ABELIAN VARIETIES OVER Q AND MODULAR FORMS

Kenneth A. Ribet

University of California, Berkeley

1. Introduction.

Let C be an elliptic curve over Q. Let N be the conductor of C. The Taniyama conjecture asserts that there is a non-constant map of algebraic curves X_{o}(N) → C which is defined over Q. Here, X_{o}(N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E having order N.

The Taniyama conjecture may be reformulated in various ways. For example, let X_{1}(M) be the modular curve associated with the problem of classifying elliptic curves E along with a point of order M on E. One knows that if there is a nonconstant map X_{1}(M) → C over Q for some M ≥ 1, then there is a non-constant map X_{o}(N) → C. (For a result in this direction, see [11].)

In a recent article [12], B. Mazur introduced another type of reformulation. Mazur says that C possesses a “hyperbolic uniformization of arithmetic type” if there is a non-constant map over the complex field π: X_{1}(M)_{C} → C_{C} for some M ≥ 1. The map π is not required to be defined over Q. In [12, Appendix], Mazur proves: Suppose that there is a non-constant map π: X_{1}(M)_{C} → C_{C}. Then there is a non-constant map π′: X_{1}(M′) → C over Q, where M′ is a suitable positive integer (which may be different from M). As mentioned above, it follows easily from the existence of π′ that C satisfies Taniyama’s conjecture. We thus arrive at the following conclusion:

(1.1) Theorem [Mazur]. Let C be an elliptic curve over Q. Then C satisfies Taniyama’s conjecture if and only if C admits a hyperbolic uniformization of arithmetic type over C.

In connection with [12], Serre asked for a conjectural characterization of elliptic curves over C which admit a hyperbolic uniformization of arithmetic type. All such curves are defined over \overline{Q}, the algebraic closure of Q in C. Thus one wishes to characterize those elliptic curves over \overline{Q} which are quotients of some modular

Typeset by AMS-TEX

This manuscript was written in conjunction with the author’s lectures at the seventh KAIST mathematics workshop in Taejon, Korea (August 11–14, 1992). The author describes work which was begun during his visits to the Hebrew University of Jerusalem and the Université de Paris XI (Orsay). It is a pleasure to thank these institutions for their hospitality. The author wishes to thank J-P. Serre for suggesting the problem and for pointing out the relevance of Tate’s theorem (Theorem 6.3 below.) The author was supported in part by NSF Grant #DMS 88-06815.

curve X_{1}(M). Equivalently, one wishes to study the set of elliptic curves which are quotients of the Jacobian J_{1}(N)\overline{Q} of X_{1}(N)\overline{Q}, for some N ≥ 1. It is well known that all complex multiplication elliptic curves have this property; this follows from [26, Th. 1]. Hence we are principally interested in elliptic curves C/\overline{Q} without complex multiplication.

In this article, we introduce the concept of an abelian variety over Q which is of “GL_{2}-type.” Roughly speaking, this is an abelian variety over Q whose algebra of Q-endomorphisms is a number field of degree equal to the dimension of the abelian variety. It is easy to see that J_{1}(N) decomposes up to isogeny over Q as a product of such abelian varieties. Hence any C/\overline{Q} which is a quotient of some J_{1}(N) is a quotient (over \overline{Q}) of an abelian variety of GL_{2}-type over Q.

We prove in this article the following facts (the third is a simple corollary of the first two):

1. Let C be an elliptic curve over \overline{Q} without complex multiplication. Then C is a quotient of an abelian variety of GL_{2}-type over Q if and only if C is isogenous to each of its Galois conjugates ^{σ}C with σ ∈ Gal(\overline{Q}/Q). (If C has this latter property, we say that C is a “Q-curve.” The terminology is borrowed from Gross [7].)

2. Assume Serre’s conjecture [24, (3.2.4_{?})] on representations of Gal(\overline{Q}/Q). Then every abelian variety of GL_{2}-type over Q is a quotient of J_{1}(N) for some N ≥ 1.

3. Let C/\overline{Q} be an elliptic curve without complex multiplication. If C is a quotient of J_{1}(N)\overline{Q} for some N, then C is a Q-curve. Conversely, suppose that C is a Qcurve. Then if the conjecture [24, (3.2.4_{?})] is correct, C is a quotient of J_{1}(N)\overline{Q} for some N.

In summary, we arrive at a conjectural characterization of elliptic curves over \overline{Q} which are quotients of modular curves X_{1}(N): they are exactly the Q-curves in the sense indicated above. This characterization was predicted by Serre.

2. Abelian varieties over Q of GL_{2}-type.

We will be concerned with abelian varieties over Q which admit actions of number fields that are “as large as possible.” We first quantify this concept.

Suppose that A is an abelian variety over Q and that E is a number field acting on A up to isogeny over Q:

E ↪ Q ⊗_{Z} End_{Q}(A).

By functoriality, E acts on the space of tangent vectors Lie(A/Q), which is a Qvector space of dimension dim A. (For an account of the Lie algebra attached to a group scheme over a field, see for example [14, §11].) The dimension of this vector space is therefore a multiple of [E: Q], so that we have [E: Q] | dim A. In particular, [E: Q] ≤ dim A.

This observation motivates the study of abelian varieties A/Q whose endomorphism algebras contain number fields of maximal dimension dim A. If E is a number field of degree dim A which is contained in Q ⊗ End_{Q}(A), then the Tate modules V_{ℓ}(A) associated with A are free of rank two over E ⊗_{Q} Q_{ℓ}. Accordingly, the action of Gal(\overline{Q}/Q) on V_{ℓ}(A) defines a representation with values in GL(2, E ⊗ Q_{ℓ}). We say that A is of “GL_{2}-type.”

Suppose that B is of GL_{2}-type, and that F ⊆ Q ⊗ End_{Q}(B) is a number field of degree dim B. Let E be a number field containing F, and let n = [E: F]. After choosing a basis for E over F, we find an embedding E ⊆ M(n, F) of E into the ring of n by n matrices over F. Since M(n, F) acts naturally up to isogeny on A := B × ⋯ × B (n factors), we obtain an embedding E ⊆ Q ⊗ End_{Q}(A). Hence A is again of GL_{2}-type. (We can summarize the situation by writing A = E ⊗_{F} B.)

We say that an abelian variety A/Q of GL_{2}-type is primitive if it is not isogenous over Q to an abelian variety obtained by this matrix construction, relative to an extension E/F of degree n > 1 (cf. [31, §8.2]).

(2.1) Theorem. Let A be an abelian variety of GL_{2}-type over Q. Then the following conditions are equivalent: (i) A is primitive; (ii) A/Q is simple; (iii) the endomorphism algebra of A/Q is a number field whose degree coincides with the dimension of A.

Proof. Let E be a number field of degree dim A which is contained in the Q-algebra X := Q ⊗_{Z} End_{Q}(A). Let D be the commutant of E in X. We claim that D is a division algebra (cf. [16, Th. 2.3]).

To prove the claim, we must show that each non-zero Q-endomorphism of A which commutes with E is an isogeny. Let λ be such an endomorphism and let B be the image of λ. Then B is a non-zero abelian subvariety of A. The field E operates on B (up to isogeny), and thereby acts by functoriality on the Q-vector space Lie(B/Q). The dimension of Lie(B/Q) is accordingly a multiple of [E: Q]; on the other hand, it coincides with dim B. Hence B = A, so that λ is an isogeny.

The Lie algebra Lie(A/Q) may now be viewed as a D-vector space. Because of this, the dimension of Lie(A/Q) is a multiple of dim_{Q}(D). In other words, we have dim(D) | dim(E). This gives the equality D = E; i.e., it shows that E is its own commutant in X.

In particular, the center F of X is a subfield of E. We have then X ≈ M(n, Q), where Q is a division algebra with center F. If Q has dimension t^{2} over F, then nt = [E: F]. This follows from the fact that E is a maximal commutative semisimple subalgebra of X, which in turn follows from the statement that E is its own commutant in X.

The structure of X shows that A is isogenous (over Q) to a product of n copies of an abelian variety B whose algebra of Q-endomorphisms is isomorphic to Q. The same Lie algebra argument we have already used shows that dim_{Q}(Q) | dim(B), so that n · dim_{Q}(Q) | dim(A). This gives the divisibility nt^{2}[F: Q] | [E: Q], which implies nt^{2} | nt. One deduces that t = 1, i.e., that Q = F, and obtains the equality n = [E: F]. Hence the dimension of B coincides with the degree of F over Q. Also, we have Q ⊗ End_{Q}(A) ≈ M(n, F).

In other words, A is obtained from B and F by the construction we outlined above. The equivalence of the three statements in the theorem is now clear: each assertion is equivalent to the equality n = 1. ■

3. ℓ-adic representations attached to primitive abelian varieties over Q of GL_{2}-type.

In what follows, we study primitive abelian varieties of GL_{2}-type over Q. Since we never encounter such abelian varieties which are not primitive in the above sense, we will often drop the word “primitive” and refer simply to abelian varieties over Q of GL_{2}-type.

To motivate the study of these varieties, we first allude to the existence of a large class of examples of (primitive) abelian varieties of GL_{2}-type over Q. Suppose that f = ∑ a_{n}q^{n} is a normalized cuspidal eigenform of weight two on a subgroup of SL(2, Z) of the form Γ_{1}(N). Then Shimura [27, Th. 7.14] associates to f an abelian variety A = A_{f} over Q together with an action on A of the field E = Q(…, a_{n}, …). The variety A_{f} may be constructed as a quotient of J_{1}(N), the Jacobian of the standard modular curve X_{1}(N) [29]. The dimension of A and the degree of E are equal. It is well known (and easy to show) that E is the full algebra of endomorphisms of A which are defined over Q [19, Cor. 4.2]. Thus, A is of GL_{2}-type over Q. For each N ≥ 1, the Jacobian J_{1}(N) is isogenous to a product of abelian varieties of the form A_{f} [19, Prop. 2.3].

In §4, we study the converse problem: Suppose that A/Q is of GL_{2}-type. Is A isogenous to a quotient of J_{1}(N), for some N ≥ 1? We show that an affirmative answer is implied by conjecture (3.2.4_{?}) of Serre’s article [24] on modular representations of Gal(\overline{Q}/Q).

We now begin the study of Galois representations attached to GL_{2}-type abelian varieties over Q. Suppose that A is such an abelian variety. Let E be the endomorphism algebra of A/Q. Then E is a number field which is either a totally real number field or a “CM field,” since each Q-polarization of A defines a positive involution on E.

Recall that for each prime number ℓ, the Tate module V_{ℓ} = V_{ℓ}(A) is free of rank two over E ⊗_{Q} Q_{ℓ}. For each prime λ | ℓ of E, let E_{λ} be the completion of E at λ, and set V_{λ} := V_{ℓ} ⊗_{E⊗Q_{ℓ}} E_{λ}. Thus V_{λ} is a two-dimensional vector space over E_{λ}, and V_{ℓ} is the direct sum of the V_{λ} with λ | ℓ. The action of Gal(\overline{Q}/Q) on V_{λ} defines a “λ-adic representation” ρ_{λ}. One knows that the collection (ρ_{λ}) (as λ ranges over the set of finite primes of E) forms a strictly compatible system of E-rational representations whose exceptional set is the set of prime numbers at which A has bad reduction. (For background on this material, see [25, §11.10] and perhaps [17, Ch. II].) We will prove some facts about the λ-adic representations ρ_{λ} and their reductions mod λ.

For each λ, let δ_{λ}: Gal(\overline{Q}/Q) → E_{λ}^{∗} be the determinant of ρ_{λ}. The δ_{λ} form a compatible system of E-rational one-dimensional representations of Gal(\overline{Q}/Q). In the case where E is totally real, we have det ρ_{λ} = χ_{ℓ}, where

χ_{ℓ}: Gal(\overline{Q}/Q) → Z_{ℓ}^{∗}

is the ℓ-adic cyclotomic character, and where ℓ is the prime of Q lying below λ [17, Lemma 4.5.1]. This formula must be modified slightly in the case where E is allowed to be a CM-field:

(3.1) Lemma. There is a character of finite order ϵ: Gal(\overline{Q}/Q) → E^{∗} such that δ_{λ} = ϵχ_{ℓ} for each finite prime λ of E. This character is unramified at each prime which is a prime of good reduction for A.

Proof. Since the abelian representations δ_{λ} arise from an abelian variety, they have the Hodge-Tate property. It follows that they are locally algebraic in the sense of [21, Ch. III]; see [21, p. III-50] and [17, Prop. 1.5.3]. One deduces that the family δ_{λ} is associated with an E-valued Grossencharacter of type A_{o} of the field Q [17, p. 761]. Since the type-A_{o} Grossencharacters of Q are just products of Dirichlet characters with powers of the “norm” character, there is an integer n and an Evalued Dirichlet character ϵ so that we have δ_{λ} = ϵχ_{ℓ}^{n} for each λ. Here, ℓ again denotes the residue characteristic of λ. (We will use the association ℓ ↔ λ from time to time without comment.) We have blurred the distinction between characters of Gal(\overline{Q}/Q) of finite order and Dirichlet characters: if ϵ is a Dirichlet character, then its Galois-theoretic avatar takes the value ϵ(p) on a Frobenius element for p in Gal(\overline{Q}/Q).

By the criterion of Néron-Ogg-Shafarevich, ρ_{λ} is ramified at a prime p ≠ ℓ if and only if A has bad reduction at p. Thus, δ_{λ} is unramified at a prime p ≠ ℓ if A has good reduction at p. In other words, if p is a prime of good reduction for A, and if ℓ ≠ p, then ϵχ_{ℓ}^{n} is unramified at p. Since χ_{ℓ} is unramified at p, ϵ is unramified at p.

It remains to check that n = 1. For this, fix a prime number ℓ. It is well known that the action of Gal(\overline{Q}/Q) on det_{Q_{ℓ}}(V_{ℓ}) is given by the character χ_{ℓ}^{dim(A)} = χ_{ℓ}^{[E: Q]}. On the other hand, the determinant of the action of Gal(\overline{Q}/Q) on V_{ℓ} is given by the character

∏_{λ|ℓ} N_{E_{λ}/Q_{ℓ}}(δ_{λ}) = N_{E/Q}(ϵ) · χ_{ℓ}^{n·[E: Q]}.

(Here, N denotes a norm.) Since χ_{ℓ} has infinite order, we deduce that N(ϵ) = 1 and that n = 1. ■

(3.2) Lemma. Each character δ_{λ} is odd in the sense that it takes the value −1 on complex conjugations in Gal(\overline{Q}/Q).

[Since χ_{ℓ} is an odd character, the Lemma may be reformulated as the statement that ϵ(−1) = +1.] For the proof, we use the comparison isomorphism

V_{λ} ≈ H_{1}(A(C), Q) ⊗_{E} E_{λ}

resulting from an embedding \overline{Q} ↪ C. In this view of V_{λ}, the complex conjugation of Gal(\overline{Q}/Q) acts on V_{λ} as F_{∞} ⊗ 1, where F_{∞} is the standard “real Frobenius” on H_{1}(A(C), Q) (cf. [4, §0.2]). In particular, δ_{λ} is odd if and only if we have det F_{∞} = −1, where the determinant is taken relative to the E-linear action of F_{∞} on H_{1}(A(C), Q).

Since F_{∞} is an involution, and H_{1}(A(C), Q) has dimension two, the indicated determinant is +1 if and only if F_{∞} acts as a scalar (= ±1) on H_{1}(A(C), Q). To prove that F_{∞} does not act as a scalar, we recall that F_{∞} ⊗ 1 permutes the two subspaces H_{0,1} and H_{1,0} of H_{1}(A(C), Q) ⊗_{Q} C in the Hodge decomposition of H_{1}(A(C), Q) ⊗_{Q} C. ■

(3.3) Proposition. For each λ, ρ_{λ} is an absolutely irreducible two dimensional representation of Gal(\overline{Q}/Q) over E_{λ}. We have End_{Q_{ℓ}[Gal(\overline{Q}/Q)]} V_{λ} = E_{λ}.

Proof. Faltings [5] proved that V_{ℓ} is a semisimple Q_{ℓ}[Gal(\overline{Q}/Q)]-module whose commutant is E ⊗ Q_{ℓ}. (This is the Tate conjecture for endomorphisms of A.)

Since V_{ℓ} is the product of the V_{λ} with λ | ℓ, each module V_{λ} is semisimple over Q_{ℓ}[Gal(\overline{Q}/Q)] and satisfies End_{Q_{ℓ}[Gal(\overline{Q}/Q)]} V_{λ} = E_{λ}. This implies that V_{λ} is simple over E_{λ}, and that End_{E_{λ}} V_{λ} = E_{λ}. The absolute irreducibility follows. ■

For each prime p at which A has good reduction, let a_{p} be the element of E such that

a_{p} = tr_{E_{λ}}(Frob_{p} | V_{λ})

whenever ℓ ≠ p. Let denote the canonical involution on E: this involution is the identity if E is totally real, and the “complex conjugation” on E if E is a CM field. The involution is the Rosati involution on E induced by every polarization of A/Q.

(3.4) Proposition. We have a_{p} = \overline{a}_{p}ϵ(p) for each prime p of good reduction.

Proof. Let ℓ be a prime number. Let σ: E ↪ \overline{Q}_{ℓ} be an embedding of fields, and let σ̄ be the conjugate embedding x ↦ σ(\overline{x}). Let V_{σ} = V_{ℓ} ⊗_{E⊗Q_{ℓ}} \overline{Q}_{ℓ}, where the tensor product is taken relative to the map of Q_{ℓ}-algebras E ⊗ Q_{ℓ} → \overline{Q}_{ℓ} induced by σ. Define Vσ̄ similarly, using σ̄.

Fix a polarization of A defined over Q. The associated e_{ℓ^{ν}} -pairings of Weil induce a bilinear map ⟨ , ⟩: V_{ℓ} × V_{ℓ} → Q_{ℓ}(1), where Q_{ℓ}(1) is (as usual) the Q_{ℓ}-adic Tate module attached to the ℓ-power roots of unity in \overline{Q}. We have ⟨ex, y⟩ = ⟨x, \overline{e}x⟩ for e ∈ E and x, y ∈ V_{ℓ}. Further, this pairing is Gal(\overline{Q}/Q)-equivariant in the sense that we have ⟨^{g}x,^{g}y⟩ = ^{g}⟨x, y⟩ for g ∈ Gal(\overline{Q}/Q). After extending scalars from Q_{ℓ} to \overline{Q}_{ℓ}, we find an isomorphism of \overline{Q}_{ℓ}[Gal(\overline{Q}/Q)]-modules

Vσ̄ ≈ Hom(V_{σ}, \overline{Q}_{ℓ}(1)).

Now (3.1) implies that the determinant of V_{σ} is a one-dimensional \overline{Q}_{ℓ}-vector space on which Gal(\overline{Q}/Q) acts by the character ^{σ}ϵχ_{ℓ}. Since V_{σ} is of dimension two, this gives

Hom(V_{σ}, \overline{Q}_{ℓ}(^{σ}ϵχ_{ℓ})) ≈ V_{σ}.

In view of the fact that Hom(V_{σ}, \overline{Q}_{ℓ}(^{σ}ϵχ_{ℓ})) is the twist by ^{σ}ϵ of Hom(V_{σ}, \overline{Q}_{ℓ}(1)), we get V_{σ} ≈ Vσ̄(^{σ}ϵ). For p ≠ ℓ a prime of good reduction, the trace of Frob_{p} acting on V_{σ} is σ(a_{p}), and similarly the trace of Frob_{p} acting on Vσ̄(^{σ}ϵ) is^{σ}ϵ(p)σ̄(a_{p}) = ^{σ}ϵ(p)σ(\overline{a}_{p}). This gives a_{p} = ϵ(p)\overline{a}_{p}, as required. ■

(3.5) Proposition. Let S be a finite set of prime numbers including the set of primes at which A has bad reduction. Then the field E is generated over Q by the a_{p} with p ∉ S.

Proof. The Proposition follows from the Tate Conjecture for endomorphisms of A by a simple argument, which we now sketch (see [17, pp. 788–789] for more details). Choose a prime number ℓ, and observe that End_{Gal(\overline{Q}/Q)} V_{ℓ} = E ⊗ Q_{ℓ} because of Faltings’s results quoted in the proof of (3.3). Let \overline{V}_{ℓ} = V_{ℓ} ⊗ \overline{Q}_{ℓ}; then as a consequence we have End_{\overline{Q}_{ℓ}[Gal(\overline{Q}/Q)]} \overline{V}_{ℓ} = E ⊗ \overline{Q}_{ℓ}. In addition, the semisimplicity of the V_{ℓ} implies that \overline{V}_{ℓ} is semisimple as a \overline{Q}_{ℓ}[Gal(\overline{Q}/Q)]-module. For each σ: E ↪ \overline{Q}_{ℓ}, let V_{σ} be as above. Then the V_{σ} are simple \overline{Q}_{ℓ}[Gal(\overline{Q}/Q)]-modules, and they are pairwise non-isomorphic. Indeed, they are a priori semisimple, but the commutant of their product is ∏_{σ} \overline{Q}_{ℓ}. It follows that their traces are pairwise distinct. Since the trace of Frob_{p} acting on V_{σ} is σ(a_{p}) (for p ∉ S ∪ {ℓ}), the Cebotarev Density Theorem implies that the functions p ↦ σ(a_{p}), p ∉ S ∪ {ℓ} are pairwise distinct. ■

For the next result, fix a finite set S as in (3.5), and let F be the subfield of E generated by the numbers a_{p}^{2}/ϵ(p) with p ∉ S.

(3.6) Proposition. The field F is totally real. The extension E/F is abelian.

Proof. Let again be the canonical complex conjugation of E. For p ∉ S, we have

\frac{\overline{a}_{p}^{2}}{\overline{ϵ}(p)} = \frac{a_{p}^{2}}{ϵ(p)^{2}} ϵ(p)

by (3.4). The first assertion of the Proposition then follows. For the second, let t_{p} = a_{p}^{2}/ϵ(p). It is clear that E is contained in the extension of F (in an algebraic closure of E) obtained by adjoining to F the square roots of all t_{p}, and all roots of unity. This gives the second assertion. ■

We now consider the reductions of the λ-adic representations ρ_{λ}. To do this directly, replace A by an abelian variety which is Q-isogenous to A and which has the property that its ring of Q-endomorphisms is the ring of integers O of E. (This process does not change isomorphism classes of the λ-adic representations ρ_{λ}.) Write simply A for this new abelian variety. For each λ, consider the kernel A[λ]: this is the group of Q-valued points of A which are killed by all elements of the maximal ideal λ of O. The action of O on A[λ] makes A[λ] into a two-dimensional vector space over the residue field F_{λ} of λ. The F_{λ}-linear action of Gal(\overline{Q}/Q) on A[λ] defines a reduction \overline{ρ}_{λ} of ρ_{λ} (cf. [17, II.2]). (There should be no confusion with the involution which appears above.)

(3.7) Lemma. For all but finitely many λ, the representation \overline{ρ}_{λ} is absolutely irreducible.

A result of Faltings [6, Theorem 1, page 204] implies that the following holds for almost all λ: A[λ] is a semisimple F_{ℓ}[Gal(\overline{Q}/Q)] module whose commuting algebra is F_{λ}. The Lemma follows directly from this statement. ■

4. Conjectural connection with modular forms.

We continue the discussion of §3, focusing on the possibility of linking A with modular forms, at least conjecturally. According to (3.7), \overline{ρ}_{λ} is absolutely irreducible for almost all λ. For each λ such that \overline{ρ}_{λ} is absolutely irreducible, conjectures of Serre [24, (3.2.3_{?}–3.2.4_{?})] state that \overline{ρ}_{λ} is “modular” in the sense that it arises from the space of mod ℓ cusp forms of a specific level N_{λ}, weight k_{λ}, and character ϵ_{λ}.

These invariants are essentially constant as functions of λ. Rather than study them for all λ, we will restrict attention to those maximal ideals λ which are odd, prime to the conductor of A, are unramified in E, and have degree one. Let Λ be the set of such ideals λ with the property that \overline{ρ}_{λ} is absolutely irreducible. Lemma 3.7 implies that Λ is an infinite set.

(4.1) Lemma. The levels N_{λ} are bounded as λ varies in Λ.

Proof. It follows from the definition of N_{λ} that this level divides the conductor of the two-dimensional ℓ-adic representation ρ_{λ}. (To compare the two conductors, one can use the Hilbert Formula of [15, §I].) The conductor of ρ_{λ} divides the conductor of the full ℓ-adic representation V_{ℓ}(A). According to results of A. Grothendieck [8, Cor. 4.6], this latter conductor is independent of ℓ. (It is by definition the conductor of A.) ■

(4.2) Lemma. For all λ ∈ Λ, we have k_{λ} = 2.

Proof. Take λ ∈ Λ. The determinant of \overline{ρ}_{λ} is the reduction mod λ of δ_{λ}. By (3.1), this determinant is the product of the mod ℓ cyclotomic character \overline{χ}_{ℓ} and the reduction mod λ of ϵ. The definition of Λ shows that ℓ is a prime of good reduction for A, so that ϵ is unramified at ℓ (Lemma 3.1). Hence, if I is an inertia group for ℓ in Gal(\overline{Q}/Q), we have det \overline{ρ}_{λ} | I = \overline{χ}_{ℓ}.

Further, suppose that D ⊂ Gal(\overline{Q}/Q) is a decomposition group for ℓ. It is clear that \overline{ρ}_{λ} | D is finite at ℓ [24, p. 189], since A has good reduction at ℓ. Indeed, the kernel of multiplication by ℓ on A_{Q_{ℓ}} extends to a finite flat group scheme G over Z_{ℓ} because of this good reduction [8, Cor. 2.2.9]. The Zariski closure of A[λ] in G then prolongs \overline{ρ}_{λ} to a group scheme of type (ℓ, ℓ) over Z_{ℓ}.

By Proposition 4 of [24, §2.8], we find that k_{λ} = 2. ■

(4.3) Lemma. For all but finitely many λ ∈ Λ, we have ϵ_{λ} = ϵ.

One checks easily that ϵ_{λ} = ϵ whenever ℓ is prime to the order of ϵ. We omit the details, since Lemma 4.3 will not be used below. ■

(4.4) Theorem. Let A be an abelian variety over Q of GL_{2}-type. Assume Serre’s conjecture [24, (3.2.4_{?})] on representations of Gal(\overline{Q}/Q). Then A is isogenous to a Q-simple factor of J_{1}(N), for some N ≥ 1.

Proof. Applying [24, (3.2.4_{?})] to the representations \overline{ρ}_{λ} with λ ∈ Λ, we find that each \overline{ρ}_{λ} arises from a newform of weight k_{λ} = 2 and level dividing N_{λ}. Since the N_{λ}’s are bounded (Lemma 4.1), there are only a finite number of such newforms.

Hence there is a fixed newform f = ∑ a_{n}q^{n} which gives rise to an infinite number of the \overline{ρ}_{λ}’s. Explicitly, we have the following situation. Let R be the ring of integers of the field Q(…, a_{n}, …). For an infinite number of λ ∈ Λ, there is a ring homomorphism φ_{λ}: R → \overline{F}_{λ} mapping a_{p} to tr(\overline{ρ}_{λ}(Frob_{p})) for all but finitely many primes p.

Let N be the level of f, and let A_{f} be the quotient of J_{1}(N) which is associated to f. Let λ be a prime for which there is a φ_{λ} as above. By the Cebotarev Density Theorem, we have

A_{f}[ℓ] ⊗_{R/ℓR} \overline{F}_{λ} ≈ A[λ] ⊗_{F_{λ}} \overline{F}_{λ},

where \overline{F}_{λ} is regarded as an R/ℓR-module via φ_{λ}. Since A_{f}[ℓ]⊗_{R/ℓR} \overline{F}_{λ} is a quotient of A_{f}[ℓ] ⊗_{F_{ℓ}} \overline{F}_{λ}, and since F_{λ} = F_{ℓ}, we have

Hom_{\overline{F}_{λ}[Gal(\overline{Q}/Q)]}(A_{f}[ℓ] ⊗_{F_{ℓ}} \overline{F}_{λ}, A[λ] ⊗_{F_{ℓ}} \overline{F}_{λ}) ≠ 0.

It follows that Hom_{F_{ℓ}[Gal(\overline{Q}/Q)]}(A_{f}[ℓ], A[λ]) ≠ 0.

Hence we have Hom_{F_{ℓ}[Gal(\overline{Q}/Q)]}(A_{f}[ℓ], A[ℓ]) ≠ 0 for an infinite number of prime numbers ℓ. By the theorem of Faltings quoted above, we get Hom_{Q}(A_{f}, A) ≠ 0. Since A is a simple abelian variety over Q, A must be a quotient of A_{f}. Since A_{f} is, in turn, a quotient of J_{1}(N), we deduce that A is a quotient of J_{1}(N). ■

5. Decomposition over \overline{Q}.

Suppose again that A/Q is an abelian variety of GL_{2}-type, and let

E = Q ⊗ End_{Q}(A).

Let X = Q ⊗ End\overline{Q}(A) be the algebra of all endomorphisms of A.

(5.1) Proposition [Shimura]. Suppose that A\overline{Q} has a non-zero abelian subvariety of CM-type. Then A\overline{Q} is isogenous to a power of a CM elliptic curve.

This is Proposition 1.5 in [28]. For a generalization of this result, see Proposition 5.2 of [9]. See also the discussion in §4 of [18].

(5.2) Proposition. Suppose that A\overline{Q} has no non-zero abelian subvariety of CM type. Then the center of X is a subfield F of E. The algebra X is isomorphic either to a matrix ring over F, or else to a ring of matrices over a quaternion division algebra over F.

Proof. We employ the same arguments used to prove (2.1) above and Theorem 2.3 of [16]. Let D be the commutant of E in X. Clearly, D is a division algebra: otherwise, we can make E act on a proper non-zero abelian subvariety of A\overline{Q}, contrary to the hypothesis that no abelian subvariety of A\overline{Q} is of CM type. Also, E is a subfield of D; it is a maximal commutative subfield because A\overline{Q} does not have complex multiplication. Hence E is its own commutant in D. Since D is the commutant of E, we get D = E. In particular, the center of X is contained in E, so that the center is a subfield F of E.

Since X is now a central simple algebra over F, we have X ≈ M(n, D), where D is now a division algebra with center F, and n is a positive integer. Suppose that D has dimension t^{2} over F; then [E: F] = nt since E is a maximal commutative subalgebra of X. Up to isogeny, A\overline{Q} is of the form B^{n}, where the endomorphism algebra of B contains D. Following an idea of J. Tunnell (cf. [20, Th. 1]), we fix an embedding \overline{Q} ↪ C and form the cohomology group H^{1}(B(C), Q). This is a Q-vector space of dimension

2 dim(B) = \frac{2}{n} dim(A) = \frac{2nt}{n}[F: Q]

with a functorial action of D. Hence the Q-dimension t^{2}[F: Q] of D divides the dimension over Q of H^{1}(B(C), Q), which is 2t[F: Q]. Thus t ≤ 2, so that either D = F, or else D is a quaternion division algebra with center F. ■

As in Proposition 3.6, let S be a finite set of primes containing the primes of bad reduction for A. Let F be the center of X. Then we have:

(5.3) Theorem. The field F is generated by the numbers a_{p}^{2}/ϵ(p) with p ∉ S.

In other words, the field F which appears in (3.6) is the same as the field F in (5.2).

To prove (5.3), we let ℓ be a prime which splits completely in E, so that all embeddings E ↪ \overline{Q}_{ℓ} take values in Q_{ℓ}. Choose a finite extension K of Q such that all endomorphisms of A are defined over K, and let H be the corresponding open subgroup of Gal(\overline{Q}/Q). Replacing H be a smaller subgroup of Gal(\overline{Q}/Q) if necessary, we may assume that H is contained in the kernel of ϵ. We have X ⊗ Q_{ℓ} = End_{Q_{ℓ}[H]} V_{ℓ}, by Faltings’s results [5]. The center of X ⊗ Q_{ℓ} is F ⊗ Q_{ℓ}.

The Tate module V_{ℓ} decomposes as a product ∏_{σ} V_{σ}, where σ runs over the set Σ of embeddings σ: E → Q_{ℓ}, and where V_{σ} = V_{ℓ} ⊗_{E⊗Q_{ℓ}} Q_{ℓ}, with Q_{ℓ} being regarded as an E ⊗ Q_{ℓ} module via σ. (Cf. the proof of (3.4).) Each V_{σ} is a simple Q_{ℓ}[H]-module because A has no CM subvariety and because the action of H on V_{ℓ} is semisimple (Faltings). Hence End_{H} V_{σ} = Q_{ℓ} for each σ.

For each prime v of K which is prime to ℓ and the set of bad primes for A, there is a “trace of Frobenius” t_{v} ∈ E associated with v. We have tr(Frob_{v} | V_{σ}) = σ(t_{v}) for each σ. One knows for σ, τ ∈ Σ that V_{σ} and V_{τ} are isomorphic Q_{ℓ}[H]-modules if and only if σ(t_{v}) = τ(t_{v}) for all v, i.e., if and only if σ|_{L} = τ|_{L}, where L = Q(…, t_{v}, …) (cf. [17, §IV.4]). This implies that the center of X ⊗ Q_{ℓ} is L ⊗ Q_{ℓ}. Thus F ⊗ Q_{ℓ} = L ⊗ Q_{ℓ} (equality inside E ⊗ Q_{ℓ}), which implies that F = L.

Suppose now that σ = τ on L, so that V_{σ} ≈ V_{τ} as representations of H. A simple argument shows that there is a character φ: Gal(\overline{Q}/Q) → Q_{ℓ}^{∗} such that V_{σ} ≈ V_{τ} ⊗ φ as Q_{ℓ}[Gal(\overline{Q}/Q)]-modules. The character φ is necessarily unramified at all primes p ≠ ℓ which are primes of good reduction for A. Taking traces, we get σ(a_{p}) = φ(p)τ(a_{p}) for all such primes. By considering determinants, we get σϵ = φ^{2} ·^{τ}ϵ. These two equations show that σ and τ agree on a_{p}^{2}/ϵ(p) for all good primes p ≠ ℓ.

It follows by Galois theory that we have a_{p}^{2}/ϵ(p) ∈ L for all good reduction primes p ≠ ℓ. Hence a_{p}^{2}/ϵ(p) ∈ F for all such p. By varying ℓ we obtain the inclusion Q(…, a_{p}^{2}/ϵ(p), …) ⊆ F, where p runs over the set of all primes of good reduction for A.

To prove the opposite inclusion, we must show that if σ(a_{p}^{2}/ϵ(p)) = τ(a_{p}^{2}/ϵ(p)) for all p, then V_{σ} and V_{τ} are H-isomorphic. By the Cebotarev Density Theorem, the hypothesis implies that the functions on Gal(\overline{Q}/Q) “tr^{2} / det” are the same for V_{σ} and V_{τ}. In particular, we have tr(h|V_{σ}) = ± tr(h|V_{τ}) for all h ∈ H. This equality implies that V_{σ} and V_{τ} become isomorphic after H is replaced by an open subgroup H_{o} of H (cf. [22, p. 324]). Since H was already chosen “sufficiently small,” we find that V_{σ} and V_{τ} are indeed isomorphic as Q_{ℓ}[H]-modules. ■

(5.4) Corollary. The center F of X is a totally real number field. The extension E/F is abelian.

Proof. The Corollary follows from (5.3) and (3.6). ■

Let g be an element of Gal(\overline{Q}/Q), and consider the automorphism x ↦ ^{g}x of X which is induced by g. This automorphism is necessarily inner (Skolem-Noether theorem), and it fixes E; therefore it is given by conjugation by an element α(g) of E^{∗} which is well defined modulo F^{∗}. The map α: Gal(\overline{Q}/Q) → E^{∗}/F^{∗} is a continuous homomorphism (cf. [20, p. 268]). It is a fact that α is unramified at all primes p at which A has semistable reduction [16, Th. 1.1].

(5.5) Theorem. We have α^{2} ≡ ϵ (mod F^{∗}). Moreover, suppose that p is a prime of good reduction for A such that a_{p} ≠ 0. Then α(Frob_{p}) ≡ a_{p} (mod F^{∗}).

Proof. To prove the first assertion, let ℓ be a prime which splits completely in E. We must prove that σ(α^{2}(g)/ϵ(g)) = τ(α^{2}(g)/ϵ(g)) whenever σ and τ are embeddings E ↪ Q_{ℓ} which agree on F.

If σ and τ have this property, then there is a character φ: Gal(\overline{Q}/Q) → Q_{ℓ}^{∗} for which V_{σ} ≈ V_{τ} ⊗ φ (the notation is as in the proof of (5.3)). The Galois group Gal(\overline{Q}/Q) acts on Hom(V_{σ}, V_{τ}) by multiplication by φ(g)^{−1}, but also by conjugation by α(g). Since α(g) acts on V_{σ} and V_{τ} by^{σ}α(g) and^{τ}α(g) (respectively), α(g) acts on Hom(V_{σ}, V_{τ}) by ^{τ}α(g)/^{σ}α(g). Hence, φ(g) = ^{σ}α(g)/^{τ}α(g) in Q_{ℓ}. On the other hand, as remarked during the proof of (5.3), we have ^{σ}ϵ = φ^{2} ·^{τ}ϵ. The two equalities give the required conclusion

σ(α^{2}/ϵ) = τ(α^{2}/ϵ).

For the second assertion, we choose ℓ ≠ p. With the notation as above, φ(p) = σα(Frob_{p})/^{τ}α(Frob_{p}). However, we have σ(a_{p}) = φ(p)τ(a_{p}), as noted during the proof of (5.3). On comparing two formulas for φ(g), we get σ(α(Frob_{p})/a_{p}) = τ(α(Frob_{p})/a_{p}). ■

Remark. It should be easy to prove that the set of p for which a_{p} = 0 has density 0, since the image of Gal(\overline{Q}/Q) in Aut V_{σ} is open in Aut V_{σ}, for any σ.

Let α̃ be a lift of α to a function Gal(\overline{Q}/Q) → E^{∗}. The function

c: (g_{1}, g_{2}) ↦ \frac{α̃(g_{1})α̃(g_{2})}{α̃(g_{1}g_{2})}

is then a 2-cocycle on Gal(\overline{Q}/Q) with values in F^{∗}. (We regard F^{∗} as a Gal(\overline{Q}/Q)- module with trivial action.) The image [c] of c in H^{2}(Gal(\overline{Q}/Q), F^{∗}) is independent of the choice of α̃. Since α^{2} lifts to the character ϵ, it is clear that [c] has order dividing two.

Consider the map

ξ: H^{2}(Gal(\overline{Q}/Q), F^{∗}) ⟶ H^{2}(Gal(\overline{Q}/F), \overline{Q}^{∗}) = Br F

obtained from the restriction map H^{2}(Gal(\overline{Q}/Q), F^{∗}) → H^{2}(Gal(\overline{Q}/F), F^{∗}) and the map H^{2}(Gal(\overline{Q}/F), F^{∗}) → H^{2}(Gal(\overline{Q}/F), \overline{Q}^{∗}) induced by the inclusion of F^{∗} into \overline{Q}^{∗}. (We view \overline{Q}^{∗} as a Gal(\overline{Q}/F)-module in the standard way; i.e., we use the Galois action. The notation “Br F ” indicates the Brauer group of F.)

(5.6) Theorem [Chi]. The class of X in Br F coincides with ξ([c]).

Proof. According to [3, Th. 3.4], the algebra X is isomorphic to a twisted matrix algebra (End_{F} E)(α). Theorem 4.8 of [3] expresses [(End_{F} E)(α)] ∈ Br F as the image of a certain two-cocycle whose values are Jacobi sums. By [20, Prop. 1], the class of this Jacobi-sum cocycle coincides with the class of the two-cocycle

(g, h) ↦ α̃(h)^{g−1} = \frac{^{g}α̃(h)}{α̃(h)}

on Gal(\overline{Q}/F). We wish to compare this with the two-cocycle

(g, h) ↦ \frac{α̃(g)α̃(h)}{α̃(gh)}.

The product of the two is the map

(g, h) ↦ \frac{^{g}α̃(h)α̃(g)}{α̃(gh)},

which is a coboundary. Hence the class of X in Br F is the negative of ξ([c]). Since [c] (and [X]) have order two, we get the Theorem as stated. ■

A final remark about X concerns the isogeny class of the simple factors of A\overline{Q}. We have X ≈ M(n, D) for some positive integer n, where D is a division algebra of dimension one or four over F. Accordingly, A\overline{Q} decomposes up to isogeny as the product of n copies a simple abelian variety B over \overline{Q} whose endomorphism algebra is isomorphic to D. Since A is defined over Q, we have

gB × ⋯ ×^{g}B ∼ ^{g}A\overline{Q} ∼ A\overline{Q} ∼ B × ⋯ × B

for each g ∈ Gal(\overline{Q}/Q). (The sign “∼” indicates an isogeny.) By the uniqueness of decomposition up to isogeny, we have ^{g}B ∼ B. One says that the isogeny class of B is “defined over Q.”

Consider the special case where n = dim A, so that B is of dimension one. The elliptic curve B is then isogenous to each of its conjugates (over \overline{Q}). Borrowing (and bending) a term used by B. H. Gross [7], we say that B is a “Q-curve.”

6. Q-curves as factors of abelian varieties of GL_{2}-type.

We have just proved: Suppose that C is an elliptic curve over \overline{Q} which occurs as a simple factor of a primitive abelian variety of GL_{2}-type over Q with no complex multiplication over \overline{Q}. Then C is a Q-curve; it is, more precisely, a Q-curve with no complex multiplication.

In this §, we prove the converse:

(6.1) Theorem. Suppose that C is an elliptic curve over \overline{Q} with no complex multiplication. Assume that C is isogenous to each of its conjugates ^{g}C with g ∈ Gal(\overline{Q}/Q). Then there is a primitive abelian variety A of GL_{2}-type over Q such that C is a simple factor of A over \overline{Q}.

(6.2) Corollary. Suppose that C is as in (6.1). Assume Serre’s conjecture [24, (3.2.4_{?})] on representations of Gal(\overline{Q}/Q). Then C is a simple factor, over \overline{Q}, of J_{1}(N) for some N ≥ 1.

The Corollary follows directly from (6.1) and (4.4). ■

Proof of (6.1). Let C be as in the statement of the Theorem. We can find a model C_{o} of C over a number field K ⊂ \overline{Q}. We may assume that K/Q is a Galois extension. For each g ∈ Gal(K/Q), C_{o} and ^{g}C_{o} are \overline{Q}-isogenous. Enlarging K if necessary, we may assume that there are isogenies μ_{g}: ^{g}C_{o} → C_{o} defined over K.

The map

c: (g, h) ↦ μ_{g}^{g}μ_{h}μ_{gh}^{−1}

may be regarded as Q^{∗}-valued, since Q ⊗ End_{K}(C_{o}) = Q. A short computation shows that c is a two-cocycle on Gal(K/Q) with values in Q^{∗}. The class of c in H^{2}(Gal(K/Q), Q^{∗}) is independent of the choices of the μ_{g}. By inflation, we may (and will) regard c as a locally constant function on Gal(\overline{Q}/Q).

(6.3) Theorem [Tate]. Let M be the discrete Gal(\overline{Q}/Q)-module \overline{Q}^{∗} with trivial action. Then H^{2}(Gal(\overline{Q}/Q), M) = 0.

This theorem of Tate is proved as Theorem 4 in Serre’s article [23], with \overline{Q} replaced by C. The proof exposed by Serre begins with the observation that

H^{2}(Gal(\overline{Q}/Q), C^{∗}) = H^{2}(Gal(\overline{Q}/Q), W),

where W is the torsion subgroup of C^{∗}, i.e., the group of complex roots of unity. This observation follows from the fact that C^{∗}/W is uniquely divisible. Since Gal(\overline{Q}/Q) acts trivially on C^{∗}, W is Gal(\overline{Q}/Q)-isomorphic to Q/Z. One proves that H^{2}(Gal(\overline{Q}/Q), Q/Z) = 0; in fact, we have H^{2}(Gal(\overline{K}/K), Q/Z) = 0 whenever K is a local or global field [23, §6.5].

Since the quotient of M by its torsion subgroup is uniquely divisible, and since the torsion subgroup of M is isomorphic to Q/Z, we obtain H^{2}(Gal(\overline{Q}/Q), M) = H^{2}(Gal(\overline{Q}/Q), Q/Z) = 0. ■

Because of Tate’s theorem, there is a locally constant function α: Gal(\overline{Q}/Q) → \overline{Q}^{∗} such that we have the identity among functions G × G → \overline{Q}^{∗}

c(g, h) = \frac{α(g)α(h)}{α(gh)}.

After again enlarging K, we may identify α with a function on Gal(K/Q) and regard g and h as elements of this Galois group.

Let E be the extension of Q generated by the values of α. The definition of c shows that we have

c(g, h)^{2} = \frac{deg μ_{g} deg μ_{h}}{deg μ_{gh}} ,

where “deg” denotes the degree of an isogeny between elliptic curves (or, more generally, of a non-zero element of Q ⊗ Hom(C_{1}, C_{2}), where C_{1} and C_{2} are elliptic curves). It follows that the function

ϵ: g ↦ \frac{α^{2}(g)}{deg μ_{g}}

is a Dirichlet character Gal(K/Q) → E^{∗}. Because α^{2} ≡ ϵ (mod Q^{∗}), the field E is an abelian extension of Q (cf. Prop. 3.6).

If one performs the analysis of §3 on the abelian variety A which we construct below, one finds (e.g., in (3.1)) another Dirichlet character called ϵ. It is very likely that the two ϵ’s are equal up to “sign,” i.e., possible inversion (cf. Lemma 7.1 below).

To simplify notation, let us write simply C for the elliptic curve C_{o} over K and C\overline{Q} for the curve originally called C. Let B be the abelian variety Res_{K/Q} C, where “Res” is Weil’s “restriction of scalars” functor [33, 13]. Then B is an abelian variety over Q of dimension [K : Q]. It represents the functor on Q-schemes S ↦ C(S_{K}); in particular, we have Hom_{Q}(X, B) = Hom_{K}(X_{K}, C) whenever X is an abelian variety over Q.

Applying this formula in the special case where X = B, we find End_{Q}(B) = Hom_{K}(B_{K}, C). On the other hand [33, p. 5],

B_{K} = ∏_{σ∈Gal(K/Q)}^{σ}C.

Hence we have

Q ⊗ End_{Q}(B) = ∏_{σ} Q ⊗ Hom_{K}(^{σ}C, C).

Since Q⊗Hom(^{σ}C, C) is the one-dimensional vector space generated by μ_{σ}, we find that R := Q ⊗ End_{Q}(B) may be written as ∏ Q · μ_{σ}. Let λ_{σ} be the element of R which corresponds to μ_{σ}: ^{σ}C → C; then R is a Q-algebra with vector space basis λ_{σ}.

(6.4) Lemma. We have λ_{σ}λ_{τ} = c(σ, τ)λ_{στ} in R for σ, τ ∈ Gal(K/Q).

Proof. Each map λ_{σ} acts on B_{K} = ∏_{g} ^{g}C as a “matrix”: it sends the factor ^{gσ}C to ^{g}C by ^{g}μ_{σ} (cf. [7. §15]). A short computation using the identity

μ_{σ}^{σ}μ_{τ} = c(σ, τ)μ_{στ}

gives the desired formula. ■

The algebra R is thus a “twisted group algebra” Q[Gal(K/Q)] in which we have the multiplication table [σ][τ] = c(σ, τ)[στ]. The obvious map of Q-vector spaces

ω: R → E, λ_{σ} ↦ α(σ)

is in fact a surjective homomorphism of Q-algebras because of (6.4).

In [2], Q ⊗ End_{Q}(B) is studied in an analogous situation where C has complex multiplication, and where K is the Hilbert class field of the field of complex multiplication. (The curve C is then a “Q-curve” in the original sense of the term.) Criteria are given for Q ⊗ End_{Q}(B) to be (i) a commutative algebra, i.e., a product of number fields, and (ii) a product of totally real fields. It might be interesting to formulate similar criteria in our context.

Let T be the abelian variety ∏_{σ} C over K. We write C_{σ} for the copy of C in the σth place, so that T becomes the product ∏ C_{σ}. We have a map R → Q ⊗ End T given as follows: For g ∈ Gal(K/Q), the element λ_{g} = [g] of R acts on T by sending C_{σ} to C_{gσ} by the map (of elliptic curves up to isogeny) “multiplication by c(g, σ).” One checks directly that this is a homomorphism of Q-algebras. The variety T will be interpreted as R ⊗_{Q} C by readers who are fond of such tensor-product constructions.

Let ι: T →^{∼} B_{K} = ∏^{σ}C be the isomorphism of abelian varieties up to isogeny which takes the factor C_{σ} of T to the factor ^{σ^{−1}}C of B_{K}, via the map ^{σ^{−1}}μ_{σ}.

(6.5) Proposition. The map ι is R-equivariant, for the action of R on T just defined and for the structural action of R on B.

Proof. The proof of this Proposition is an uninteresting computation, which is omitted.

(6.6) Corollary. The Lie algebra Lie(B/Q) is a free R-module of rank one.

Proof. The statement to be proved is true if and only if Lie(B_{K}/K) is free of rank one over R⊗_{Q}K. By (6.5), Lie(B_{K}/K) may be identified with R⊗_{Q}Lie(C/K), with R operating trivially on the second factor. Hence Lie(B_{K}/K) is indeed free of rank one over R ⊗_{Q} K. ■

To complete the proof of (6.1), we let A be the abelian variety E ⊗_{R} B, where E is viewed as a R-module via ω. Explicitly, use the fact that R is a semisimple Q-algebra to write R as a direct sum of its quotient E with the kernel of the map ω. Let π ∈ R be the projector onto E, and let A ⊆ B be the image of π, viewed as an endomorphism of A up to isogeny. (In other words, A is the image of m · π, where m is a positive integer chosen so that m · π is a true endomorphism of B.) Then A is an abelian subvariety of B, defined over Q, whose algebra of Q-endomorphisms is E. By (6.6), E acts without multiplicity on Lie(B) and therefore, in particular, without multiplicity on Lie(A). Hence A has dimension equal to [E: Q], which means that A is of GL_{2}-type. (It is clear that A is non-zero because π is non-zero.)

Since B_{K} is isogenous to a product of copies of C, the same holds true for A_{K}. Thus C is a quotient of B_{K}. ■

7. Q-curves over quadratic fields.

Suppose that C is a Q-curve as above and that K is a quadratic field. Let σ be the non-trivial automorphism of K over Q. Then, by hypothesis, there is a K-isogeny μ = μ_{σ}: ^{σ}C → C. We take the identity map for μ_{1}, where “1” is the identity automorphism of K. The cocycle c takes the value 1 on all elements of Gal(K/Q) × Gal(K/Q) other than (σ, σ). Its value on that pair is the non-zero integer m such that μ^{∘}^{σ}μ is multiplication by m on C. The algebra R may be written Q[X]/(X^{2} − m), where X corresponds to the element of R we have been calling [σ].

Let us split c by defining α: Gal(K/Q) → \overline{Q}^{∗} to be the map taking 1 to 1 and σ to a square root of m. The character

θ: g ↦ \frac{α^{2}(g)}{deg μ_{g}}

is then trivial if m is positive and of order two if m is negative. In the case where θ is of order two, it is an isomorphism Gal(K/Q) →^{∼} {±1}.

If m is a perfect square, then we have E = Q in the notation of §6. The abelian variety A is then a model of C over Q.

Assume for the rest of this § that m is not a perfect square. Then R = E is a quadratic number field, and we have B = A in the notation of §6. The field E is real if m is positive and imaginary if m is negative. Thus E is real if and only if θ is trivial.

The λ-adic representations of A define a Dirichlet character ϵ (Lemma 3.1). According to (3.2), ϵ is an even character. Also, ϵ is non-trivial if and only if E is imaginary (3.4). Thus ϵ if non-trivial if and only if θ is non-trivial.

(7.1) Lemma. The characters ϵ and θ are equal.

Proof. By (6.5), A_{K} is K-isogenous to the abelian variety “E⊗C,” i.e., to the product of two copies of C with E acting through a regular representation E ↪ M(2, Q). In particular, the λ-adic representations of A_{K} are just the ℓ-adic representations of C, viewed as taking values in GL(2, E_{λ}) rather than in GL(2, Q_{ℓ}). This implies that the determinants of the ρ_{λ}|_{Gal(\overline{Q}/K)} are the cyclotomic characters χ_{ℓ}. Hence ϵ is trivial on Gal(\overline{Q}/K), and therefore may be identified with a character of the group Gal(K/Q), whose order is two. Since θ is also a character of this latter group, and since the two characters are simultaneously non-trivial, they are equal. ■

As mentioned above, it seems very likely that Lemma 7.1 (quite possibly in the form ϵ = θ^{−1}) generalizes to the situation of §6.

(7.2) Proposition [Serre]. At least one of the two quadratic fields E, K is a real quadratic field.

Proof. We give two proofs, the first of which was communicated to the author by Serre: Assume that K is a complex quadratic field. After we embed K in C, the automorphism σ of K becomes the restriction to K of ¯ , complex conjugation on C. Choose a holomorphic differential ω on C. Then C_{C} may be identified with the curve C/L, with L ⊂ C the period lattice of ω. The curve ^{σ}C becomes the complex conjugate C/L̄ of C. The isogeny μ is induced by the map “multiplication by γ” on C, for some non-zero complex number γ. The integer m may be identified with γγ̄, and is therefore positive. This means that E is a real quadratic field.

Another proof can be given as follows. Assume that E is an imaginary quadratic field. Then ϵ is a non-trivial character of Gal(K/Q). This character is even by (3.2). Therefore K is real. ■

In connection with Proposition 7.2, it is natural to ask whether each of the three possibilities allowed by the Proposition do, in fact, occur. The case where E is imaginary, so that K is real, was treated in detail by Shimura [28, §10], who constructed a family of numerical examples. This case was later studied by Serre [24, p. 208], who pointed out that Theorem 4.4 holds in this context. A situation where E is a real quadratic field and K is an imaginary quadratic field is described by Koike in [10, §1]. In this example, the abelian variety A is associated to a weight-two newform on Γ_{o}(81). One has E = Q(\sqrt{3}) and K = Q(\sqrt{−3}).

To exhibit an example where both E and K are real, we consider the eightdimensional complex vector space S of weight-two cusp forms on Γ_{o}(169). According to [1, Table 5], one can find a (normalized) newform f = ∑ a_{n}q^{n} ∈ S whose coefficients generate a quadratic field E. Moreover, the only two such forms are f and ^{σ}f = ∑^{σ}a_{n}q^{n}, where σ is the non-trivial automorphism of E over Q. Since f is a form with trivial “Nebentypus” character, E is a priori a real quadratic field. In fact, the author has learned from D. Zagier that tables constructed by Cohen and Skoruppa show that E = Q(\sqrt{3}).

Let φ be the quadratic Dirichlet character with conductor 13. Then f ⊗ φ := ∑ φ(n)a_{n}q^{n} is again a normalized newform in S; its coefficients generate the same quadratic field E as the coefficients of f. Hence we have either f = φ ⊗ f, or else σf = φ ⊗ f. The former possibility is excluded by [18, Th. 4.5], since φ is an even character (i.e., since φ corresponds to a real quadratic field). Hence f is a form with an “extra twist” by φ: we have φ ⊗ f = ^{σ}f. Moreover, again by Theorem 4.5 of [18], f is not a form with complex multiplication.

The abelian variety A = A_{f} associated to f satisfies Q⊗End_{Q}(A) = E. According to [19] (see especially Theorem 5.1 of [19]), the full algebra X = Q ⊗ End\overline{Q}(A) of A coincides with its subalgebra Q ⊗ End_{K}(A) = E, where K = Q(\sqrt{13}) is the extension of Q cut out by φ. This algebra is a quaternion algebra over Q. Since K may be embedded in R, a well known theorem of Shimura [30, Th. 0] implies that X is isomorphic to the matrix algebra M(2, Q). (Once we know that E = Q(\sqrt{3}), we can give an alternative proof that X is a matrix algebra which is based on the explicit description of X given in [19].) Hence A becomes isogenous over K to a product C × C, where C is an elliptic curve defined over K. Therefore, we are in the situation described above, with E = Q(\sqrt{3}) and K = Q(\sqrt{13}).

8. Descent of abelian varieties up to isogeny.

Suppose that L/K is a Galois extension of fields and that A is an abelian variety over L. A well-known theorem of Weil [32] states that A has a model over K if and only if there are isomorphisms μ_{σ}: ^{σ}A →^{∼} A (σ ∈ Gal(L/K)) which satisfy the compatibility condition μ_{σ}^{σ}μ_{τ} = μ_{στ} . (8.1)

As an application of the techniques encountered in §6, we will prove an analogous criterion for abelian varieties up to isogeny. Namely, suppose that A is isogenous (over L) to an abelian variety B/L which has a model over K. Then one finds isomorphisms μ_{σ}: ^{σ}A →^{∼} A of abelian varieties up to isogeny over L which satisfy the compatibility condition (8.1). Conversely, one has

(8.2) Theorem. Suppose that there are isomorphisms of abelian varieties up to isogeny over L, μ: ^{σ}A →^{∼} A, which satisfy (8.1). Then there is an abelian variety B over K such that A is L-isogenous to B_{L}.

Proof. We can, and do, assume that L is a finite extension of K. Let X be the abelian variety X = Res_{L/K} A. Recalling the discussion of §6, we find a decomposition

Q ⊗ End_{K}(X) = ∏_{σ∈Gal(L/K)} Q ⊗ Hom_{L}(^{σ}A, A).

The homomorphism μ_{σ} in the “σ^{th} factor” corresponds to an element [σ] of Q ⊗ End_{K}(X). The element [g] operates on ∏^{σ}A as a matrix, sending ^{τg}A to ^{τ}A by ^{τ}μ_{g} for each τ ∈ Gal(L/K).

Because the analogue of c(σ, τ) is 1 in this context, we have simply [σ][τ] = [στ] for σ, τ ∈ Gal(L/K): the identity (8.1) shows that the product of the matrices representing [σ] and [τ] is the matrix representing [στ]. Hence R := Q[Gal(L/K)] operates on X.

An analogue of Proposition 6.5 shows that we have

X_{L} ≈ R ⊗_{Q} A

in the category of abelian varieties over L up to isogeny. To see this explicitly, we let A_{σ} be a copy of A indexed by σ and consider the isomorphism of abelian varieties over L up to isogeny

ι: ∏_{σ} A_{σ} →^{∼} ∏_{σ}^{σ}A = X_{L}

which takes A_{σ} to ^{σ^{−1}}A via the map ^{σ^{−1}}μ_{σ}. Via this isomorphism, the automorphism [σ] of X is transported to the permutation which takes each factor A_{g} of ∏_{g} A_{g} to the factor A_{σg}, via the identity map A → A.

Let B be the image of η := ∑_{σ}[σ], so that B is an abelian subvariety of X which is defined over K. (The sum η need not be a literal endomorphism of X, so that, strictly speaking, one should consider the image of a suitable multiple of η.) The isomorphism ι makes B_{L} correspond to the diagonal image of A in ∏ A_{σ} = A × ⋯ × A. Hence B_{L} is isogenous to A. ■

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UC Mathematics Department, Berkeley, CA 94720 USA

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