pacific journal of mathematics Vol. 181, No. 3, 1997

IMAGES OF SEMISTABLE GALOIS REPRESENTATIONS

Kenneth A. Ribet

In memory of Olga Taussky-Todd

1. Introduction.

Let p be an odd prime number. Suppose first that E is a semistable elliptic curve over Q. The action of Gal(\overline{Q}/Q) on the group of p-division points of E defines a representation

ρ_{E,p}: Gal(\overline{Q}/Q) → GL(2, F_{p}).

Serre has shown that this representation is surjective whenever it is irreducible [26, Prop. 21], [29, §3.1]. Serre’s arguments prove more generally the surjectivity of all continuous irreducible representations

ρ: Gal(\overline{Q}/Q) → GL(2, F_{p})

which are semistable in the sense of [15, §1]. (Recall that ρ is semistable if the determinant of ρ is the mod p cyclotomic character χ, the Serre conductor of ρ is square free, and the Serre weight of ρ is either 2 or p+1. Here, the weight and conductor are the invariants defined in [28].)

In this article, we treat the situation where F_{p} is replaced by a finite field of characteristic p, or more generally by a finite product F = ∏ F_{i} of finite fields of characteristic p. A continuous representation ρ: Gal(\overline{Q}/Q) → GL(2, F) may then be viewed as a product of components ρ_{i}: Gal(\overline{Q}/Q) → GL(2, F_{i}). We shall assume that each ρ_{i} is irreducible and semistable. Since the determinant of each ρ_{i} is then the mod p cyclotomic character, the image of ρ is contained in the group

A := { M ∈ GL(2, F) | det M ∈ F_{p}^{∗} }.

Making the supplementary hypothesis p ≥ 5, we show below that the image of ρ is GL(2, F)-conjugate to

{ M ∈ GL(2, F′) | det M ∈ F_{p}^{∗} },

where F′ is the subalgebra of F generated by the traces tr(ρ(σ)) for σ ∈ Gal(\overline{Q}/Q). (See Theorem 3.2 and Corollary 3.3.) In particular, ρ( Gal(\overline{Q}/Q)) = A if and only if F is generated as an F_{p}-algebra by the traces.

Applying the lifting techniques of [25], we deduce an analogous result for certain p-adic representations of Gal(\overline{Q}/Q).

Our results have evident relevance to the study of the Galois representations defined by semistable abelian varieties over Q which are products of abelian varieties with large fields of endomorphisms. (In [23], the author referred to these as abelian varieties of “GL_{2} type.”) While the computation of F′ seems to be difficult to perform in certain cases, it can be carried out for the mod p representations coming from J_{0}(N) if N is a prime and p is a prime number satisfying some mild conditions. In fact, our original goal was to find the image of the p-adic representation attached to J_{0}(N), thus answering questions which were formulated by R. Coleman and B. Kaskel in connection with [10] and [2]. It is a pleasure to thank them for their continuing encouragement and interest in this work.

2. Representations over a finite field.

Let F be a finite field, and let p be the characteristic of F. We will assume that p is odd; most of our results will require that p be at least 5. Suppose that

ρ: Gal(\overline{Q}/Q) → GL(2, F)

is an irreducible representation whose determinant is the mod p cyclotomic character χ: Gal(\overline{Q}/Q) → F_{p}^{∗} ↪ F^{∗}. Let k(ρ) and N(ρ) be the weight and conductor of ρ in the sense of Serre [28, §§1-2]. As we recalled above, Oesterlé [15, §1] has defined the notion of semistability: ρ is semistable if the conductor N(ρ) of ρ is square free and Serre’s weight k(ρ) is either 2 or p+1.

The definition of the conductor shows that N(ρ) is square free if and only if ρ(σ) is unipotent whenever σ belongs to an inertia subgroup of Gal(\overline{Q}/Q) for a prime ℓ ≠ p. To illuminate the condition on k(ρ), we let I be an inertia subgroup of Gal(\overline{Q}/Q) for the prime p. The semisimplification of ρ_{|I} is described by a pair of characters φ, φ′: I ⇉ \overline{F}^{∗}, cf. [28, §2]. Since det ρ is the cyclotomic character χ, we have in particular φφ′ = χ. If k(ρ) is one of 2, p+1, then {φ, φ′} is either {1, χ} or else the set of fundamental characters ψ, ψ′: I → \overline{F}^{∗} of level 2 ([28, §2]). It follows that the order of φ′φ^{−1} (a character which is defined only up to inversion) is either p−1 or p+1.

Lemma 2.1. Assume that p > 3 and that ρ is a semistable irreducible representation as above. Let ϵ: Gal(\overline{Q}/Q) → F^{∗} be a continuous character. If ϵ ⊗ ρ is semistable, then ϵ is trivial.

Proof. By definition, the representation ϵ ⊗ ρ sends each σ ∈ Gal(\overline{Q}/Q) to the product ϵ(σ)·ρ(σ) ∈ GL(2, F). If ρ and ϵ⊗ρ are both semistable, then ϵ is certainly unramified outside p, since both ρ(σ) and ϵ(σ)·ρ(σ) are unipotent whenever σ belongs to an inertia group for a prime ℓ ≠ p. Moreover, ϵ^{2} is the trivial character, since the determinants of ρ and ϵ ⊗ ρ coincide. Hence ϵ is either the trivial character, as desired, or the quadratic character χ^{\frac{p−1}{2}} . One checks easily, however, that no pair of characters drawn from the set { 1, χ, ψ, ψ′ } have a quadratic ratio.

Remark. The lemma does not extend to cover the case p = 3. To see this, consider the modular forms f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} which correspond to the two strong modular elliptic curves of conductor 89. As B. Gross explains in the last paragraphs of [8], these forms define irreducible mod 3 representations ρ_{f} and ρ_{g} of Gal(\overline{Q}/Q) whose restrictions to a decomposition group for 3 are direct sums of two characters. Further, although ρ_{f} and ρ_{g} are semistable, the two representations are twists of each other by the mod 3 cyclotomic character. (In other words, f and g are “companions” of each other in Serre’s language.)

To verify this latter fact, we can cite [8, Th. 13.10] along with Gross, or perform a numerical calculation with gp [1] to check that the mod 3 congruence b_{p} ≡ (\frac{p}{3})a_{p} holds for 3 ≤ p ≤ 200. Using the results of J. Sturm [33], we can then conclude that it holds for all p ≥ 3.

It perhaps is worth recalling at this juncture that an irreducible mod p semistable representation ρ is absolutely irreducible. This follows easily from the fact that ρ( Gal(\overline{Q}/Q)) contains a matrix with distinct eigenvalues in F_{p}^{∗}. Such a matrix is obtained by taking ρ(c), where c is a complex conjugation because det ρ is the cyclotomic character, which is odd, the eigenvalues of ρ(c) are +1 and −1. These are distinct because p is odd.

Proposition 2.2. If ρ is semistable, then the image of ρ has order divisible by p.

Proof. Let G = ρ(Gal(\overline{Q}/Q)), and assume that the order of G is prime to p. Since the semistability hypothesis implies that ramification subgroups of G for primes ℓ ≠ p are unipotent, these ramification groups are forced to be trivial. In other words, we have N(ρ) = 1 and the representation ρ is unramified outside p. As Serre remarks in a note on page 710 of [27, Vol. III], an analogue of the argument of Tate [34] shows that there are no irreducible representations ρ with this property when p = 3.

Now assume that p ≥ 5 and write \overline{G} for the image of G in PGL(2, F). Group theory shows that \overline{G} is either cyclic or dihedral, or else one of the three exceptional groups S_{4}, A_{4}, A_{5} [26, §2.5]. In fact, \overline{G} cannot be cyclic, since the cyclicity of \overline{G} would imply that G is abelian, and hence that ρ is not absolutely irreducible.

To rule out the other cases, one considers an inertia subgroup \overline{I} of \overline{G} for the prime p. We know that \overline{I} is a cyclic group of order either p + 1 or p − 1. Indeed, \overline{I} may be viewed as the image of I ⊂ Gal(\overline{Q}/Q) under the character φ/φ′ which we introduced above. This character has order either p−1 or p+1; its image is cyclic because it is a finite subgroup of \overline{F}^{∗}.

Assume now that \overline{G} is dihedral, and let Z be the center of \overline{G}. It is evident that \overline{I} is contained in Z, since \overline{I} is a cyclic subgroup of a dihedral group and the order of \overline{I} is greater than 2. Accordingly, the quadratic extension of Q corresponding to Z is everywhere unramified this contradiction excludes the dihedral case and shows that \overline{G} must be one of the three exceptional groups.

However, as observed in the proof of [26, Prop. 21], the fact that \overline{I} has an element of order p ± 1 rules out the groups S_{4}, A_{4}, A_{5} in case p ≥ 7. Thus we are left only with the possibility that p = 5, in which case \overline{G} is either S_{4} or A_{4}, since its order is prime to 5 by assumption. The group \overline{I} is then cyclic of order 4, since S_{4} has no element of order 6. Also, we have \overline{G} ≈ S_{4}, since A_{4} has no element of order 4. Consider the quotient S_{3} of S_{4}. This quotient allows us to produce an S_{3}-extension of Q which is ramified only at 5 and such that the inertia groups for 5 in the extension have order 2. However, there certainly is no such extension, since the class number of Q(\sqrt{5}) is 1.

Corollary 2.3 (cf. [29, Prop. 1]). Let ρ be as in Proposition 2.2, and suppose that p > 2. Then the image of ρ contains a subgroup isomorphic to SL(2, F_{p}). In particular, if F = F_{p}, then ρ is surjective.

Proof. The second statement is a consequence of the first, since the cyclotomic character χ: Gal(\overline{Q}/Q) → F_{p}^{∗} is surjective. To prove the first statement, we proceed as in [26, §2.4]. Namely, let g ∈ G be an element of order p, and let v be a non-zero vector in F ⊕ F which is fixed by g. Since ρ is irreducible, G cannot fix the line generated by v; therefore, there is an r ∈ G such that v and rv form a basis of F ⊕ F. With respect to this

basis, g has the form (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}), while rgr^{−1} has the form (\begin{matrix}1 & 0 \\ b & 1 \end{matrix}). Scaling one

of the vectors v, rv, we may assume that a = 1. Hence, in an appropriate

basis, G contains the elements (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and (\begin{matrix}1 & 0 \\ b & 1 \end{matrix}). A well known theorem of L. E. Dickson (cf. [7, Th. 2.8.4]) states that the group generated by these elements has a subgroup isomorphic to SL(2, F_{p}). In fact, a more precise

statement is true for p ≠ 3: (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and (\begin{matrix}1 & 0 \\ b & 1 \end{matrix}) generate the group SL(2, F′),

where F′ = F_{p}(b).

Our aim now is to complement the Corollary by determining the exact image of ρ in a situation generalizing that where F = F_{p}. The situation which we have in mind is that where F is a minimal field of definition for ρ in the sense that it is generated over F_{p} by the numbers tr(ρ(σ)) for σ ∈ Gal(\overline{Q}/Q). We first show in this case that F is also a minimal field of definition for ρ as a projective representation, at least when p ≥ 5.

For the following lemma, let \overline{F} be an algebraic closure of F and consider an arbitrary subfield K of \overline{F}. We view G and \overline{G} inside GL(2, \overline{F}) and PGL(2, \overline{F}), respectively.

Lemma 2.4. Suppose that ρ semistable and that p ≥ 5. Then \overline{G} lies in PGL(2, K) if and only if G lies in GL(2, K).

Proof. If G lies in GL(2, K), then it is evident that \overline{G} is contained in PGL(2, K). Conversely, suppose \overline{G} ⊆ PGL(2, K). Then certainly G ⊆ \overline{F}^{∗} · GL(2, K). Let α: Gal(\overline{Q}/Q) → \overline{F}^{∗}/K^{∗} be the composite homomorphism

Gal(\overline{Q}/Q) →^{ρ} G ↪ \overline{F}^{∗}GL(2, K) → (\overline{F}^{∗}GL(2, K))/GL(2, K) = \overline{F}^{∗}/K^{∗}.

For σ ∈ Gal(\overline{Q}/Q), we have α(σ) = 1 whenever the trace of ρ(σ) is a nonzero element of K. Indeed, write ρ(σ) = t · M, with M ∈ GL(2, K) and t ∈ \overline{F}^{∗}. Then tr (ρ(σ)) = t · tr(M) and we have tr M ∈ K. If tr (ρ(σ)) belongs to K^{∗}, then t lies in K, so that ρ(σ) is an element of GL(2, K). In particular, α(σ) = 1 whenever the trace of ρ(σ) is a non-zero element of F_{p}.

Let M now be the finite abelian extension of Q which is cut out by α. We seek to show that α is identically 1, i.e., that M = Q. We first prove that α is unramified outside p by using the remark about traces. If σ belongs to an inertia subgroup of Gal(\overline{Q}/Q) for a prime ℓ ≠ p, then ρ(σ) is unipotent, so that its trace is 2. Since p is odd, 2 is a non-zero element of F_{p}, and we may conclude that α(σ) = 1.

Thus M is a finite abelian extension of Q which is unramified outside p. Moreover, [M: Q] is prime to p, since \overline{F}^{∗} has no elements of order p. Hence one has M ⊆ Q(μ_{p}); equivalently, α factors through the mod p cyclotomic character χ.

Now let I be an inertia group for p in Gal(\overline{Q}/Q). It suffices to show that there is an element σ of I for which α(σ) = 1 and for which χ(σ) is a generator of the cyclic group F_{p}^{∗}. The semisimplification of ρ_{|I} is described by a pair of characters φ, φ′: I ⇉ \overline{F}^{∗}, cf. [28, §2]. As we mentioned above, one has either {φ, φ′} = {1, χ} or {φ, φ′} = {ψ, ψ^{p}}, where ψ and ψ^{p} are the fundamental characters of level 2. Suppose first that we are in the former case, and let σ ∈ I be such that t = χ(σ) is a generator of F_{p}^{∗}. Then tr (ρ(σ)) = 1 + t is non-zero since p ≠ 3. Thus α(σ) = 1, as required. In the latter case, choose σ ∈ I so that t = ψ(σ) generates ψ(I) ≈ F_{p^{2}}^{∗}. Then χ(σ) = t^{p+1} is a generator of F_{p}^{∗}; note that χ = ψψ′ = ψ^{p+1}. On the other hand, tr (ρ(σ)) = t + t^{p}. The number t^{p−1} cannot be −1, since it has order p + 1. Since tr (ρ(σ)) is non-zero, we may conclude α(σ) = 1.

Theorem 2.5. Assume that ρ is semistable, that p ≥ 5 and that F is generated over F_{p} by the set \{ tr(ρ(σ)) | σ ∈ Gal(\overline{Q}/Q) \}. Then

ρ(Gal(\overline{Q}/Q)) = \{ M ∈ GL(2, F) | det M ∈ F_{p}^{∗} \}.

Proof. It is clear that G = ρ(Gal(\overline{Q}/Q)) is contained in the indicated matrix group because det ρ is F_{p}^{∗}-valued. Since the image of det ρ is precisely F_{p}^{∗}, the theorem amounts to the statement that G contains SL(2, F). An equivalent assertion is that the commutator subgroup of G contains a subgroup isomorphic to SL(2, F).

Let k be a finite extension of F_{p} which contains F and which has even degree over F_{p}. Since det(G) ⊆ F_{p}^{∗}, the subgroup \overline{G} of PGL(2, k) lies in PSL(2, k). Dickson [4, Ch. XII] has enumerated all subgroups of PSL(2, k); his list is summarized in [4, §260].

To situate \overline{G} within Dickson’s list, we recall that the order of \overline{G} is divisible by p by Proposition 2.2, and that the identity representation G → GL(2, \overline{F}) is irreducible. (The representation ρ is irreducible over F by hypothesis. It is then absolutely irreducible, as was noted above.) It follows that \overline{G} is one of the groups enumerated in [4, §251–§253]. Since we have assumed p ≥ 5, the final conclusion is easy to state: After replacing \overline{G} by a conjugate of \overline{G} inside PGL(2, k), we have either \overline{G} = PSL(2, K) or \overline{G} = PGL(2, K), for some subfield K of k.

Thus, in either case one has \overline{G} ⊆ PGL(2, K). From the Lemma, G ⊆ GL(2, K). In particular, tr(ρ(σ)) ∈ K for all σ ∈ Gal(\overline{Q}/Q). Since these numbers generate F over F_{p}, F ⊆ K. On the other hand, SL(2, K) ⊆ k^{∗} · G because \overline{G} contains PSL(2, K). On taking commutators, we obtain SL(2, K) ⊆ [G, G], and therefore SL(2, F) ⊆ [G, G]. As indicated above, this proves the theorem.

In the spirit of [26, Th. 4], let us choose an inertia group I for p in Gal(\overline{Q}/Q) and define X to be the smallest closed normal subgroup of Gal(\overline{Q}/Q) which contains I.

Theorem 2.6. In the situation of Theorem 2.5

ρ(X) = { M ∈ GL(2, F) | det M ∈ F_{p}^{∗} }.

Proof. By Theorem 2.5, the group G = ρ( Gal(\overline{Q}/Q)) is the group of matrices in GL(2, F) with determinants in F_{p}^{∗}. Let H = ρ(X); thus H is a normal subgroup of G. Let H̄ and Ḡ be the images of H and G in PGL(2, F). The group Ḡ is either PSL(2, F) or PGL(2, F), according as the degree of F over F_{p} is even or odd. The discussion above shows that the order of H is at least p − 1 ≥ 4. Therefore, the intersection H ∩ PSL(2, F) has order at least 2. Since PSL(2, F) is normal in Ḡ, H ∩ PSL(2, F) is a non-trivial normal subgroup of PSL(2, F). Accordingly, it is all of PSL(2, F); in other words, H̄ contains PSL(2, F). On taking commutators as above, we see that H contains SL(2, F). Because the mod p cyclotomic character maps I onto F_{p}^{∗}, it follows now that H = G.

Remark 1. In the context of Theorem 2.6, one may consider the more general situation where F is not necessarily generated by the traces tr(ρ(σ)) for σ ∈ Gal(\overline{Q}/Q). Let F′ be the subfield of F which is generated by these traces. Because complex conjugation acts in ρ as a matrix with distinct rational eigenvalues, a well known theorem of I. Schur [24, IX a] (cf. [35, Lemme I.1]) implies that ρ can be conjugated into a representation with values in GL(2, F′). The theorem applies to this latter representation, and shows that its image is the group of matrices in GL(2, F′) whose determinants lie in the multiplicative group of the prime field F_{p}.

To prove the well known statement that there is a model for ρ over F′, one may proceed alternatively by direct computation, along the lines suggested by Wiles [36, p. 483]. Remark 2. The referee has asked whether Theorem 2.5 extends to the case p = 3. The answer is negative; in fact, a counterexample is furnished by the abelian surface J_{0}(23). Recall that J_{0}(23) has “real multiplication” by the Hecke ring T associated with the space of weight-two cusp forms on Γ_{0}(23). The algebra T ⊗ Q is the real quadratic field Q(\sqrt{5}), and T is the ring of integers of T ⊗ Q. The group V of 3-division points on J_{0}(23) is a twodimensional vector space over the field F := T/3T, which has nine elements. The action of Gal(\overline{Q}/Q) on V is given by a homomorphism ρ: Gal(\overline{Q}/Q) → Aut_{F} V ≈ GL(2, F). As we shall see in §5, this representation is semistable; it is irreducible because 3 is prime to 23 − 1. Moreover, the traces tr(ρ(σ)) generate F; indeed, the trace of a Frobenius element Frob_{2} for the prime 2 satisfies x^{2} + x − 1 = 0. In view of Theorem 2.5, it is very tempting to guess that the image of ρ contains SL(2, F). Equivalently, consider the composite of ρ and the natural homomorphism GL(2, F) → PGL(2, F); let Ḡ be the image of this composite, so that Ḡ is a subgroup of PSL(2, F). The guess that the image of ρ contains SL(2, F) means that Ḡ = PSL(2, F).

What information do we have about Ḡ? The group Ḡ is “irreducible” (i.e., acts transitively on P^{1}(F)) since ρ is irreducible. Also, the order of Ḡ is divisible by 3, as one sees by considering an inertia subgroup of G for the prime 23. Finally, the order of Ḡ is divisible by 5 because of the information concerning tr(ρ(Frob_{2})), which implies that ρ(Frob_{2}) has order 5. The results of Dickson used above thus permit only two possibilities for Ḡ: either Ḡ is all of PSL(2, F), or else the alternating group A_{5}.

Rather to the author’s surprise, calculations based on [14, Table B] suggested strongly that Ḡ is in fact the smaller of these two groups. The author’s suspicion that this was the case deepened when he learned that there is an A_{5}-extension of Q which is ramified only at 3 and 23: the second line of [6, Table 1] shows that the splitting field of the polynomial x^{5} +3x^{3} +6x^{2} +9 is such an extension. Subsequently, Jean-François Mestre carried out computations which confirm that this latter A_{5}-extension is indeed the extension of Q which is cut out by the projective representation deduced from ρ. In particular, one has Ḡ ≈ A_{5}.

3. Products.

We next consider finite products of representations as above, keeping fixed the prime number p. Thus let F_{1}, …, F_{t} (t ≥ 1) be finite fields of characteristic p, where p is a prime which is different from 2 and 3. Let F be the finite étale F_{p}-algebra F_{1}×⋯×F_{t}. Suppose that ρ: Gal(\overline{Q}/Q) → GL(2, F) is a continuous homomorphism, so that ρ is a product of representations ρ_{i} (i = 1, …, t) as above. We will assume that each ρ_{i} is semistable and irreducible, and also that det ρ_{i} = χ for i = 1, …, t. With the evident convention, the latter hypothesis may be summarized by the formula det ρ = χ.

For each σ ∈ Gal(\overline{Q}/Q), we obtain an element tr(ρ(σ)) of F by considering the trace of the matrix ρ(σ). Motivated by the remark above, we let F′ be the subalgebra of F generated by the tr(ρ(σ)).

Lemma 3.1. The representation ρ has a model over F′.

Proof. As we have seen, each component ρ_{i} has a model over the subfield of F_{i} generated by the traces tr(ρ_{i}(σ)). After replacing ρ_{i} by such a model (and F_{i} by the trace field in question), we arrive at a situation in which F′ maps surjectively onto each factor F_{i}.

As one knows, F′ is then isomorphic to a partial product of the factors F_{i}. To see this explicitly, we let π: F → F_{i} be the projections (x_{1}, …, x_{t}) ↦ x_{i} and consider the following relation on the set { 1, …, t }: i ∼ j if and only if the map π_{i} × π_{j} : F′ → F_{i} × F_{j} is not surjective. It is easy to see that this relation is an equivalence relation and that i ∼ j if and only if there is an isomorphism σ: F_{i} →^{∼} F_{j} so that π_{j} = σ^{∘}π_{i} on F′. If there is such an isomorphism, it is unique; we denote it σ_{ji}. One shows that

F′ = {(x_{1}, …, x_{t}) ∈ F | x_{j} = σ_{ji}(x_{i}) for all pairs (i, j) such that i ∼ j}.

In particular, F′ is isomorphic to the product ∏_{i∈I} F_{i}, where I is a set of representatives for the equivalence ∼.

By the Brauer-Nesbitt theorem, ρ_{j} and ^{σ_{ji}}ρ_{i} are isomorphic whenever i and j are equivalent. (The two representations have the same trace and determinant.) Replace ρ_{j} by ^{σ_{ji}}ρ_{i} for all equivalent pairs (i, j) with i ∈ I. Then the representation ρ, a priori with values in GL(2, F), takes values in GL(2, F′).

Theorem 3.2. One has

ρ(Gal(\overline{Q}/Q)) = \{M ∈ GL(2, F) | det M ∈ F_{p}^{∗}\}

if and only if F′ = F.

Proof. The necessity is clear, since the trace function SL(2, F) → F is surjective. For the sufficiency, as in the proof of Theorem 2.5, one must show that ρ(Gal(\overline{Q}/Q)) contains SL(2, F). Let H = ρ(Gal(\overline{Q}/Q)) ∩ SL(2, F), so that H is a subgroup of the product SL(2, F_{1})×⋯×SL(2, F_{t}). By Theorem 2.5, H projects onto each factor SL(2, F_{i}). Because each group SL(2, F_{i}) is its own commutator subgroup, the “two principle” [19, 3.3] implies that H is the full product of the SL(2, F_{i}) if and only if H maps onto each product SL(2, F_{i}) × SL(2, F_{j}) for i ≠ j.

Assume, then, that we have i ≠ j, and suppose that the image of the product ρ_{i} ×ρ_{j} does not contain SL(2, F_{i})×SL(2, F_{j}). By exploiting results of Dieudonné and Hua, one may construct: (i) An isomorphism ω: F_{i} →^{∼} F_{j}, and (ii) a continuous homomorphism ϵ: Gal(\overline{Q}/Q) → F_{j}^{∗} such that ωρ_{i} and ρ_{j} ⊗ ϵ are isomorphic representations of Gal(\overline{Q}/Q) over F_{j}. This statement was proved during the course of the proof of [19, Th. 3.8] — the theorem itself states merely that ^{ω}ρ_{i} and ρ_{j} ⊗ ϵ have equal traces and determinants.

By Lemma 3.1, however, we have ϵ = 1; thus ^{ω}ρ_{i} ≈ ρ_{j}. Accordingly, we have ω(tr(ρ_{i}(σ))) = tr(ρ_{j}(σ)) for all σ ∈ Gal(\overline{Q}/Q). This means that the image of tr(ρ(σ)) in F_{i}×F_{j} lies in the subalgebra { (x, ωx) | x ∈ F_{i} } of F_{i}×F_{j}. Thus the quantities tr(ρ(σ)) fail to generate F_{i}×F_{j} over F_{p}. This contradicts the hypothesis that they generate the full product F over F_{p}.

Corollary 3.3. Assume that ρ is as in Theorem 3.2. Then the image of ρ is conjugate in GL(2, F) to the group

{M ∈ GL(2, F′) | det M ∈ F_{p}^{∗}}.

Proof. By Lemma 3.1, there is a model for ρ over F′: after conjugating by a matrix in GL(2, F), we make ρ take values in GL(2, F′). Applying the Theorem to this model, we arrive at the desired conclusion.

We continue the discussion begun with Theorem 2.6, letting X be the subgroup of Gal(\overline{Q}/Q) which appears in the statement of that result.

Theorem 3.4. In the situation of Theorem 3.2, suppose that F = F′. Then ρ(X) = { M ∈ GL(2, F) | det M ∈ F_{p}^{∗} }.

Proof. The group ρ(X) is a normal subgroup of G = ρ( Gal(\overline{Q}/Q)). By Theorem 3.2, G = { M ∈ GL(2, F) | det M ∈ F_{p}^{∗} }. The intersection ρ(X) ∩ SL(2, F) is then normal in SL(2, F); it maps onto each factor SL(2, F_{i}) by Theorem 2.6. On taking commutators with elements of SL(2, F) of the form (1, …, 1, α, 1, …, 1), we see that ρ(X)∩SL(2, F) contains SL(2, F_{i}) (viewed as a subgroup of SL(2, F)) for each i. It follows that ρ(X) contains SL(2, F). We then obtain ρ(X) = G, since the mod p cyclotomic character is totally ramified at p.

4. Lifts.

Again suppose that p is a prime ≥ 5 and let O_{1}, …, O_{t} be integer rings of finite-degree unramified extensions of Q_{p}. For each i, let ρ̃_{i}: Gal(\overline{Q}/Q) → GL(2, O_{i}) be a continuous representation whose determinant is the p-adic cyclotomic character χ̃: Gal(\overline{Q}/Q) → Z_{p}^{∗}. Let ρ̃ be the product of the ρ̃_{i}, so that ρ̃ is a p-adic representation Gal(\overline{Q}/Q) → GL(2, O), where O := ∏ O_{i}. The diagonal embedding Z_{p} ↪ O induces an inclusion Z_{p}^{∗} ↪ O^{∗}. We clearly have ρ̃(Gal(\overline{Q}/Q)) ⊆ A, where

A = { M ∈ GL(2, O) | det M ∈ Z_{p}^{∗} }.

Let F = O/pO, and let ρ: Gal(\overline{Q}/Q) → GL(2, F) be the reduction of ρ̃ mod p.

In the following statement, X is again the closed normal subgroup of Gal(\overline{Q}/Q) which is generated by the inertia groups for p in Gal(\overline{Q}/Q).

Theorem 4.1. Suppose that the mod p reduction of each ρ̃_{i} is semistable and irreducible. Assume that F is generated as an F_{p}-algebra by the traces tr ρ(g) with g ∈ Gal(\overline{Q}/Q). Then ρ̃(X) = ρ̃(Gal(\overline{Q}/Q)) = A.

Remark. Using Nakayama’s Lemma, one may reformulate the trace hypothesis as the apparently stronger assertion that O is generated as a Z_{p}- algebra by the traces tr ρ̃(g) with g ∈ Gal(\overline{Q}/Q).

Proof. The group ρ̃(Gal(\overline{Q}/Q)) and its subgroup ρ̃(X) are closed subgroups of A whose determinants are equal to all of Z_{p}^{∗}. It suffices to show that ρ̃(X) contains SL(2, O). We recall the following fact (see [25, p. IV-23] and [19, Th. 2.1]):

Proposition 4.2. Let G be a closed subgroup of GL(2, O) and let Ḡ be the image of G in GL(2, O/pO). If Ḡ contains SL(2, O/pO), then G contains SL(2, O).

Proof. On taking G = ρ̃(X), we now obtain the required inclusion ρ̃(X) ⊇ SL(2, O) from Theorem 3.4.

5. Semistable abelian varieties of GL_{2}-type over Q.

Let A be an abelian variety over Q for which Q ⊗ End_{Q} A is a number field of degree equal to the dimension of A. Suppose that the ring R = End_{Q} A is the full ring of integers in the field Q ⊗ End_{Q} A. (After replacement of A by an isogenous abelian variety, this hypothesis is always verified.) Let 𝔪 be a maximal ideal of R and let A[𝔪] be the kernel of 𝔪 on A, i.e., the group of points in A(\overline{Q}) which are annihilated by all elements of 𝔪. It is easy to check that A[𝔪] is free of rank two over F = R/𝔪, cf. [32, Prop. 10, p. 56]. In fact, the representation

ρ: Gal(\overline{Q}/Q) → GL(2, F)

defined by the action of Gal(\overline{Q}/Q) on A[𝔪] is just the mod 𝔪 reduction of the 𝔪-adic representation of Gal(\overline{Q}/Q) which is attached to A. As usual, we denote by p the characteristic of F.

Proposition 5.1. Assume that p is odd, that ρ is irreducible, and that A is semistable over Q. Then ρ is a semistable representation.

Proof. We first show that the determinant of ρ is the mod p cyclotomic character χ. According to the statement of [23, Lemma 3.1], we have det ρ = ϵχ, where ϵ is a Dirichlet character Gal(\overline{Q}/Q) → R^{∗} which depends only on A (i.e., is independent of 𝔪) and which is ramified only at primes of bad reduction for A. In fact, the proof of this Lemma shows that ϵ is unramified at a prime q whenever the following condition is satisfied: there is a prime λ of R such that the determinant of the λ-adic representation of A is unramified at q. Now fix q and take any prime λ not dividing q. By a well known result of Grothendieck [9, Prop. 3.5], each element of an inertia subgroup for q in Gal(\overline{Q}/Q) acts unipotently in the λ-adic representation for A. In particular, the determinant of this representation is unramified at q. Thus ϵ is unramified at every prime number q, so that ϵ is the trivial character.

The same proposition of Grothendieck, applied to the 𝔪-adic representation for A, shows that the conductor of ρ is square free. We may paraphrase this statement by saying that ρ is semistable outside p. Furthermore, results of Raynaud [16] imply that k(ρ) = 2, so that ρ is semistable at p, whenever ρ is finite at p. (See [28, Prop. 4, p. 189].) It remains only to show that k(ρ) = p+1 if ρ is not finite at p.

For this, we consider A[𝔪] as a subgroup of A[p], and view both as modules for a decomposition group D_{p} in Gal(\overline{Q}/Q) for the prime p. As explained in the proof of [21, Lemma 6.2], one may deduce from Grothendieck’s study in SGA7I that A[p] is an extension of an unramified D_{p}-module by a subgroup A[p]^{f} which is finite, i.e., which extends to a finite flat group scheme over Z_{p}. Were A[𝔪] contained in A[p]^{f}, A[𝔪] would be finite, contrary to assumption. Hence A[𝔪] has an unramified quotient. This implies that the restriction

of ρ to D_{p} has the form (\begin{matrix}θ_{1}χ & ∗ \\ 0 & θ_{2} \end{matrix}), where the θ_{i} are unramified characters.

The recipe for k(ρ) given in [28, §2] then sets k(ρ) = p+1. (Compare Remarque (1) on page 188 of [28].)

6. Application to J_{0}(N) for prime N.

Let N be a prime number, and consider the abelian variety J = J_{0}(N) over Q. In this section and the next, we will study the action of Gal(\overline{Q}/Q) on torsion points of J. The first work in this direction was the investigation of Shimura [30], which concerns the mod p representation of Gal(\overline{Q}/Q) defined by J_{0}(11), when p lies between 7 and 97. Shimura’s discussion was completed by Serre [26, §5.5], who determined for all p the image of the mod p representation attached to J_{0}(11). Subsequently, Lang-Trotter [12], Part I, §8; calculated the image of the (adelic) representation of Gal(\overline{Q}/Q) defined by all torsion points of J_{0}(11).

In what follows, we generalize some of the results of Serre and Lang-Trotter to the case where 11 is replaced by an arbitrary prime. We exploit the insights of Mazur’s article [13], which makes an extensive study of the action of Gal(\overline{Q}/Q) on torsion points of J_{0}(N), and the recent thesis of B. Kaskel [10], which determines the image of the adelic representation defined by J_{0}(37). There is a certain amount of overlap with the joint article [2].

Let T be the ring of Hecke operators Z[…, T_{n}, …], considered as a ring of endomorphisms of J_{0}(N). To orient the reader, we recall that T is the full ring of endomorphisms of J_{0}(N) [13, p. 95]. To avoid the situation where N is very small, we will assume that J_{0}(N) ≠ 0. This means that N = 11 or that N ≥ 17.

Let p be a prime number ≥ 5.

Proposition 6.1. The quotient T/pT is generated by the operators T_{n} with n prime to pN.

Proof. Let S be the space of cusp forms of weight 2 on Γ_{0}(N) over F_{p}. We view S as the mod p reduction of the space of weight-two forms on Γ_{0}(N) with integral q-expansions. (See [13, Ch. II, §4] for a comparison of several possible definitions of S.) We consider the bilinear pairing α: T/pT × S → F_{p} which maps (T, f) to the initial coefficient of q in the Fourier expansion of f|T. It is well known that α is a perfect pairing; this is explained, for example, in [20, §1]. To prove the Proposition, then, it suffices to prove that there is no non-zero element f = ∑ a_{n}q^{n} of S which satisfies a_{n} = 0 for all n prime to pN.

Suppose that f = ∑ a_{n}q^{n} satisfies the condition. We will show first that a_{n} = 0 for all n prime to N. If N = p there is nothing extra to prove, so we will assume for the moment that N and p are distinct. Let g be the form ∑_{(N,n)=1} a_{n}q^{n}, i.e., the sum

∑_{n=1}^{∞} b_{n}q^{n}, b_{n} = \begin{cases}a_{n} & if (n, N) = 1, \\ 0 & otherwise. \end{cases}

Then b_{n} = 0 whenever n is not a multiple of p; our aim is to show that g = 0. Now the point is that g may be considered as a weight-two mod p modular form on Γ_{0}(N^{2}); this follows from [31, Prop. 3.64]. The hypothesis about the vanishing of the b_{n} means that g is annihilated by the operator θ = q \frac{d}{dq}. Since p > 2, this forces g = 0 as desired, since θg has “filtration” p + 3 if g is non-zero. (This is Katz’s generalization of the Serre-Swinnerton-Dyer theorem [11, p. 55].)

To complete the proof, we must show that f = 0. This follows from Proposition 4.10, Lemma 5.9 and Lemma 5.10 of [13, Ch. II].

Remark. The argument we have given is essentially that of [13], Ch. II, Prop. 14.13. The exploitation of q \frac{d}{dq} to deal with the absence of T_{p} is a familiar ploy it was used by the author in [22, Prop. 2] and by Wiles in [36, Lemma, p. 491].

Suppose now that 𝔪 is a maximal ideal of T of residue characteristic p. (We continue to assume p ≥ 5.) Let J[𝔪] denote the kernel of 𝔪 on J(\overline{Q}). By the results of [13], J[𝔪] defines the two-dimensional semisimple representation ρ_{𝔪} of Gal(\overline{Q}/Q) which is associated with 𝔪; this representation is irreducible if and only if 𝔪 is not an Eisenstein prime of T. When 𝔪 is non-Eisenstein, ρ_{𝔪} is a semistable representation: this follows from the results of Deligne-Rapoport [3] to the effect J has multiplicative reduction at N, together with Proposition 5.1.

Proposition 6.2. The image of ρ_{𝔪}: Gal(\overline{Q}/Q) → GL(2, T/𝔪) is the group of elements of GL(2, T/𝔪) having determinant in F_{p}^{∗}.

Proof. This mild strengthening of [13, Ch. II, Prop. 14.12] may be derived directly from Theorem 2.5. Indeed, for each prime ℓ different from p and N, the image under ρ_{𝔪} of a Frobenius element for ℓ in Gal(\overline{Q}/Q) has trace T_{ℓ} mod 𝔪. It follows from the Lemma above that the T_{ℓ} mod 𝔪 generate T/𝔪.

Remark. By using Theorem 2.6 in place of Theorem 2.5, we obtain the stronger statement that ρ_{𝔪}(X) is the group of matrices in GL(2, T/𝔪) whose determinants lie in F_{p}^{∗}. A similar remark applies to the Proposition which follows.

Suppose next that p ≥ 5 is such that none of the 𝔪|p in T is an Eisenstein prime. This means simply that N ≢ 1 mod p. Let F be the F_{p}-algebra ∏_{m|p} T/𝔪, and let

ρ: Gal(\overline{Q}/Q) → GL(2, F)

be the product of the ρ_{𝔪}.

Proposition 6.3. The image of ρ is the group of matrices in GL(2, F) having determinant in F_{p}^{∗}.

Proof. Indeed, the natural map T/pT → F is surjective by the Chinese remainder theorem. Accordingly, the Lemma implies that F is generated by the images of the T_{ℓ} with ℓ prime to pN. In the language of Theorem 3.2, this means that F′ = F. Applying that theorem, we find that the image of ρ is as stated.

Continuing the discussion, we add the hypothesis that the surjection T/pT → F is an isomorphism, i.e., that p is unramified in T. Then ρ is the representation giving the action of Gal(\overline{Q}/Q) on the group J[p], viewed as a free T/pT-module of rank two. Proposition 6.3 thus furnishes a description of the Galois group of the field cut out by the p-division points of J. Further, as we have seen in Proposition 4.2, the result of Proposition 6.3 is equivalent to an analogous statement about the p-adic Galois representation defined by J. More precisely, because p in unramified in T, T ⊗ Z_{p} is a product of discrete valuation rings. The Tate module Ta_{p}(J) of J is then automatically free of rank two over T ⊗ Z_{p}. After choosing a basis of Ta_{p}(J), we may summarize the action of Gal(\overline{Q}/Q) on the p-power division points of J by a homomorphism

ρ̃: Gal(\overline{Q}/Q) → GL(2, T ⊗ Z_{p}).

By Propositions 4.2 and 6.3, we find that the image of ρ̃ is “as large as possible”:

Theorem 6.4. Suppose that p is unramified in T and that p is prime to 6 · (N−1). Then ρ̃( Gal(\overline{Q}/Q)) = \{ M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗} \}.

Remark. We obtain the more precise equality

ρ̃(X) = { M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗} }

from Theorem 3.4 and Proposition 4.2.

7. Complements.

We continue our study of J = J_{0}(N), where N is a prime number by studying products of p-adic representations attached to J. We are motivated by the discussions of [25, Ch. IV, §3] and [26, §4.4], whose tools serve very well in this context.

For each prime p, let Ta_{p}(J) be the Z_{p}-adic Tate module of J and write ρ_{p} (rather than ρ̃, as above) for the p-adic representation of Gal(\overline{Q}/Q) which is associated to J:

ρ̃: Gal(\overline{Q}/Q) → Aut_{T⊗Z_{p}} (Ta_{p}(J)) ⊂ Aut_{T⊗Q_{p}} (Ta_{p}(J) ⊗ Q) .

Let G_{p} be the image of ρ_{p}. This group was determined in Theorem 6.4 for most prime numbers p. We shall describe the p-adic Lie algebra of G_{p} in general; thus we determine G_{p} “up to finite groups” even when (or especially when) p does not satisfy the hypothesis to Theorem 6.4.

We recall that Mazur proved [13, Ch. II, §15-§17] that Ta_{p}(J) is free of rank two over T ⊗ Z_{p} when p is odd, and also in many circumstances when p = 2. In this favorable situation, we may view G_{p} as a closed subgroup of GL(2, T ⊗ Z_{p}). In any case, Ta_{p}(J) ⊗ Q_{p} is free of rank two over the Q_{p}- algebra T ⊗ Q_{p} by [13, Ch. II, Lemma 7.7]. Hence we may always regard G_{p} as a closed subgroup of GL(2, E), where E = T ⊗ Q_{p}. Note that E is a commutative semisimple Q_{p}-algebra, i.e., the product of fields which are finite extensions of Q_{p}. Because the determinant of ρ_{p} is the p-adic cyclotomic character, we have G_{p} ⊆ H_{p}, where

H_{p} := {M ∈ GL(2, E) | det M ∈ Q_{p}^{∗}}.

Proposition 7.1. The group G_{p} is open in H_{p}.

Proof. Let 𝔤 and 𝔥 be the p-adic Lie algebras of G_{p} and H_{p}, respectively. Thus

𝔤 ⊆ 𝔥 = Q_{p} × 𝔰𝔩_{2}(E),

where 𝔰𝔩_{2}(E) = [𝔥, 𝔥] is the Lie algebra of two-by-two matrices over E with trace 0. Since the p-adic cyclotomic character has infinite order, g is not contained in 𝔰𝔩_{2}(E). The proposition states that 𝔤 = 𝔥.

The equality 𝔤 = 𝔥 is proved as [18, Th. 4.5.4] in the special case where T⊗Q is a field, i.e., where J is a simple abelian variety. (Note that End\overline{Q} J = End_{Q} J, as was proved in [17]; hence J is simple over Q if and only if it is absolutely simple.) In the general case, J is isogenous over Q to a product A_{1} × ⋯ × A_{t} of simple abelian varieties to which [18, Th. 4.5.4] applies. Thus 𝔤 and 𝔥 have equal images in End(Ta_{p}(A_{i}) ⊗ Q_{p}) for each i. Moreover, one knows that End_{𝔤}(Ta_{p}(J)) = E = End_{𝔥}(Ta_{p}(J)). Indeed, the Tate conjecture for abelian varieties, which was proved in [5], implies that End_{𝔤}(Ta_{p}(J)) = (End\overline{Q} J) ⊗ Q_{p}. On the other hand, one knows by [17] that (End\overline{Q} J) ⊗ Q = T ⊗ Q; hence (End\overline{Q} J) ⊗ Q_{p} = E, as was claimed. The proof of [18, Th. 4.4.10] now yields the required equality 𝔤 = 𝔥.

For each set of prime numbers S, let

ρ_{S}: Gal(\overline{Q}/Q) → ∏_{p∈S} Aut(Ta_{p}(J))

be the product of the ρ_{p} for p in S. Let G_{S} be the image of ρ_{S}, so that G_{S} is a closed subgroup of the product ∏_{p∈S} G_{p}.

Corollary 7.2. If S is a finite set of primes, then G_{S} is open in H_{S} := ∏_{p∈S} H_{p}.

Proof. The assertion to be proved follows from Proposition 7.1 and an argument due to Serre [25, Lemma 4, p. IV-24]:

Fix p ∈ S for the moment, and ensure by a change of basis if necessary that G_{p} is a subgroup of GL(2, R), where R is the ring of integers of E. Let F be the product of the residue fields of the factors of R, and let G′_{p} be the kernel of the composition

G_{p} ↪ GL(2, R) → GL(2, F).

Clearly, G′_{p} has finite index in G_{p}, so that G′_{p} is open in H_{p} by Proposition 7.1. Further, G′_{p} is a pro-p group; in fact, it is a projective limit of nilpotent groups of p-power order.

Let G′_{S} be the inverse image of ∏_{p∈S} G′_{p} in G_{S}. Since G′_{S} is a subgroup of ∏_{p∈S} G′_{p}, it is pro-nilpotent. Thus G′_{S} is the product of its Sylow subgroups. Now the p-Sylow subgroup of G′_{S} has finite index in G_{p}. Thus G′_{S} has finite index in ∏_{p∈S} G_{p}, a group which is open in H_{S} by Proposition 7.1. Hence G_{S} is open in H_{S}.

Theorem 7.3 (Kaskel [10]). Assume that N does not belong to S. Then G_{S} = ∏_{p∈S} G_{p}.

Proof. We first consider the case where S is finite, arguing by induction on the size of S. The statement to be proved is evident if S has at most one element, so we may assume that S = T ∐{p} and that the statement is true with S replaced by T. We must show that the natural injection G_{S} ↪ G_{T} × G_{p} is an isomorphism, or equivalently that the Galois extensions of Q cut out by ρ_{T} and ρ_{p} are linearly disjoint. Let K be the intersection of these two fields. Then K is ramified only at N, since ρ_{p} is ramified only at N and at p, while ρ_{T} is unramified at p. Further, the inertia groups for the prime N in the image of ρ_{p} are pro-p groups since J_{0}(N) is semistable at N. Similarly, the inertia groups for N in the image of ρ_{T} are profinite groups of order prime to p; indeed, p is not a member of T. Hence the inertia groups for N in Gal(K/Q) are trivial, so that K is an everywhere unramified extension of Q. This gives K = Q and proves the linear disjointness.

The case where S is not necessarily finite now follows, since G_{S} is closed and dense in ∏_{p∈S} G_{p}.

Corollary 7.4. Let S be the set of prime numbers p which are prime both to 6(N − 1) and to the discriminant of T. Then

G_{S} = ∏_{p∈S}{M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗}}.

Proof. We shall give two proofs of this result, the first in spirit of Kaskel’s theorem. For each p ∈ S, we have G_{p} = { M ∈ GL(2, T⊗Z_{p}) | det M ∈ Z_{p}^{∗} } by Theorem 6.4. Let T be the set of primes p ∈ S which are different from N; thus S = T ∐{N} if N is prime to the discriminant of T and S = T if not.

We clearly have G_{T} = ∏_{p∈T} { M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗} } by Kaskel’s theorem and Theorem 6.4. This gives what is wanted if S = T. In what follows, we suppose to the contrary that N belongs to S.

The idea now is to analyze the subgroup G_{S} of G_{T} × G_{N} as in the proof above. Let K/Q now be the obstruction to the equality G_{S} = G_{T} × G_{N}. In other words, K is the intersection of the two extensions of Q whose Galois groups are the images of G_{T} and G_{N}. Clearly, K is ramified only at N. Moreover, I is an inertia group for N in Gal(K/Q), then the order of I (as a supernatural number) is divisible only by the primes in T. In particular, this order is prime to N(N − 1). On the other hand, it is easy to see that only N and primes dividing N − 1 can intervene in the order of an inertia group for N in the image of ρ_{N}. Indeed, the restriction of ρ_{N} to an inertia group

for N in Gal(\overline{Q}/Q) has the form (\begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix}), where χ is the N-adic cyclotomic

character. Hence K is unramified at N, and we obtain the required equality G_{S} = G_{T} × G_{N} as in the proof of Kaskel’s theorem.

To make a second proof of the equality

G_{S} = ∏_{p∈S}{M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗}},

we fix a prime p in T and let X be the group which we considered in Theorem 2.6 and Theorem 3.4. We have ρ_{p}(X) = { M ∈ GL(2, T⊗Z_{p}) | det M ∈ Z_{p}^{∗} } as explained in the remark at the end of the preceding section. On the other hand, if p′ is an element of S which is different from p, ρ_{p′} is unramified at p and therefore ρ_{p′}(X) = {1}. Thus ρ_{S}(X) contains

{ M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗} },

viewed as a subgroup of the product. Hence we have

G_{S} ⊇ ∏_{p∈T}{M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗}}.

This gives what is wanted if S = T, so we assume once again that N belongs to S. Then G_{S} is a subgroup of the product

∏_{p∈S}{ M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗} }

which maps onto { M ∈ GL(2, T ⊗ Z_{N}) | det M ∈ Z_{N}^{∗} } by Theorem 6.4. Moreover G_{S} contains the kernel of the natural projection

∏_{p∈S}{M ∈ GL(2, T ⊗ Z_{p}) | det M ∈ Z_{p}^{∗}}

→ {M ∈ GL(2, T ⊗ Z_{N}) | det M ∈ Z_{N}^{∗}}.

Therefore G_{S} is the full product.

For our final result, we consider the product ρ_{f} of all of the representations ρ_{p}, i.e., the representation ρ_{S} where S is the set of all prime numbers. We view ρ_{f} as taking values in

{M ∈ GL(2, T⊗Q_{2}) | det M ∈ Q_{2}^{∗}}×∏_{p≠2}{M ∈ GL(2, T⊗Z_{p}) | det M ∈ Z_{p}^{∗}},

a group that we will call A. (We separate 2 from the odd primes since it is not known that Ta_{2} is free of rank two over T ⊗ Z_{2}.)

Theorem 7.5. The image of ρ_{f} is an open subgroup of A.

Proof. Let A_{p} be the pth component of A, so that we have A = ∏ A_{p}, with the product extended over all primes. Let S be the set of those p which are prime to 6(N − 1)N and to the discriminant of T. Fix p ∈ S and let X be the subgroup of Gal(\overline{Q}/Q) which we have considered repeatedly: the closed subgroup of Gal(\overline{Q}/Q) generated by all inertia groups for p. As we have seen, ρ_{p}(X) = A_{p}; on the other hand ρ_{p′}(X) = {1} for p′ ≠ p. Hence ρ_{f}(X) = A_{p}, where A_{p} is considered as a subgroup of the product A. On varying p, we find that ρ_{f} (Gal(\overline{Q}/Q)) contains the group ∏_{p∈S} A_{p}, i.e., the kernel of the projection A → ∏_{p∉S} A. On the other hand, the image of Gal(\overline{Q}/Q) in this finite product is open by Corollary 7.2. Hence ρ_{f} (Gal(\overline{Q}/Q)) is open in the full product.

References

[1] C. Batut, D. Bernardi, H. Cohen and M. Olivier, GP/PARI, Available by anony-

      mous ftp from megrez.math.u-bordeaux.fr or math.ucla.edu, in the directory
      /pub/pari.

[2] R. Coleman, B. Kaskel and K. Ribet, Torsion points on X_{0}(N), Contemporary Math., to appear.

[3] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Lecture Notes in Math., Vol. 349, Springer-Verlag, Berlin and New York, (1973), 143-316.

[4] L.E. Dickson, Linear groups with an exposition of the Galois field theory, Dover, New York, 1958.

[5] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.

[6] G. Frey (Ed.) On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., Vol. 1585, Springer-Verlag, Berlin-Heidelberg-New York, 1994.

[7] D. Gorenstein, Finite Groups, Chelsea, New York, 1980.

[8] B.H. Gross, A tameness criterion for Galois representations associated to modular forms mod p, Duke Math. J., 61 (1990), 445-517.

[9] A. Grothendieck, SGA7 I, Exposé IX, Lecture Notes in Math., Vol. 288, Springer-Verlag, Berlin and New York, (1972), 313-523.

[10] B. Kaskel, The adelic representation associated to X_{0}(37), PhD. thesis, UC Berkeley, May, 1996.

[11] N.M. Katz, A result on modular forms in characteristic p, Lecture Notes in Math., Vol. 601, Springer-Verlag, Berlin and New York, (1977), 53-61.

[12] S. Lang and H. Trotter, Frobenius distributions in GL_{2}-extensions, Lecture Notes in Math., Vol. 504, Springer-Verlag, Berlin and New York, 1976.

[13] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES, 47 (1977), 33-186.

[14] T. Miyake, Modular forms, Springer-Verlag, Berlin and New York, 1989.

[15] J. Oesterlé, Travaux de Wiles (et Taylor, …), Partie II, Séminaire Bourbaki, 804 (1995).

[16] M. Raynaud, Schémas en groupes de type (p, …, p), Bull. Soc. Math. France, 102 (1974), 241-280.

[17] K.A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Annals of Math., 101 (1975), 555-562.

[18] , Galois action on division points of abelian varieties with many real multiplications, Am. J. Math., 98 (1976), 751-804.

[19] , On ℓ-adic representations attached to modular forms, Invent. Math., 28 (1975), 245-275.

[20] , Mod p Hecke operators and congruences between modular forms, Invent. Math., 71 (1983), 193-205.

[21] , On modular representations of Gal(\overline{Q}/Q) arising from modular forms, Invent. Math., 100 (1990), 431-476.

[22] , Multiplicities of p-finite mod p Galois representations in J_{o}(Np), Boletim da Sociedade Brasileira de Matemática, Nova Série, 21 (1991), 177-188.

[23] , Abelian varieties over Q and modular forms, 1992 Proceedings of KAIST Mathematics Workshop, Taejon, Korea Advanced Institute of Science and Technology, (1992), 53-79.

[24] I. Schur, Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen, Sitz. Pr. Akad. Wiss., (1906), 164-184; Gesam. Abhl., I, 177-197, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1973.

[25] J.-P. Serre, Abelian ℓ-adic representations and elliptic curves, Addison-Wesley Publ. Co., Redding, Mass., 1989.

[26] , Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15 (1972), 259-331.

[27] , Œuvres, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986.

[28] , Sur les représentations modulaires de degré 2 de Gal(\overline{Q}/Q), Duke Math. J., 54 (1987), 179-230.

[29] , Travaux de Wiles (et Taylor, …), Partie I, Séminaire Bourbaki, 803 (1995).

[30] G. Shimura, A reciprocity law in non-solvable extensions, Journal für die reine und angewandte Mathematik, 221 (1966), 209-220.

[31] , Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, 1971.

[32] G. Shimura and Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Second printing corrected, Math. Society of Japan, Tokyo, 1975.

[33] J. Sturm, On the congruence of modular forms, Lecture Notes in Math., Vol. 1240, Springer-Verlag, Berlin and New York, (1987), 275-280.

[34] J. Tate, The non-existence of certain Galois extensions of Q unramified outside 2, Contemporary Mathematics, 174 (1994), 153-156.

[35] J.-L. Waldspurger, Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2), Compositio Math., 54 (1985), 121-171.

[36] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Annals of Math., 141 (1995), 443-551.

This article was begun while the author was a research professor at the MSRI, where research was supported in part by NSF grant DMS-9022140. This work was further supported by the investigator’s NSF Grants DMS 88-06815 and DMS 93-06898. The author thanks the referee for a number of interesting suggestions and questions and Ian Kiming and Jean-François Mestre for carrying out computations in connection with the example discussed at the end of §2.

University of California Berkeley, CA 94720-3840 USA E-mail address: ribet@math.berkeley.edu