MULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES

BY

KENNETH A. RIBET^{∗}

Mathematics Department, University of California, Berkeley CA 94720 USA

Let p and q be distinct primes. The new part of J_{o}(pq) (defined, according to one’s taste, as a quotient or subvariety of J_{o}(pq)) is known to be isogenous to the Jacobian J of the Shimura curve derived from the rational quaternion algebra of discriminant pq. We show that J and the new subvariety of J_{o}(pq) differ in one significant respect. Namely, certain kernels which are 2-dimensional for the latter variety may be either 2- or 4-dimensional for the former. We give a simple criterion to decide whether the dimension is 2 or 4, and exhibit a procedure for constructing examples where the dimension is 4. We illustrate this procedure by constructing a 4-dimensional kernel with pq = 11 · 193.

To Professor I.I. Piatetski-Shapiro

Introduction

Let p and q be distinct primes. Consider the modular curve X_{o}(pq) over Q. This curve is endowed with a Hecke correspondence T_{ℓ} for each prime number ℓ. (The operators T_{p} and T_{q} are often denoted U_{p} and U_{q}, respectively.) The correspondence T_{ℓ} induces an endomorphism of J_{o}(pq) = Pic^{o}(X_{o}(pq)) which is again denoted T_{ℓ}. Let T = T_{pq} be the (commutative) subring of End(J_{o}(pq)) which is generated by the family of endomorphisms T_{ℓ}.

∗Partially supported by the National Science Foundation

Suppose that m is a maximal ideal of T, and let k = T/m be its residue field. Let

ρ_{m} : Gal(\overline{Q}/Q) → GL(2, k)

be the usual two-dimensional semisimple representation of Gal(\overline{Q}/Q) attached to m ([3], Th. 6.7 or [12], §5). This representation is characterized, up to isomorphism, by the fact that it is unramified at each prime number r different from p, q, and the characteristic of k, together with the condition that the matrix ρ_{m}(Frob_{r}) have trace T_{r} mod m and determinant r mod m, for each such r. (Here Frob_{r} is a Frobenius element for the prime r in Gal(\overline{Q}/Q).)

Let J_{o}(pq)[m] be the kernel of m on J_{o}(pq)(\overline{Q}), i.e., the intersection of the kernels of all elements of m on J_{o}(pq)(\overline{Q}). The following result is a special case of Theorem 5.2(b) of [12].

Theorem 1 Assume that m is prime to 2pq and that the representation ρ_{m} is irreducible. Then J_{o}(pq)[m] is two-dimensional over k, and the k-linear action of Gal(\overline{Q}/Q) on J_{o}(pq)[m] defines a representation equivalent to ρ_{m}.

Let J_{new} be the new subvariety of J_{o}(pq). This abelian variety may be defined as the connected component of the kernel of the natural “trace” map

J_{o}(pq) ⟶ J_{o}(p) × J_{o}(p) × J_{o}(q) × J_{o}(q)

([7], §2(b)) or, alternatively, as the dual of the new quotient of J_{o}(pq) as in [16]. The variety J_{new} is a T-stable subvariety of J_{o}(pq); the image \overline{T} of T in End(J_{new}) is the new quotient of T. We say that m is new if it arises from a maximal ideal of \overline{T} by pullback, relative to the quotient map T → \overline{T}. Equivalently, m is new if and only if the kernel J_{new}[m] of m on J_{new} is non-zero. Using the irreducibility of J_{o}(pq)[m], we deduce

Theorem 2 Suppose that m satisfies the hypothesis to Theorem 1, and that m is new. Then the representation J_{new}[m] is equivalent to ρ_{m}.

Now let O be a maximal order in a rational quaternion algebra of discriminant pq. (Up to isomorphism, there is only one such order.) Let C be the Shimura curve associated with the moduli problem of classifying abelian surfaces with an action of O. Let J = Pic^{o}(C) be the Jacobian of C. As recalled in [12], §4, the curve C carries a family of Hecke correspondences T_{ℓ}, which again lead to commuting endomorphisms T_{ℓ} ∈ End(J). The ring \overline{T} acts faithfully on J in such a way that each Hecke operator T_{ℓ} ∈ \overline{T} acts on J as the corresponding endomorphism T_{ℓ} ([12], Cor. 4.2). Using this action, we define the kernel J[m] of m on J(\overline{Q}) for each new maximal ideal m of T. The group J[m] is non-zero because the action of \overline{T} on J is faithful.

View J[m] as a k[Gal(\overline{Q}/Q)]-module. If ρ_{m} is irreducible, a standard argument of Mazur ([6], II, Prop. 14.2) shows that the semisimplification of J[m] is a direct sum of copies of the simple k[Gal(\overline{Q}/Q)]-module corresponding to ρ_{m}. Call ν the number of summands. The k-dimension of J[m] is then 2ν. Our aim is to calculate the integer ν, which we call the multiplicity of ρ_{m} in J[m].

We achieve this goal whenever m satisfies the hypotheses to Theorem 2, i.e., whenever m is new and prime to 2pq and the representation ρ_{m} is irreducible. To explain our result, we remark that the representation ρ must be ramified at at least one of the primes p, q. Indeed, were ρ_{m} unramified (or “finite”) at both p and q, the main theorem of [12] would show that ρ_{m} is modular of level 1. In particular, ρ_{m} would be realized as a module of division points on the abelian variety J_{o}(1), which is zero. This, clearly, would be a contradiction. Permuting p and q if necessary, we therefore may assume that ρ_{m} is ramified at p. Let us do so. Also, we fix a Frobenius element Frob_{q} for q in Gal(\overline{Q}/Q).

The following Theorem will be proved in §2:

Theorem 3 Assume that m satisfies the hypotheses of Theorem 2. Then the multiplicity of ρ_{m} in J[m] is 1 unless ρ_{m} is unramified at q and the Frobenius element Frob_{q} acts in ρ_{m} as ±1. In this latter case, the multiplicity is 2.

By the main theorem of [12], the hypothesis that ρ_{m} is unramified at q implies that ρ_{m} is modular of level p in the sense that it arises from a maximal ideal of the Hecke algebra T_{p} associated to J_{o}(p). The maximal ideal m is then old as well as new: it is an “ideal of fusion” between old and new at level pq.

Conversely, we can make examples of m for which ρ_{m} occurs with multiplicity 2 via the following construction, in which p is considered as fixed, but q is allowed to vary. Start with a representation ρ of Gal(\overline{Q}/Q) which is modular of level p. To fix ideas, assume that ρ is irreducible and that it is a representation over a finite field of characteristic prime to 2p. Using the Cebotarev density theorem, find a prime q for which Frob_{q} acts in ρ as ±1. Then there is a new maximal ideal m ⊂ T_{pq} for which ρ_{m} and ρ are isomorphic. Indeed, the trace and determinant of Frob_{q} are respectively ±2 and 1. Since the determinant of Frob_{q} is q, we have q ≡ 1 mod m. Therefore, the key congruence

tr ρ(Frob_{q}) ≡ ±(q + 1) (mod m)

is satisfied. According to Theorem 1 of [15], ρ arises from a maximal ideal m of T_{pq} which is “q-new.” Such a maximal ideal is genuinely new (and not just q-new) unless the corresponding representation ρ_{m} of Gal(\overline{Q}/Q) is finite at p. In our situation, ρ_{m} ≈ ρ cannot be finite at p, since it would otherwise be modular of level 1 as in the discussion above.

For a concrete example, take p = 11, and let

ρ : Gal(\overline{Q}/Q) → GL(2, F_{3})

be the representation of Gal(\overline{Q}/Q) given by the group of 3-division points on the elliptic curve J_{o}(11). The image of ρ is all of GL(2, F_{3}) (see [18]). The Galois extension K of Q which is cut out by ρ contains the field Q(Δ^{1/3}, μ_{3}), where Δ = −11^{5} is the discriminant of J_{o}(11) ([17], p. 305). We seek a criterion to determine whether a not a prime number q ≠ 3, 11 satisfies the condition ρ(Frob_{q}) = ±1. This condition implies that det(ρ(Frob_{q})) = 1, which means the congruence q ≡ 1 (mod 3) is satisfied. Similarly, the trace of ρ(Frob_{q}) must be ±2 (mod 3), which translates to the condition that the coefficient c_{q} of q in the expansion of X · ∏_{n≥1}(1 − X^{n})^{2}(1 − X^{11n})^{2} must be non-zero mod 3. (See [18], where the c_{q} are tabulated for q ≤ 2000.)

Assume that these necessary conditions are satisfied. Then ρ(Frob_{q}) is

either ±1, as desired, or else is of the form ± ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) with x ≠ 0 in F_{3}. Con-

sider the image of ρ(Frob_{q}) in the Galois group Gal(Q(Δ^{1/3}, μ_{3})/Q), which is isomorphic to the symmetric group S_{3}. This image is trivial or of order 3

according as ρ(Frob_{q}) = ±1 or ρ(Frob_{q}) = ± ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) (with x ≠ 0). Con-

cretely, this means that ρ(Frob_{q}) = ±1 if 11 is a cube mod q. Calculation shows that the prime q = 193 satisfies this condition, as well as the necessary congruences q ≡ 1 and c_{q} ≡ ±2 (mod 3). (We have c_{193} = 4.) Therefore, there is a new maximal ideal m in T_{11·193}, of residue characteristic 3, such that the multiplicity of ρ_{m} in J[m] is 2.

Our motivation in proving Theorem 3 is to “contrast and compare” the two abelian varieties J_{new} and J. According to a theorem of [10] (now easier to prove using [4], Satz 4), J_{new} and J are isogenous over Q. There has been some speculation that a natural T-equivariant isogeny (in one direction or the other) should link these two abelian varieties. Theorems 2 and 3 seem to show that the support of the kernel of such an isogeny must, in general, contain maximal ideals m for which ρ_{m} is irreducible. This suggests that any construction of such an isogeny must be relatively elaborate.

Our proof of Theorem 3 is based on a number of the ideas of [12]. In particular, it relies on the exact sequence (see [12], Th. 4.1) relating the mod p bad reduction of J with the mod q bad reductions of J_{o}(q) and J_{o}(pq).

The author wishes to thank Nigel Boston, Ling San, and Jean-François Mestre for helpful conversations and correspondence during the preparation of this paper.

1 Multiplicity one for character groups

In this §, q is a prime number and M is a positive integer prime to q. Set N = qM. Let J be the Néron model of J_{o}(N)_{/Q_{q}}, and write J_{o}(N)_{/F_{q}} for the special fiber of J. As is well known, the connected component of 0 in J_{o}(N)_{/F_{q}}, call it J_{o}(N)_{/F_{q}}^{o}, sits in an exact sequence

0 → T → J_{o}(N)_{/F_{q}}^{o} →^{π} J_{o}(M)_{/F_{q}} × J_{o}(M)_{/F_{q}} → 0,  (1)

where T is a torus over F_{q}. (This follows from results of Raynaud [8] and Deligne-Rapoport [2], as explained in the appendix to [6].) The character group

L = Hom_{\overline{F}_{q}}(T, G_{m})

of T may be described explicitly as the group of degree-0 divisors on the supersingular points on X_{o}(M)_{/\overline{F}_{q}} ([6], Appendix).

Recall [7] that the two degeneracy maps δ_{1}, δ_{q} : X_{o}(N) \overrightarrow{→} X_{o}(M) induce by functoriality a pair of maps J_{o}(M) \overrightarrow{→} J_{o}(N). By composing their direct product with the sum map J_{o}(N)^{2} → J_{o}(N), we obtain the standard degeneracy map α : J_{o}(M)^{2} → J_{o}(N), cf. [7, 11, 15, 16]. Passing to characteristic q, we find a map

α_{/F_{q}} : J_{o}(M)_{/F_{q}}^{2} → J_{o}(N)_{/F_{q}}^{o}.

The composition of π with α_{/F_{q}} is then an endomorphism of J_{o}(M)_{/F_{q}}^{2}, which we may represent as a 2 × 2 matrix of endomorphisms of J_{o}(M)_{/F_{q}}.

Lemma 1 The matrix π ∘ α_{/F_{q}} is the matrix ( \begin{matrix}id & Ver \\ Ver & id \end{matrix} ), where id is the

identity endomorphism of J_{o}(M) and Ver is the Verschiebung endomorphism of J_{o}(M), i.e., the transpose of the Frobenius endomorphism of J_{o}(M).

Proof . The map α_{/F_{q}} is obtained by functoriality from two degeneracy maps X_{o}(N) \overrightarrow{→} X_{o}(M), whereas the map π is deduced by functoriality from two maps in the other direction, which are peculiar to characteristic q ([2], V, §1). The four possible compositions are determined in [2], loc. cit. Two of them are the identity map of X_{o}(M)_{/F_{q}}, while the other two are the Frobenius of X_{o}(M)_{/F_{q}}. Passing to Jacobians (via the functor Pic^{o}), we obtain the indicated formula.

Let T = T_{N} be the subring of End(J_{o}(N)) generated by the Hecke operators T_{ℓ} for prime numbers ℓ. The ring T acts by functoriality on T and on L. These actions factor through faithful actions of the “q-new quotient” of T, which is denoted T_{1} in [12], §3. (See [12], Theorem 3.10.) We say that a maximal ideal m of T is q-new if it arises by pullback from this quotient.

Similarly, the q-old quotient T_{o} of T may be defined in any of several equivalent ways. In ([12], §3), T_{o} is defined as the image of T in End(S_{o}), where S_{o} is the direct sum of two copies of the space of weight-2 cusp forms on Γ_{o}(M). (This direct sum is naturally a T-module because T preserves the direct sum after it has been identified with a subspace of the space of weight-2 cusp forms on Γ_{o}(N), as in the work of Atkin-Lehner [1].) An equivalent geometric definition is the following one. There is a unique operation of T which is compatible with α and the operation of T on J_{o}(N). The ring T acts on J_{o}(M)^{2} through its quotient T_{o}, which acts faithfully. Finally, consider the functorial action of T on J_{o}(N)_{/F_{q}}^{o}. Then, in the exact sequence (1), there is a unique operation of T on J_{o}(M)^{2} which is compatible with π and the action of T on J_{o}(N)_{/F_{q}}^{o}. In this latter operation, T again acts through T_{o}, which acts faithfully ([12], Th. 3.11). In analogy with the terminology introduced for T_{1}, we say that a maximal ideal m of T is q-old if it arises by pullback from T_{o}.

Theorem 4 Let m be a maximal ideal of T which is prime to 2N and such that the corresponding representation ρ_{m} of Gal(\overline{Q}/Q) is irreducible. Assume that ρ_{m} is unramified at q. Then m is q-old.

Proof . To prove this theorem, we review the proof given for Theorem 8.2 of [12]. First of all, m is either q-old or q-new (but might be both), so we may assume that m is q-new. We do so. Pick an “auxiliary” prime number Q: this is a prime number which is prime to 2N and the characteristic of T/m, and which is such that a Frobenius element for Q in Gal(\overline{Q}/Q) maps under ρ_{m} to the image of a complex conjugation in Gal(\overline{Q}/Q). Following the proof of Theorem 8.2 of [12], we introduce the Hecke algebra T_{QN} associated with the space of weight-2 cusp forms on Γ_{o}(NQ). We then pick a maximal ideal M of T_{NQ} which is compatible with m. To explain this notion, we view T_{N} and the Q-old quotient of T_{NQ} as subrings of the commutative ring R which they generate together in the ring of endomorphisms of the direct sum of two copies of the space of weight-2 cusp forms on Γ_{o}(N). (In this ring, the operators T_{ℓ} coming from T_{N} and from T_{NQ} are equal for all primes ℓ, except for the case ℓ = Q. The ring R is generated by the shared Hecke operators T_{ℓ} (with ℓ ≠ Q), together with the Hecke operator T_{Q} coming from T_{N} and the Q^{th} Hecke operator U_{Q} coming from T_{NQ}.) The compatibility means that m and the projection of M in the Q-old quotient of T_{NQ} are contained in a common maximal ideal.

Looking carefully at the proof of Theorem 8.2 of [12], one sees that the following property of M is proved: the ideal M is “qQ-old” in the sense that it arises from the “Qq-old quotient” of T_{NQ}, defined (for example) as a subring of End(J_{o}(M)^{4}). For the convenience of the reader, §3 provides more details concerning this point.

The indicated property of M becomes much simpler if we jettison (temporarily) the two different Hecke operators T_{Q}. Namely, consider the subring R of T which is generated by all the T_{ℓ} with the exception of T_{Q}. Let m_{R} be the intersection of m with R. What is proved about M gives, in particular, that m_{R} is q-old in the sense that it arises by pullback from the subring R_{o} of T_{o} which is generated by the T_{ℓ} with ℓ ≠ Q. By the going-up theorem of Cohen-Seidenberg, this shows that there is some maximal ideal m′ of T which is q-old and which contains m_{R}. To prove the theorem, it is enough to verify that m and m′ coincide.

Let F be the residue field of m_{R}, so that F is a subfield of each of T/m and T/m′. By the Chinese Remainder Theorem, to prove that m = m′ is to prove that the image of T in T/m × T/m′ is a field. The image of R in T/m × T/m′ is F, embedded “diagonally” in the product. It thus suffices to show that the image of T_{Q} in the product lies inside the image of F in the product. However, by applying the Cebotarev Density Theorem to ρ_{m} × ρ_{m′}, we see that there are infinitely many primes ℓ, ℓ ≠ Q, so that T_{ℓ} and T_{Q} have the same image in T/m × T/m′.

In the following discussion, we assume that m is a maximal ideal of T which is prime to 2N and such that the representation ρ_{m} of Gal(\overline{Q}/Q) is irreducible. According to [12], Theorem 5.2(b), the kernel J_{o}(N)[m] of m on J_{o}(N) is multiplicity free in the sense that J_{o}(N)[m] is two-dimensional (and therefore isomorphic to ρ_{m} as a representation of Gal(\overline{Q}/Q)). We shall calculate the dimension of L/mL. By the characterization of T_{1} which was cited, L/mL is non-zero if and only if m is q-new. We say that L/mL is multiplicity free if it is of dimension ≤ 1 over the residue field k = T/m. With this definition, L/mL is automatically multiplicity free if m is not qnew.

Proposition 1 Assume that m is q-new, prime to 2N, and such that ρ_{m} is irreducible. Then L/mL is multiplicity free unless ρ_{m} is unramified at q and a Frobenius element Frob_{q} for q in Gal(\overline{Q}/Q) acts in ρ_{m} as ±1. In this latter case, L/mL is of dimension 2 over k.

Proof . Let D be the Galois group Gal(\overline{Q}_{q}/Q_{q}), thought of as a decomposition group for q in Gal(\overline{Q}/Q). We consider that Frob_{q} is a Frobenius element of D. Let ℓ be the characteristic of k. As discussed in §6 of [12], the work of Grothendieck in SGA7I [5] establishes a natural inclusion of k[D]-modules

Hom(L/mL, μ_{ℓ}) ↪ J_{o}(N)[m].  (2)

The left-hand module is unramified, with Frob_{q} acting as qT_{q} ([12], Proposition 3.8). Further, the Hecke operator T_{q} is the negative of the Atkin-Lehner involution w_{q} on the torus T ([12], Proposition 3.7). Therefore, Frob_{q} acts on Hom(L/mL, μ_{ℓ}) as +q or as −q.

In our situation, the dimension of Hom(L/mL, μ_{ℓ}) is at least 1, while the dimension of J_{o}(N)[m] is exactly 2, as remarked above. Hence, if L/mL is not multiplicity free, it has dimension 2, and Hom(L/mL, μ_{ℓ}) coincides with J_{o}(N)[m]. Since the latter module is a model for the representation ρ_{m}, this latter representation is unramified. (Here, we could say “unramified at q,” but this qualification is somewhat superfluous since the coincidence of Hom(L/mL, μ_{ℓ}) and J_{o}(N)[m] gives merely an equality of k[D]-modules.) It follows also that Frob_{q} acts in ρ_{m} as one of the two scalars ±q. To see that q ≡ 1 (mod m), we note that the determinant of Frob_{q} in ρ_{m} is q, while the determinant of either scalar ±q is q^{2}. Therefore, if L/mL is not multiplicity free, then Frob_{q} acts as ±1, as claimed.

Suppose, conversely, that ρ_{m} is unramified at q and that Frob_{q} acts in ρ_{m} as +1 or −1. By considering the determinant, we see again that q ≡ 1 (mod m). Also, m is q-old, by Theorem 4. This implies that the kernel (J_{o}(M) × J_{o}(M))[m] of m on J_{o}(M)^{2} is non-zero. On the other hand, the kernel of α : J_{o}(M)^{2} → J_{o}(N) has trivial intersection with J_{o}(M)^{2}[m], since the kernel of α is “Eisenstein,” whereas ρ_{m} is irreducible. (See Theorem 1 of [13], Theorem 4.3 of [11], and Theorem 5.2(c) of [12].) By the irreducibility of V = J_{o}(N)[m], α identifies J_{o}(M)^{2}[m] with J_{o}(N)[m].

Since m is q-new, T_{q} mod m is ±1 and the Atkin-Lehner involution w_{q} acts on J_{o}(N)[m] as ∓1 (with the opposite sign). By comparing the actions of Frob_{q} on J_{o}(N)[m] and its submodule Hom(L/mL, μ_{ℓ}), we see that the sign “±” for the action of Frob_{q} on J_{o}(N)[m] is + if T_{q} (mod m) is +1, and is − if T_{q} (mod m) is −1. Finally, the degeneracy map α is equivariant with respect to the Atkin-Lehner involution w_{q} on J_{o}(N) and the involution (x, y) ↦ (y, x) on J_{o}(M)^{2}. Putting all this together, we find that J_{o}(M)^{2}[m] lies inside the diagonal image of J_{o}(M) in J_{o}(M)^{2} if Frob_{q} acts on J_{o}(N)[m] as −1, and in the antidiagonal image of J_{o}(M) in J_{o}(M)^{2} if Frob_{q} acts on J_{o}(N)[m] as +1. (By the “antidiagonal image,” we mean the graph of multiplication by −1.)

To fix ideas, we will suppose from now on that Frob_{q} acts as −1 on J_{o}(N)[m], so that J_{o}(N)[m] lies in the image of J_{o}(M) in J_{o}(N) under the composition of α with the diagonal embedding of J_{o}(M) in J_{o}(M)^{2}. One, extremely minor, advantage of this choice is that the composition is a map J_{o}(M) → J_{o}(N) which has trivial kernel. In the antidiagonal situation, the analogous composition has (in general) a non-trivial kernel. However, the kernel is finite; and, as remarked above, it is “Eisenstein.” This implies, as already noted, that the kernel does not intersect the groups which interest us.

Pulling back J_{o}(N)[m] to J_{o}(M), we find a subgroup W of J_{o}(M)[ℓ] such that

J_{o}(N)[m] = { α(x, x) | x ∈ W }.

The Frobenius element Frob_{q} of Gal(\overline{Q}/Q) acts on W as −1. Viewing W as a group of points on J_{o}(M)_{/F_{q}}, we see that the Frobenius endomorphism of J_{o}(M)_{/F_{q}} acts on W as −1. Since q ≡ 1 (mod ℓ), the Verschiebung endomorphism of J_{o}(M)_{/F_{q}} acts, as well, as −1 on W. By Lemma 1, all points (x, x) with x ∈ W lie in the kernel of π ∘ α. Thus J_{o}(N)[m] (viewed as a group of points of J_{o}(N)_{/F_{q}}) lies in the kernel of π_{/F_{q}}, which is the torus T. Therefore, we have J_{o}(N)[m] ⊂ Hom(L/mL, μ_{ℓ}). It follows that L/mL has dimension greater than 1.

2 Shimura Curves

The principal aim of this section is to prove Theorem 3. We return to the notations introduced in the Introduction. Thus p and q are distinct primes, J is the Jacobian of the Shimura curve introduced earlier, T = T_{pq} is the Hecke algebra at level pq, and m is a maximal ideal of T. We assume that m is new and prime to 2pq, and that the representation ρ_{m} is irreducible. We suppose that p and q have been ordered so that ρ_{m} is ramified at p. Our goal is to investigate the dimension of J[m] over k = T/m.

To achieve this goal, we will apply the discussion of §1 in the case where M = p, i.e., N = pq. We shall give J_{o}(q)^{2} the unique T-action which is compatible with the action of T on J_{o}(pq) and the analogue β : J_{o}(q)^{2} → J_{o}(pq) of the degeneracy map α : J_{o}(p)^{2} → J_{o}(pq). Then T acts on each of the abelian varieties J, J_{o}(pq) and J_{o}(q)^{2}.

Use the notation J, previously employed for the Néron model of J_{o}(N), for the Néron model of J over Q_{p}. Its special fiber J_{/F_{p}} is an extension of a finite group Ψ by its connected component J_{/F_{p}}^{o}, which is a torus (cf. [12], §4). Write Y for the character group of this torus (computed over an algebraic closure \overline{F}_{p} of F_{p}), so that the torus itself becomes Hom(Y, G_{m}). Let L be the character group which is introduced in that §, and let X be the analogue of L for the abelian variety J_{o}(q)^{2}. Note that X and L are defined with reference to an algebraic closure of F_{q}, while Y has just been defined with reference to an algebraic closure of F_{p}. We suppose from now on that these algebraic closures have in fact been selected as algebraic closures of the residue fields F_{q^{2}} and F_{p^{2}} of the maximal order O which was used in the definition of J. The ring T acts by functoriality on L, Y and X.

Theorem 5 There is a natural T-equivariant exact sequence

0 → Y → L →^{β^{∗}} X → 0.  (3)

This theorem is proved as Theorem 4.1 of [12], although the notations there are somewhat different. In [12], the group X is called “X ⊕ X,” the symbol X being reserved for the character group attached to J_{o}(q). Further, Y is defined as the kernel of the map from L to X; the content of the theorem is that the character group attached to J may be identified with this kernel. Finally, the Hecke algebra appearing in [12], Th. 4.1 is of necessity the formal polynomial ring T̃ on symbols representing Hecke operators, since it is only after proving this theorem that one knows that T̃ operates on J through its quotient T.

In [12], the map Y → L appears to depend on a certain number of choices. This is not very surprising in view of the fact that the very definition of Y depends on the choice of an algebraic closure of F_{p}, while L is defined in terms of an algebraic closure of F_{q}. On the other hand, all characters in Y and L are in fact defined over the quadratic subfields of \overline{F}_{p} and \overline{F}_{q}, respectively. Thus, it is reasonable to hope that a canonical map Y → L can be defined with these quadratic subfields as initial data. The main content of [14] is to realize this hope. In particular, [14] shows that a canonical map Y → L can be given when F_{q^{2}} and F_{p^{2}} are chosen as the residue fields of O. This is the choice that we have made.

Proposition 2 We have 1 ≤ dim_{k} Y/mY ≤ 2. The dimension is 2 if and only if ρ_{m} is unramified at q and Frob_{q} acts in ρ_{m} as ±1.

Proof . The localization X_{m} of X at m is 0. Indeed, suppose that X_{m} is nonzero. Then m is p-old in the sense that it arises by pullback from the p-old quotient of T (which can be defined as the image of T in the endomorphism ring of J_{o}(q)^{2}). This shows that ρ_{m} is modular of level q, which implies that ρ_{m} is unramified at p. However, according to our ordering of p and q, ρ_{m} is ramified at p.

The sequence (3) thus shows that L_{m} and Y_{m} are isomorphic. In particular, we have L/mL ≈ Y/mY. Our proposition thus follows from Proposition 1.

We next recall the interpretation of Ψ in terms of the monodromy pairing on Y [5]. As recalled in [12], §1, there is an interpretation of Ψ as the cokernel of an injection μ : Y′ → Hom(Y, Z), where Y′ is the analogue of the group Y for the abelian variety dual to J. Since J = Pic^{o}(C) is naturally isomorphic to the dual of J (i.e., to the Albanese variety of C), we have an isomorphism Y ≈ Y′. This isomorphism is compatible with the functorial actions of T on Y and Y′. Indeed, to check this one must show that all Hecke operators T_{r} (with r prime) are fixed under the Rosati involution on End(J) coming from the usual principal polarization on the Jacobian J. This is a standard fact for the generators T_{r} of T with r prime different from p and q. For r = p and r = q, we use the interpretation of T_{r} as −w_{r}, plus the fact that the involution w_{r} is its own transpose.

We thus have an exact sequence of T-modules

0 → Y →^{μ} Hom(Y, Z) → Ψ → 0.  (4)

To say that μ is T-equivariant amounts to the statement that the bilinear pairing ( , ) associated with μ satisfies (T y_{1}, y_{2}) = (y_{1}, T y_{2}) for T ∈ T and y_{1}, y_{2} ∈ Y.

We recall that ℓ is the characteristic of k = T/m.

Proposition 3 The k-dimensions of Y/mY and (Y/ℓY)[m] are equal.

Proof . Consider the map “multiplication by ℓ” on the terms of the exact sequence (4). Using the Snake Lemma, we get a four-term sequence

0 → Ψ[ℓ] → Y/ℓY → Hom(Y/ℓY, Z/ℓZ) → Ψ/ℓΨ → 0.  (5)

Further, the localization of Ψ at m is 0. Indeed, the exact sequence of [12], Theorem 4.3 gives after localization a sequence

X_{m} → Ψ_{m} → E_{m} → 0,  (6)

where E is a quotient of the component group attached to the Néron model of J_{o}(pq) at q. Since this component group is Eisenstein ([12], Th. 3.12), we have E_{m} = 0. Also, X_{m} = 0, as was already noted.

It follows that localization of (5) gives an isomorphism of the localizations at m of the two middle terms. Now take kernels of m (i.e., “[m]”) on both sides, and use the easily verified fact that these kernels may be identified with the kernels on the two middle terms before localization. This gives an isomorphism between (Y/ℓY)[m] and Hom(Y/mY, Z/ℓZ), which implies, in particular, the desired dimension equality.

Let ν again be the multiplicity of ρ_{m} in J[m], so that dim_{k} J[m] = 2ν. In view of Proposition 2, the following result will complete the proof of Theorem 3.

Proposition 4 We have ν = dim_{k} Y/mY; i.e., dim_{k} J[m] = 2 dim_{k} Y/mY.

Proof . Because J is purely toric at p, there is an exact sequence of T-modules

0 → Hom(Y/ℓY, μ_{ℓ}) → J(\overline{Q}_{p})[ℓ] → Y/ℓY → 0.  (7)

This sequence is noted in [9], (3.3.1), and can certainly be deduced easily from [5]. In particular, the inclusion of Hom(Y/ℓY, μ_{ℓ}) in J(\overline{Q}_{p})[ℓ] is an analogue for J of the inclusion (2) for J_{o}(N). The sequence (7) is compatible with the natural action of Gal(\overline{Q}_{p}/Q_{p}) on all terms; in particular, the action of Gal(\overline{Q}_{p}/Q_{p}) on Hom(Y/ℓY, μ_{ℓ}) is unramified. Taking kernels of m yields

0 → Hom(Y/mY, μ_{ℓ}) → J[m] → (Y/ℓY)[m].  (8)

By Propositions 2 and 3, the dimensions of Hom(Y/mY, μ_{ℓ}) and (Y/ℓY)[m] are equal, with the common value being 1 or 2. Clearly, if ν > 1, then ν = 2 and the common dimension is 2. Assume that ν = 1, so that the dimension of J[m] is 2. If Y/mY does not have dimension 1, then (8) shows that Hom(Y/mY, μ_{ℓ}) and J[m] are equal inside Hom(Y/ℓY, μ_{ℓ}). In particular, this gives J[m] ⊆ Hom(Y/ℓY, μ_{ℓ}), which forces the action of Gal(\overline{Q}_{p}/Q_{p}) to be unramified. Since p and q were ordered so that this action is ramified, we may conclude that dim Y/mY = 1.

3 Appendix

The purpose of this Appendix is to justify the assertion about M which is made in the course of the proof of Theorem 4. The justification was postponed until now because it is relatively technical. Also, it uses material which is close in spirit to that of §2. For example, it makes use of a generalization of the exact sequence (3) to the case of the Jacobian of a Shimura curve made with level structure.

To be more precise, let M, q, and Q be the positive integers occurring during the course of the proof of Theorem 4. Then Q and q are distinct primes, and they are prime to M. Let O be a maximal order in a rational quaternion algebra of discriminant qQ, and let C now be the Shimura curve associated with the moduli problem of classifying abelian surfaces A which are furnished with an action of O and an O-stable subgroup of A which is isomorphic to (Z/MZ)^{2} as an abelian group. The Hecke ring T_{MqQ} acts on J = Pic^{o}(C), and this action cuts out the “qQ-new quotient” of T_{MqQ} ([12], 4.2).

Now the maximal ideal M which was constructed in the proof of Theorem 4 is qQ-new by Theorem 7.3 of [12]. This implies that the kernel J[m] is non-zero. The residue field k of m coincides with the residue field of M; therefore J[M] is naturally a k[Gal(\overline{Q}/Q)]-module. By the Eichler-Shimura relations for J, and the argument of Mazur which was repeatedly cited, the semisimplification of J[M] is a direct sum of copies of the representation V giving ρ_{m}. In particular, we may realize V as a submodule of J[M]. Because V is by assumption unramified at q, we may identify V with a group of division points on the fiber J_{/F_{q}} at q for the Néron model for J.

This fiber is an extension of a component group Ψ by a torus Hom(Y, G_{m}). Let L be the character group of the toric part of J_{o}(MqQ)_{/F_{Q}}, and let X be the character group of the toric part of J_{o}(MQ)_{/F_{Q}}^{2}. Then the exact sequence (3) again may be constructed with these new definitions of Y, X, and L. Further, the group Ψ fits into an exact sequence which gives after localization at M the following analogue of (6):

X_{M} → Ψ_{M} → E_{M} → 0.

As in the proof of Proposition 3, the group E_{M} vanishes.

These considerations imply that X_{M} is non-zero. Indeed, suppose that X_{M} is 0. Then Ψ[M] = 0, so that the kernel of M on the fiber J_{/F_{q}} coincides with Hom(Y/MY, G_{m}). This kernel is of k-dimension at least 2 because it contains the group V. Hence dim Y/MY ≥ 2. From (3) and the vanishing of X_{M}, we then get dim L/ML ≥ 2. As in the proof of Proposition 1, this inequality implies that Frob_{Q} acts in ρ_{m} as a scalar ±1. However, we chose Q so that Frob_{Q} acts in ρ_{m} as a complex conjugation; in particular, it does not act as a scalar.

Knowing that X_{M} is non-zero gives us the information that M arises by pullback from the image of T_{MqQ} in End(J_{o}(MQ)^{2}). (We term this image the “q-old quotient” of T_{MqQ}.) In particular, we may realize the T_{MqQ}[Gal(\overline{Q}/Q)]-module V as a group of division points of J_{o}(MQ)^{2}. Since this group is unramified at Q, we may identify it with a group of division points on J_{o}(MQ)_{/F_{Q}}^{2}. The group of division points in fact lives in the connected component of 0 in J_{o}(MQ)_{/F_{Q}}^{2}, since the component group of J_{o}(MQ)_{/F_{Q}}^{2} is Eisenstein ([12], Theorem 3.12). It clearly cannot lie in the toric part Hom(X, G_{m}) of J_{o}(MQ)_{/F_{Q}}^{2}, since Frob_{Q} does not act on V as a scalar. Hence it maps non-trivially to the abelian variety quotient J_{o}(M)_{/F_{Q}}^{4} of J_{o}(MQ)_{/F_{Q}}^{2}. Therefore M arises by pullback from the quotient of T_{MqQ} which is cut out by J_{o}(M)_{/F_{Q}}^{4}. As in [12], Theorem 3.11, this quotient is the same quotient which acts faithfully on J_{o}(M)^{4}, considered in the usual way as an abelian subvariety of J_{o}(MqQ); in other words, it is the “qQ-old quotient” of T_{MqQ}.

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