an isomorphism ^{g}λ: ^{g}A →^{∼} B. After setting μ_{g} := λ^{−1}^{∘}^{g}λ, we find that c is identically 1, so that γ is trivial.

Conversely, suppose that γ = 1. Let L and A_{o} be as above; to simplify notation, we shall write A for A_{o}. After replacing L by a finite extension of L, we can, and do, assume that the μ_{σ} are defined over L. Further, the hypothesis that γ = 1 means that there is a locally constant function α: G → F^{∗} so that c(σ, τ) = α(σ)α(τ)/α(στ); we enlarge L, if necessary so that α is defined modulo Gal(\overline{K}/L), and so that L is a Galois extension of K. We then replace μ_{g} by (1/α(g)) · μ_{g} for each g ∈ G. The new μ’s satisfy μ_{στ} = μ_{σ}^{σ}μ_{τ}, and μ_{g} depends only on the image of g in the finite group Δ = Gal(L/K). Finally, μ_{g} may be viewed as an F -equivariant isomorphism up to isogeny ^{g}A →^{∼} A which is defined over L.

Let R be the abelian variety Res_{L/K} A, where “Res” denotes Weil’s “Restriction of scalars” functor. In other words, R represents the functor C ↦ Hom(C_{/L}, A) from the category of abelian varieties up to isogeny over K to the category of Q-vector spaces. Thus R is an abelian variety over K which is furnished with a structural homomorphism (up to isogeny) λ : R_{/L} → A. Given C over K and a homomorphism (of abelian varieties up to isogeny) φ: C_{/L} → A, there is a unique homomorphism θ: C → R over K such that λ^{∘}θ = φ.

One constructs R by considering the product ∏_{g∈Gal(L/K)} ^{g}A and using obvious patching data to descend this product to K. In particular, R has dimension [L: K] · dim(A). From this point of view, the map λ: R_{/L} → A is the projection of ∏_{g∈Gal(L/K)} ^{g}A onto its factor A. Alternatively, given R with its structural map λ, and an element g of Gal(L/K), we can view ^{g}λ as a map R_{/L} → ^{g}A. One shows that the map R_{/L} → ∏_{g∈Gal(L/K)} ^{g}A induced by the family of ^{g}λ is an isomorphism.

By functoriality, F acts on R over K. Thus we have F ⊆ X, where X = End_{K}(R). Further, the universal property satisfied by R makes it easy to compute End_{K}(R) as a Q-vector space. Indeed, we have

End_{K}(R) = Hom_{K}(R, R) = Hom_{L}(R_{/L}, A) = ⊕_{g} Hom_{L}(^{g}A, A) = ⊕_{g} F · μ_{g}.

Let [g] be the element of End_{K}(R) which corresponds to μ_{g} in the gth factor of the direct sum. Then [g] is the unique element in End_{K}(R) such that λ^{∘}[g] = μ_{g}^{∘}^{g}λ. The unicity implies that each [g] commutes with the action of F on R. A short computation, based on the unicity and the formula μ_{στ} = μ_{σ}^{σ}μ_{τ}, shows that [σ][τ] = [στ] for σ, τ ∈ Δ. Also, the [g] commute with F. Hence X is in the end the group algebra F [Δ].

The field F is a direct summand of the algebra X = F [Δ] via the inclusion a ∈ F ↦ a · [1] and the augmentation map [g] ↦ 1. Let B be the subvariety of R corresponding to the direct summand F of X. Then B is an abelian variety over K with an induced action of F. We claim that B is a Hilbert-Blumenthal abelian variety over K, i.e., that dim(B) = dim(A), and that in fact A and B are isomorphic Hilbert-Blumenthal abelian varieties up to isogeny over L. To see this, we remark that R_{/L} is isogenous to a product of copies of A (since the ^{g}A are each isogenous to A), so that B_{/L} is certainly isogenous to a product of some number of copies of A. To determine the dimension of B, we remark that

F = End_{K}(B) = Hom_{K}(B, R) = Hom_{L}(B_{/L}, A).

Since F is the endomorphism algebra of A, B_{/L} is isogenous to A. ■

Suppose now that A is a K-HBAV over \overline{K}. Fix a polarization θ_{A}: A → Aˇ of A as an abelian variety. The associated Rosati involution of the endomorphism algebra of A may be viewed as a positive involution of the totally real field F and therefore is forced to be the identity.

Let B again be a Hilbert-Blumenthal abelian variety over \overline{K}, and fix a polarization θ_{B} of B. Then for each F -equivariant map μ: B → A, we define the “degree” deg(μ) by the formula

deg(μ) = μ^{∘}θ_{B}^{−1}^{∘}μ ^{ˇ ∘}θ_{A},

so that deg(μ) is an element of the endomorphism algebra of A. Identifying this algebra with F, we consider that deg(μ) is an element of F. If \overline{K} is a subfield of C,

then θ_{A} and θ_{B} identify ⋀_{F}^{2} H_{1}(A(C), Q) and ⋀_{F}^{2} H_{1}(B(C), Q) with F. The number

deg(μ) is the element of F describing the map ⋀_{F}^{2} H_{1}(B(C), Q) → ⋀_{F}^{2} H_{1}(A(C), Q)

induced by μ.

If μ: A → A is given by an element c of F, then deg(μ) = c^{2}. Also, it is easy to check that deg is multiplicative in the following sense. Suppose that (C, θ_{C}) is a third polarized Hilbert-Blumenthal abelian variety, and that λ: C → B is F - equivariant. Then deg(μ^{∘}λ) = deg(μ) deg(λ), provided of course that the “degrees” are computed with respect to a fixed set of polarizations.

(3.2) Proposition. Let A be a K-HBAV over \overline{K}. Then the order of the associated cohomology class γ ∈ H^{2}(G, F^{∗}) is at most two.

Proof. Fix a polarization θ: A → Aˇ of A and for each g ∈ G let ^{g}θ denote the associated polarization of ^{g}A. Choose a family (μ_{g}) as above, and let c be the F^{∗}-valued two-cocycle on G defined by this family. For each g, let

d_{g} := μ_{g}^{∘}^{g}θ^{−1}^{∘}μ_{g}^{ ˇ ∘}θ

be the degree of μ_{g}, calculated with respect to the polarizations θ and ^{g}θ. This element of F is canonical in the following sense: if we replace θ by another polarization θ′ of A, then d_{g} will remain unchanged provided that we replace ^{g}θ by ^{g}θ′.

By construction,

c(σ, τ) = μ_{σ}^{∘}^{σ}μ_{τ}^{∘}μ_{στ}^{−1}

for σ, τ ∈ G. On taking degrees, we find the formula

c(σ, τ)^{2} = \frac{d_{σ}d_{τ}}{d_{στ}}

which expresses the square of c as a coboundary. ■

(3.3) Theorem. Let γ be an element of order dividing two in H^{2}(G, F^{∗}). Then there is an open normal subgroup H of G such that G/H is an elementary abelian two-group and such that the image of γ in H^{2}(H, F^{∗}) is trivial. In other words, γ becomes trivial after K is replaced by a finite (2, …, 2)-extension of K.

Proof. Let P = F^{∗}/{±1}, so that there is a tautological exact sequence

0 → {±1} → F^{∗} → P → 0. (3.4)

In the case where [F: Q] is odd, there is a natural splitting of this exact sequence: we may view P as the subgroup of F^{∗} consisting of elements with positive norm to Q^{∗}. In the general case, (3.4) is still split, but apparently in no natural way. This follows directly from:

(3.5) Lemma. The group P is a free abelian group, of countable rank.

To prove the lemma, we consider the map ϕ: a ↦ (a) which takes an element a of P to the fractional ideal of F generated by a lift of a to F^{∗}. The image of ϕ is a subgroup (of finite index) of the group of all fractional ideals of F; this latter group is the free abelian group on the set of non-Archimedean places of F. Hence the image of ϕ is a free abelian group, so that P is abstractly isomorphic to the direct sum of the image of ϕ and the kernel of ϕ. On the other hand, ker(ϕ) is the group U/{±1}, where U is the group of units of F. According to Dirichlet’s theorem, ker(ϕ) is a free abelian group of rank n−1, where n = [F: Q]. It follows that P is the direct sum of two free abelian groups, one finitely generated and one countably generated. This proves the lemma.

Returning to the proof of (3.3), we fix a splitting of (3.4). We shall assume that this is the indicated natural splitting if [F: Q] is odd. The splitting fixes an isomorphism of abelian groups F^{∗} ≈ {±1} × P. This isomorphism induces in turn a decomposition

H^{2}(G, F^{∗})[2] ≈ H^{2}(G, {±1}) × H^{2}(G, P)[2],

where the notation “[2]” indicates the kernel of multiplication by 2. The element γ of H^{2}(G, F^{∗}) is then the product of its two projections γ_{±} ∈ H^{2}(G, {±1}) and \overline{γ} ∈ H^{2}(G, P)[2]. The factor \overline{γ} is independent of the chosen splitting of (3.4): it is the image of γ under the map on cohomology induced by the quotient map F^{∗} → P. On the other hand, the “sign” component γ_{±} of γ depends on the splitting. We thus consider that γ_{±} is defined intrinsically only in the case where [F: Q] is odd.

To prove (3.3), we must show that both γ_{±} and \overline{γ} become trivial when K is replaced by a (2, …, 2)-extension of K.

To treat \overline{γ}, we consider the exact sequence

1 → P ⟶^{x↦x^{2}} P → P/P^{2} → 1;

note that P is torsion free. The associated long exact cohomology sequence then gives an isomorphism

Hom(G, P/P^{2}) →^{∼} H^{2}(G, P)[2].

Suppose that \overline{γ} corresponds to the homomorphism φ: G → P/P^{2}. Then ker(φ) is a subgroup of G which corresponds to a (2, …, 2)-extension K_{P} of K. It is clear that \overline{γ} becomes trivial over this extension.

It remains to split γ_{±}. It is known that H^{2}(G, {±1}) = 0 if K has characteristic 2, cf. [6, p. II-5]. Assume, then, that the characteristic of K is different from 2. The group H^{2}(G, {±1}) may then be identified with Br(K)[2], where Br(K) is the Brauer group of K. A theorem of S. A. Merkur’ev [3] states that Br(K)[2] is generated by the classes of quaternion algebras over K. Since each quaternion algebra over K is split by a quadratic extension of K, it follows that γ_{±} is split by a (2, …, 2)-extension of K, as required. ■

4. The odd-dimensional situation

In this section, we assume that K has characteristic 0. (An assumption concerning the parity of [F: Q] will be made later.)

We begin with some preliminary comments concerning the class γ defined by a K-HBAV. First, we note that the tensor product F ⊗_{Q} K decomposes as some direct sum of fields ⊕_{ω∈Ω} K_{ω}, where the K_{ω} are finite extensions of K. The index set Ω is the set Hom(F, \overline{K}) of field embeddings F ↪ \overline{K}, modulo the action of G on Hom(F, \overline{K}).

Next, suppose that L is a finite extension of K, and consider M := L⊗_{K} \overline{K} as a G-module, with G acting trivially on the first factor. Choose an embedding σ of L into \overline{K}, and let H = Gal(\overline{K}/σ(L)). Then L ⊗_{K} \overline{K} is the induced representation Ind_{H}^{G} \overline{K}. Indeed, M may be written as the group of functions f: Σ → \overline{K}, where Σ is the set of embeddings L ↪ \overline{K}. In this optic, G acts via (g · f)(τ) = g(f(g^{−1}τ)) for g ∈ G and τ ∈ Σ. It is clear that M = ⊕_{τ∈Σ}M_{τ}, where M_{τ} consists of those functions which vanish outside of τ. Further, the subgroups M_{τ} of M are permuted transitively by G, since G acts transitively on Σ. The formula M = Ind_{H}^{G} \overline{K} then follows by a well known criterion (see, e.g., [1, Ch. III, 5.4]).

Finally, we consider F ⊗_{Q} \overline{K} as a G-module, with G acting trivially on the first factor and in the natural way on the second. Then

F ⊗_{Q} \overline{K} = (F ⊗_{Q} K) ⊗_{K} \overline{K} = ⊕_{ω}(K_{ω} ⊗_{K} \overline{K}).

Therefore,

F ⊗_{Q} \overline{K} = ⊕_{ω} Ind_{H_{ω}}^{G} \overline{K}.

In this latter formula, we have chosen for each ω a K-embedding K_{ω} ↪ \overline{K} and have put H_{ω} := Gal(\overline{K}/K_{ω}).

In a similar vein, one proves

(F ⊗_{Q} \overline{K})^{∗} = ⊕_{ω} Ind_{H_{ω}}^{G} \overline{K}^{∗}.

Shapiro’s lemma [1, Ch. III, 6.2] provides an identification

H^{i}(G, Ind_{H_{ω}}^{G} \overline{K}^{∗}) = H^{i}(H_{ω}, \overline{K}^{∗})

for each i ≥ 0. (The induced module Ind_{H_{ω}}^{G} \overline{K}^{∗} may be regarded as a co-induced module because the index of H_{ω} in G is finite.) Thus

H^{i}(G, (F ⊗_{Q} \overline{K})^{∗}) = ⊕_{ω} H^{i}(H_{ω}, \overline{K}^{∗})

for each i. Consequently,

H^{1}(G, F ⊗_{Q} \overline{K})^{∗}) = 0

by Hilbert’s Theorem 90. Similarly,

H^{2}(G, (F ⊗_{Q} \overline{K})^{∗}) = ⊕_{ω} Br(K_{ω}) (4.1)

because H^{2}(H_{ω}, \overline{K}^{∗}) is the Brauer group of K_{ω}.

Now consider the exact sequence of G-modules

0 → F^{∗} → (F ⊗ \overline{K})^{∗} → (F ⊗ \overline{K})^{∗}/F^{∗} → 0, (4.2) where G acts in the usual way on \overline{K} and the indicated tensor products are taken over Q. Exploiting the vanishing of H^{1}(G, (F ⊗ \overline{K})^{∗}), we obtain

0 → H^{1}(G, (F ⊗ \overline{K})^{∗}/F^{∗}) →^{δ} H^{2}(G, F^{∗}) → H^{2}(G, (F ⊗ \overline{K})^{∗}) → · · · .

Here, δ is the indicated connecting homomorphism in the long cohomology sequence arising from (4.2).

(4.3) Lemma. Let A be a K-HBAV. Then the element γ of H^{2}(G, F^{∗}) defined by A lies in the image of δ. Equivalently, the image of γ in H^{2}(G, (F ⊗\overline{K})^{∗}) is zero.

Proof. We must exhibit an element β of H^{1}(G, (F ⊗ \overline{K})^{∗}/F^{∗}) such that γ = δ(β). Let V = Lie(A/\overline{K}). For each g ∈ G, the map μ_{g}: ^{g}A → A induces a (F ⊗\overline{K})-linear homomorphism Lie(^{g}A/\overline{K}) → Lie(A/\overline{K}), or equivalently an F -linear homomorphism λ_{g}: V → V which is g-linear in the sense that it satisfies λ(a · v) = g(a)λ(v) for a ∈ \overline{K} and v ∈ V. Now it is well known, and easy to verify, that the Lie algebra Lie(A/\overline{K}) is free of rank one over F ⊗ \overline{K}. Let v be a basis of V, considered as a free rank-one F ⊗ \overline{K}-module. Then one has λ_{g}(v) = a_{g} · v for some element a_{g} in (F ⊗ \overline{K})^{∗}. The relations among the μ_{g} provide the formula c(σ, τ)a_{στ} = a_{σ}^{σ}a_{τ} for σ, τ ∈ G. It follows that the function G → (F ⊗ \overline{K})^{∗}/F^{∗} induced by g ↦ a_{g} is a 1-cocycle, and that the corresponding class β in H^{1}(G, (F ⊗ \overline{K})^{∗}) maps to γ under δ. ■

(4.4) Proposition. Suppose that γ is the element of H^{2}(G, F^{∗}) arising from a K-HBAV and that \overline{γ} = 0. Then γ = 0 provided that no element of order two in the Brauer group of K is split by all extensions of K of the form K_{ω}.

Proof. In view of the hypothesis \overline{γ} = 0, we have γ = γ_{±} ∈ H^{2}(G, \{±1\}). According to (4.3), γ lies in the kernel of the map

j: H^{2}(G, \{±1\}) → H^{2}(G, (F ⊗ \overline{K})^{∗})

induced by the inclusion of \{±1\} in (F ⊗ \overline{K})^{∗}. This map may be viewed as the natural map

Br(K)[2] → ⊕_{ω} Br(K_{ω}),

which is injective by hypothesis. Thus γ is indeed 0. ■

(4.5) Corollary. Suppose that γ ∈ H^{2}(G, F^{∗}) arises from a K-HBAV. Suppose that [F : Q] is odd. Then the class γ becomes zero after K is replaced by the extension K_{P} of K defined by \overline{γ}. In particular, if \overline{γ} = 0, then γ = 0.

Proof. The two assertions of the corollary are equivalent, since \overline{γ} becomes trivial after K is replaced by K_{P}. Because of (4.4), to prove the second assertion it is enough to prove that the map j which occurs in the proof of (4.4) is injective whenever [F: Q] is odd.

However, [F: Q] = ∑_{ω}[K_{ω}: K]. Thus, if [F: Q] is odd, then there is at least one index ω for which [K_{ω}: K] is odd. Further, if [K_{ω}: K] is odd, then it is evident that the map Br(K)[2] → Br(K_{ω}) is injective since there is a corestriction map cor: Br(K_{ω}) → Br(K) whose composition with the natural map Br(K) → Br(K_{ω}) is multiplication by [K_{ω}: K]. ■

References

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2. N. Elkies, Remarks on elliptic K-curves, Preprint, 1993.

3. A. S. Merkur’ev, On the norm residue symbol of degree 2, Dokl. Akad. Nauk. SSSR 261 (1981), 542–547; Soviet Math. Dokl. 24 (1981), 546–551.

4. K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), 43–62.

5. , Abelian varieties over Q and modular forms, 1992 Proceedings of KAIST Mathematics Workshop, Korea Advanced Institute of Science and Technology, Taejon, 1992, pp. 53–79.

6. J-P. Serre, Cohomologie galoisienne, Lecture Notes in Math., vol. 5, 4^{e} édition, Springer-Verlag, Berlin and New York, 1973.

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UC Mathematics Department, Berkeley, CA 94720 USA

   E-mail address: ribet@math.berkeley.edu