Elliptic Curves of Odd Modular Degree

Frank Calegari^{∗}  Matthew Emerton^{†}

December 17, 2007

1 Introduction

Let E be an elliptic curve over Q of conductor N. Since E is modular [3], there exists a surjective map π : X_{0}(N) → E defined over Q. There is a unique such map of minimal degree (up to composing with automorphisms of E), and its degree m_{E} is known as the modular degree of E. This degree has been much studied, both in relation to congruences between modular forms [38] and to the Selmer group of the symmetric square of E [14], [15], [35]. Since this Selmer group can be considered as an elliptic analog of the class group, one might expect in analogy with genus theory to find that m_{E} satisfies certain divisibility properties, especially, perhaps, by the prime 2. More precisely, we remind the reader that the class number of an imaginary quadratic extension K of Q is odd if and only if the absolute discriminant D_{K} of K is equal either to 4 or else to an odd prime (and similarly for the strict class number of a real quadratic extension). In this paper we consider the analogous question of the parity of m_{E}. In particular we establish a conjecture of Watkins (see theorem 1.7 below) to the effect that if E has no rational 2-torsion points, then m_{E} is odd only if the conductor N of E is a prime congruent to 3 mod 8. In fact we prove the following more precise theorem:

1.1 Theorem. If E/Q is an elliptic curve of odd modular degree, then:

1. The conductor N of E is divisible by no more than two odd primes,

2. E is of even analytic rank, and

3. One of the following holds:

(a) E has a rational point of order 2 (equivalently, admits a rational 2-isogeny);

(b) E has prime conductor and supersingular reduction at 2, and Q(E[2]) is totally complex (equivalently, E(R) is connected);

(c) E has complex multiplication, and N = 27, 32, 49, or 243.

∗Supported in part by the American Institute of Mathematics

†Supported in part by NSF grant DMS-0401545

1.2 Example. The following examples of elliptic curves with odd modular degree should serve to illustrate conditions (3a), (3b) and (3c). The curve X_{0}(15) has modular degree one and a rational two torsion point, and thus satisfies condition (3a). Another example is given by the curve

y^{2} + xy = x^{3} − x^{2} − 58x − 105

(2537E in Cremona’s tables) of conductor 43 · 59 with modular degree 445 and torsion subgroup Z/4Z. The curves X_{0}(11) and X_{0}(19) both have modular degree one and satisfy condition (3b). An example of larger conductor is given by

y^{2} + y = x^{3} + x^{2} − 4x − 10

of conductor 24859 and modular degree 3979. Finally, there are exactly four curves of odd modular degree with complex multiplication, namely X_{0}(27), X_{0}(32), X_{0}(49) (all of modular degree one) and

y^{2} + y = x^{3} + 2

of modular degree 9, conductor 243 and j-invariant 0.

1.3 Remark. Each of the conditions appearing in Theorem 1.1 is invariant under isogeny, other than the condition that E(R) be connected, which however is invariant under isogenies of odd degree. Since the modular parameterization of E factors through the optimal member of the isogeny class of E (that is, the member of its isogeny class having minimal modular degree; in older terminology, a strong Weil curve), it is therefore no loss of generality in the proof of Theorem 1.1 to assume that E is optimal.

1.4 Remark. Cremona and Watkins have computed the modular degree of every optimal elliptic curve of conductor ≤ 25,000 [8] and these computations suggest that there may be even stronger limitations on the conductor of a curve of odd modular degree than those imposed by Theorem 1.1. Indeed, in the range of Watkins’ computations, every curve of odd modular degree has conductor divisible by at most two primes, and the conductor always has one of the following forms: 2p, 4p, or pq, where p and q are odd primes.

1.5 Remark. The statement of the Theorem regarding the analytic rank of E is consistent with the conjecture of Birch and Swinnerton-Dyer and with the rank conjecture of Watkins [35, Conj. 4.1] that 2^{r}|m_{E} where r = rank(E(Q)).

1.6 Remark. In [32] the parity of m_{E} was determined for a very particular explicit class of elliptic curves, namely, the Neumann–Setzer curves, which are the curves of prime conductor > 17 which have a rational 2-torsion point. (See also the remark following Theorem 5.1 below.)

The following result, conjectured by Watkins ([32, Conj. 4.3], [35, Conj. 4.2]), is a simple consequence of Theorem 1.1 (see Lemma 3.3):

1.7 Theorem. Let E/Q be an elliptic curve of prime conductor N, and suppose that E is neither a Neumann–Setzer curve, nor X_{0}(17) (equivalently, E does not have a rational 2-torsion point). If m_{E} is odd, then N ≡ 3 mod 8.

One technique for proving that an elliptic curve E has even modular degree is to show that the map π factors through X_{0}(N)/w for some non-trivial Atkin–Lehner involution w. We use this approach in section 2 to prove Theorem 2.1, which in turn implies parts (1) and (2) of Theorem 1.1, and shows that (3a) holds if N is divisible by at least two primes. It remains to prove (3) in the case when N is a prime power. The most difficult case to handle is when N is actually prime, and in this case we deduce Theorem 1.1 from the following result, proved in section 3.

1.8 Theorem. Let N be prime, let T denote the Hecke algebra over Z acting on weight two cuspforms on Γ_{0}(N), and let 𝔪 be a maximal ideal of T such that T/𝔪 = F_{2}, and such that the associated semi-simple Galois representation \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{2}) is irreducible. If the completion T_{𝔪} = Z_{2}, then

1. 𝔪 is supersingular at 2 (i.e. T_{2} ∈ 𝔪), or equivalently, \overline{ρ}|_{D_{2}} is absolutely irreducible.

2. \overline{ρ} is totally complex.

The relevance of this result to Theorem 1.1 is that, since N is prime in the context of Theorem 1.8, a result of Ribet [38] shows (assuming, as we may, that E is optimal) that the modular degree of E is even if and only if 2 is a congruence prime for the newform of level N attached to E.

1.9 Remark. The equivalence between the supersingularity of 𝔪 and the absolute irreducibility of \overline{ρ}|_{D_{2}} can be deduced (in the standard way) as follows: if V_{/Z_{2}} denote a finite group scheme subquotient of J[𝔪]_{/Z_{2}} whose generic fibre (regarded as a Galois module) is isomorphic to \overline{ρ}|_{D_{2}}, then both conditions are equivalent to V being local-local. For the first condition, this follows from the Eichler-Shimura relation and a calculation with Dieudonné modules; see [23, p. 113]. For the second, note that if V is not local-local, then it admits either a multiplicative or an étale subgroup scheme, either of which gives rise to an étale subgroup scheme of the generic fibre V_{/Q_{2}}. Thus \overline{ρ}|_{D_{2}} admits an unramified subrepresentation, and hence is not absolutely irreducible. Conversely, if \overline{ρ}|_{D_{2}} is absolutely reducible, then (extending scalars to F_{4} if necessary) it contains a one-dimensional subrepresentation, which (by local class field theory at 2) must be an unramified character. The Zariski closure in V (or in F_{4} ⊗_{F_{2}} V) of this subrepresentation gives rise to a rank one subgroup scheme (or subvector space scheme) which is either étale or multiplicative. Thus V is not local-local. Of course, suitably modified forms of these arguments apply to any maximal ideal in T of residue characteristic prime to N (a not necessarily prime level); see for example [13, Thms. 2.5, 2.6].

The proof of Theorem 1.8 is motivated by the following considerations: If p is an odd prime and \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{p}) is a surjective modular representation, then theorems of Wiles and Taylor–Wiles [37, 34] show that the universal minimal deformation ring R_{∅} attached to \overline{ρ} is isomorphic to the universal minimal modular deformation ring T_{∅} (= T_{𝔪}, since N is prime). Since T_{∅} is a finite W (F_{p}) = Z_{p} algebra with residue field F_{p}, it is exactly equal to Z_{p} if and only if it is an étale Z_{p}-algebra. On the other hand, since R_{∅} ≅ T_{∅}, this is equivalent to R_{∅} being étale over Z_{p}, which is in turn equivalent to the reduced Zariski cotangent space of R_{∅} being trivial. Since by construction R_{∅} represents the minimal deformation functor, its reduced Zariski cotangent space considered as a set has cardinality equal to the number of minimal deformations

ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{p}[x]/(x^{2}))

of \overline{ρ}. Thus to prove that T_{𝔪} ≠ Z_{p} it suffices to show that there exists a non-trivial minimal deformation of \overline{ρ} to GL_{2}(F_{p}[x]/(x^{2})).

In spirit, the proof of Theorem 1.8 follows this strategy; in other words, we determine whether or not T_{𝔪} = Z_{2} by a calculation on tangent spaces. A significant problem arises, however, since we are working in the case p = 2, whilst the method of Wiles and Taylor– Wiles applies only to p > 2. This is not a mere technical obstruction; many phenomena can occur when p = 2 that do not occur for odd p. To name two such: the possible failure of T_{𝔪} to be Gorenstein and the consequent failure of multiplicity one [21], and the fact that \overline{ρ} can arise from a totally real extension of Q. Calculations in the second case suggest that the Taylor–Wiles strategy for proving R = T in the minimal case will not work without some significant new idea, since the numerical coincidences that occur for odd p whilst balancing the Selmer and dual Selmer groups in the Greenberg–Wiles product formula (see for example the remarks of de Shalit [9], top of p. 442) do not occur in the case p = 2. Mark Dickinson [11] has proved an R = T theorem for p = 2; however, his result requires many non-trivial hypotheses, and indeed does not apply to any of the representations considered in Theorem 1.8, since the Taylor–Wiles auxiliary prime arguments fail when p = 2 and the image of \overline{ρ} is dihedral. (The main application of [11] to date has been to representations with image SL_{2}(F_{4}) ≃ A_{5}.) The recent preprint [20] establishes modularity lifting results at the prime p = 2 in some generality; nevertheless, these results do not apply to the situation we consider. Indeed, at least as stated the results of [20] require that \overline{ρ} has non-solvable image, and in any case, they do not give the precise R = T statements that would be needed for applications to Theorem 1.8.

Thus, instead of appealing to any general modularity results, we show that T_{𝔪} is bigger than Z_{2} by explicitly constructing (in certain situations) non-trivial deformations of \overline{ρ}_{𝔪} to F_{2}[x]/(x^{2}) that are demonstrably modular (and hence contribute to the reduced cotangent space of T_{𝔪}). The most difficult point is to show that these deformations are modular of the correct (minimal) level. We prove this via a level-lowering result for modular forms with values in Artinian Z_{2}-algebras (Theorem 3.14 below). This level lowering result may be of independent interest; for example, it provides evidence that an R = T theorem should hold for those \overline{ρ} of characteristic two to which it applies.

The proof of (3) when N is a prime power (but is not actually prime) is given in section 4. In section 5 we make some concluding remarks.

Let us close this introduction by pointing out that recently Dummigan [12] has provided a heuristic explanation for Watkins’ rank conjecture that also relies on a hypothetical R = T theorem for the residual Galois representation \overline{ρ} arising from the 2-torsion on an elliptic curve E: he uses the symmetric square map from \overline{ρ} to Sym^{2}\overline{ρ} to lift elements from the 2-Selmer group of E to the tangent space to the deformation ring of \overline{ρ}. He also shows that the resulting tangent space elements can never be “trapped” (in the words of [37, p. 450]) by the Taylor–Wiles method of introducing auxiliary primes. Thus, although the experimental work of Watkins on the parity of modular degrees, together with the results of this paper and of [12], suggests the validity of an R = T theorem for (at least certain) residual Galois representations arising from the 2-torsion on elliptic curves, the proof of such a theorem seems out of the reach of current techniques.

2 N composite with at least two distinct prime factors

In this section we prove the following theorem.

2.1 Theorem. If E is an elliptic curve of odd modular degree, then the conductor N of E is divisible by at most two odd primes, and E is of even analytic rank. Furthermore, if N is divisible by at least two primes, then E contains a rational 2-torsion point.

We begin with a preliminary lemma. Let E be an elliptic curve over a field k; let O denote the origin of E. We let A denote the group of automorphisms of E as a curve over k (i.e. k-rational automorphisms of E that do not necessarily fix O), and suppose that W is a finite elementary abelian 2-subgroup of A.

2.2 Lemma. The order of W divides twice the order of E[2](k).

Proof. Let A_{0} denote the subgroup of A consisting of automorphisms of E as an elliptic curve over k (i.e. k-rational automorphisms of E that do fix O). The action of E(k) on E via translation realizes E(k) as a normal subgroup of A which has trivial intersection with A_{0}, and which together with A_{0} generates A. Thus A sits in the split short exact sequence of groups

0 → E(k) → A → A_{0} → 1.  (1)

(This is of course well known. The surjection A → A_{0} may also be regarded as the map A = Aut(E) → Aut(Pic^{0}(E)) induced by the functoriality of the formation of Picard varieties — the target being the group of automorphisms of Pic^{0}(E) as a group variety — once we identify E and Pic^{0}(E) as group varieties in the usual way.)

The short exact sequence (1) induces a short exact sequence

0 → W ∩ E(k) → W → W_{0} → 1,

where W_{0} denotes the projection of W onto A_{0}. The known structure of A_{0} shows that W_{0} is either trivial or of order 2. Since W ∩ E(k) ⊂ E[2](k), the lemma follows. □

Proof of Theorem 2.1. The discussion of remark 1.3 shows that it suffices to prove the theorem under the additional assumption that E is an optimal quotient of X_{0}(N).

Let W denote the group of automorphisms of X_{0}(N) generated by the Atkin–Lehner involutions; W is an elementary abelian 2-group of rank equal to the number of primes dividing N. Since E is an optimal quotient of X_{0}(N), the action of W on X_{0}(N) descends to an action on E. If w ∈ W were to act trivially on E, then the quotient map X_{0}(N) → E would factor through X_{0}(N)/w, contradicting our assumption that E is of odd modular degree. Thus Lemma 2.2 shows that W has order at most 8, and hence that N is divisible by at most 3 primes. Furthermore, if N is divisible by more than one prime, then it shows that E[2](Q) is non-trivial.

Suppose now that N is odd, so that X_{0}(N) and E both have good reduction at 2. We may then apply the argument of the preceding paragraph over F_{2}, and so conclude from Lemma 2.2 that W has order at most 4. Hence N is divisible by at most two primes.

Recall that the sign of the functional equation of f is −w_{N}. If E is of odd analytic rank, and if f_{E} denotes the normalized newform of level N attached to E, then w_{N}f_{E} = f_{E}, and so the automorphism of E induced by w_{N} has trivial image in A_{0}. Thus w_{N} acts on E via translation by an element P ∈ E(Q). Since w_{N} interchanges the cusps 0 and ∞ on X_{0}(N), we see that P = π(0) − π(∞) (where π : X_{0}(N) → E is a modular parameterization of E, chosen so that π(∞) = O).

The assumption that E has odd analytic rank also implies that L(f_{E}, 1) = 0. Since this L-value can be computed (up to a non-zero factor) by integrating f_{E} from 0 to ∞ in the upper half-plane, we conclude that P = O, and thus that w_{N} acts trivially on E. Hence π factors through the quotient X_{0}(N)/w_{N} of X_{0}(N), and so must be of even modular degree, a contradiction. □

3 N prime

3.1 Reductions

3.1 Lemma. Theorem 1.8 implies part (3) of theorem 1.1 for N prime.

Proof. Suppose that E is an elliptic curve of conductor N, assumed to be optimal in its isogeny class. Let f_{E} be the associated Hecke eigenform of level Γ_{0}(N) and weight 2. From a theorem of Ribet [38], 2|m_{E} if and only if f_{E} satisfies a congruence mod 2 with another cuspidal eigenform of level N. The set of cuspidal eigenforms (in characteristic zero) congruent to f is indexed by Hom(T_{𝔪} ⊗ Q_{2}, \overline{Q}_{2}). Thus f_{E} satisfies no non-trivial congruences if and only if T_{𝔪} ⊗ Q_{2} = Q_{2}, or equivalently if and only if T_{𝔪} = Z_{2}. □

Of use to us will be the following theorem of Grothendieck on Abelian varieties with semistable reduction [18, Exposé IX, Prop. 3.5]:

3.2 Theorem (Grothendieck). Let A be an Abelian variety over Q with semistable reduction at ℓ. Let I_{ℓ} ⊂ Gal(\overline{Q}/Q) denote a choice of inertia group at ℓ. Then the action of I_{ℓ} on the p^{n}-division points of A for p ≠ ℓ is rank two unipotent; i.e., as an endomorphism, for σ ∈ I_{ℓ},

(σ − 1)^{2}A[p^{n}] = 0.

In particular, I_{ℓ} acts through its maximal pro-p quotient, which is procyclic.

Shimura proved that a modular form f of weight 2 and level Γ_{0}(N) gives rise to a modular Abelian variety A_{f} in such a way that the p-adic representations ρ_{f} attached to f arise from the torsion points of A_{f}. For prime N, these varieties are semistable at N, and so we may apply the theorem above to deduce that for p = 2, such representations ρ restricted to I_{N} factor through a pro-cyclic 2-group. For representations \overline{ρ} with image inside GL_{2}(F_{2}) ≃ S_{3}, this means in particular that the order of inertia at N is either 1 or 2.

Let us now consider a Galois representation \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{2}), arising from a cuspidal Hecke eigenform of level Γ_{0}(N), whose image is not contained in a Borel subgroup. (This is equivalent to \overline{ρ} being irreducible, and also to the image of \overline{ρ} having order divisible by 3.) Let L be the fixed field of the kernel of \overline{ρ}; the extension L/Q is unramified outside 2 and N. If L/Q is unramified at N, then \overline{ρ} has Serre conductor 1, contradicting a theorem of Tate [33]. Thus by the discussion above we see that inertia at N factors through a group of order 2, that L/Q is an S_{3}-extension, and (hence) that \overline{ρ} is absolutely irreducible. Let K/Q be a cubic subfield of L, and let F be the quadratic extension inside L. Since \overline{ρ} is finite flat at 2, it follows from Fontaine’s discriminant bounds [16] that the power of 2 dividing the discriminant of F/Q is at most 4. Thus F/Q must be Q(\sqrt{±N}) (as it is ramified at N). The extension L/F is unramified at the prime above N, since as explained above, \overline{ρ}|_{I_{N}} has order dividing two. Moreover, the extension L/K is ramified at 2 if and only if \overline{ρ} is supersingular, as follows from Remark 1.9.

3.3 Lemma. If \overline{ρ} is supersingular at two and L is not totally real, then N ≡ 3 mod 8. In particular, Theorem 1.1 implies Theorem 1.7.

Proof. By class field theory the quadratic field F/Q admits a degree three extension ramified precisely at 2 only if 2 is unramified and inert in F. This occurs if and only if N ≡ 3 mod 8 and F = Q(\sqrt{−N}), or N ≡ 5 mod 8 and F = Q(\sqrt{N}). Moreover if F = Q(\sqrt{N}), then K and L are totally real. □

We shall prove Theorem 1.8 by showing in the following subsections that if \overline{ρ} satisfies at least one of the following conditions:

1. \overline{ρ} is totally real;

2. \overline{ρ} is unramified at 2;

3. \overline{ρ} is ordinary, complex, and ramified at 2;

then T_{𝔪} ≠ Z_{2}.

3.2 \overline{ρ} is totally real

The theory of modular deformations is not well understood when \overline{ρ} is totally real. Thus our arguments in this section are geometric. We use the following theorem, due to Merel [25, Prop. 5]. (This interpretation of Merel’s result is due to Agashe [1, Cor. 3.2.9]).

3.4 Theorem. Let N be prime. Then J_{0}(N)(R) is connected.

If we let g denote the dimension of J := J_{0}(N) it follows that J(R) ≃ (R/Z)^{g}, J(R)^{tors} ≃ (Q/Z)^{g} and J[2](R) = (Z/2Z)^{g}.

Let J[2^{∞}] = lim_{⟶} J[2^{m}]. Then J[2^{∞}] is a 2-divisible group over Q admitting an action of T_{2}.

Since T_{2} is finite and flat over the complete local ring Z_{2} there exists a decomposition

T_{2} = ∏ T_{𝔪},

where the product is taken over the maximal ideals 𝔪 of T of residue characteristic two. If g(𝔪) denotes the rank of T_{𝔪} over Z_{2}, then

∑_{𝔪}g(𝔪) = rank(T_{2}/Z_{2}) = g.

If J[𝔪^{∞}] := J[2^{∞}] ⊗_{T_{2}} T_{𝔪}, then J[2^{∞}] ≃ ∏ J[𝔪^{∞}] (compare [23] §7, p. 91). From lemma 7.7 of [23] we see that Ta_{𝔪}J ⊗ Q_{2} is free of rank two over T_{𝔪} ⊗ Q_{2} (where Ta_{𝔪}J := Hom(Q_{2}/Z_{2}, J[𝔪^{∞}](\overline{Q})) is the 𝔪-adic Tate module of J), and thus that

J[𝔪^{∞}](C) ≅ (Q_{2}/Z_{2})^{2g(𝔪)}.  (2)

Let J[2]_{𝔪} := J[2] ⊗_{T_{2}} T_{𝔪} be the 2-torsion subgroup scheme of J[𝔪^{∞}].

3.5 Lemma. For all maximal ideals 𝔪 of residue characteristic two there is an equality

dim_{Z/2Z}(J[2]_{𝔪}(R)) = g(𝔪)

Proof. The isomorphism (2) induces an isomorphism J[2]_{𝔪}(C) ≅ (Z/2Z)^{2g(𝔪)}. Let σ ∈ Gal(C/R) denote complex conjugation. Then (σ − 1)^{2}J[2]_{𝔪}(C) = 0. Thus J[2]_{𝔪}(R) (which is the kernel of σ − 1) has dimension at least g(𝔪). If dim_{Z/2Z}(J_{𝔪}[2](R)) > g(𝔪) for some 𝔪, then since

J[2](R) = ∏ J[2]_{𝔪}(R),

and since (as was noted above) dim_{Z/2Z}(J[2](R)) = g, we would deduce the inequality:

g = ∑ dim_{Z/2Z}(J[2]_{𝔪}(R)) > ∑_{𝔪}g(𝔪) = g,

which is absurd. □

Now let \overline{ρ} be a totally real (absolutely) irreducible continuous modular representation of Gal(\overline{Q}/Q) into GL_{2}(F_{2}) of level Γ_{0}(N), and let 𝔪 be the corresponding maximal ideal of T. The main result of [2] shows that the Gal(\overline{Q}/Q)-representation J[𝔪](\overline{Q}) is a direct sum of copies of \overline{ρ}. Thus, since \overline{ρ} is totally real, we find that

dim_{Z/2Z} J[𝔪](R) = dim_{Z/2Z} J[𝔪](C) ≥ dim_{Z/2Z} \overline{ρ} = 2.

Combining this inequality with the inclusion J[𝔪](R) ⊆ J[2]_{𝔪}(R) and Lemma 3.5 we find that g(𝔪) ≥ 2, and thus that T_{𝔪} ≠ Z_{2}.

3.3 Rings of definition for modular forms

In this section, we explain what we mean by a modular form with coefficients in a ring R. Given a level structure Γ := Γ_{0}(N) or Γ_{1}(N), for some N ≥ 1, and a weight k, the space of cuspforms S_{k}(Γ, C) is unambiguously defined. For any subring R of C, we let S_{k}(Γ, R) denote the R-submodule of S_{k}(Γ, C) consisting of cuspforms whose q-expansion coefficients lie in R. It is well know that the natural map R⊗_{Z} S_{k}(Γ, Z) → S_{k}(Γ, R) is an isomorphism, and so for an arbitrary ring R we define S_{k}(Γ, R) := R ⊗_{Z} S_{k}(Γ, Z).

In the case when Γ := Γ_{1}(N), the nebentypus action of (Z/NZ)^{×} on S_{k}(Γ, C) preserves S_{k}(Γ, Z), and hence induces a nebentypus action on S_{k}(Γ, R) for any R. If χ : (Z/NZ)^{×} → C^{×} is a character, then, letting Z[χ] denote the subring of C generated by the values of χ, we let S_{k}(Γ, χ, Z[χ]) denote the subspace of S_{k}(Γ, Z[χ]) consisting of cuspforms with nebentypus character χ. If R is any Z[χ]-algebra, then we let S_{k}(Γ, χ, R) denote the image of R⊗_{Z[χ]} S_{k}(Γ, χ, Z[χ]) under the tautological isomorphism R⊗_{Z[χ]} S_{k}(Γ, Z[χ]) ≅ S_{k}(Γ, R). (This image is contained in the R-submodule of S_{k}(Γ, R) consisting of cuspforms on which (Z/NZ)^{×} acts through the character χ. However, if R is not flat over Z[χ], then it need not coincide with this submodule.)

The following lemma confirms that certain elementary manipulations with the spaces S_{k}(Γ, R) are permissible in the context of the above definitions.

3.6 Lemma. If R is an F_{2}-algebra, then the following maps are well defined.

1. A map S_{1}(Γ_{1}(M), χ, R) → S_{2}(Γ_{0}(M), R) which induces the identity on q-expansions, for any M ≥ 1 and and odd character χ of conductor dividing M and of order two.

2. The level lowering map U_{2} : S_{2}(Γ_{0}(2^{k+1}M), R) → S_{2}(Γ_{0}(2^{k}M), R), for any M ≥ 1 and k ≥ 1.

Proof. It suffices to treat the case R = F_{2}, since there is an isomorphism R ⊗_{F_{2}} S_{k}(Γ, F_{2}) ≅ S_{k}(Γ, R) by definition. For part (3), lift f ∈ S_{1}(Γ_{1}(M), χ, F_{2}) to characteristic zero (possible, by definition), and then multiply by the Eisenstein series E_{1,χ}. We obtain a form g of level Γ_{0}(M) which we then reduce to obtain the desired map. Similarly, for part (4), lift f to characteristic zero. The operator U_{2} reduces the level (provided that k ≥ 1) and preserves the integrality of q-expansions at ∞. Hence the reduction of g = U_{2}f lies in S_{2}(Γ_{0}(2^{k}N), F_{2}). □

Let T denote the Z-algebra of endomorphisms of S_{k}(Γ, C) (or equivalently, of S_{k}(Γ, Z)) generated by the Hecke operators. The algebra T acts on S_{k}(Γ, Z), and hence acts on S_{k}(Γ, R) for any ring R. Furthermore, the usual pairing ⟨f, T ⟩ := a_{1}(T f) (f ∈ S_{k}(Γ, R), T ∈ T) induces an isomorphism S_{k}(Γ, R) ≅ Hom(T, R) [29, Thm. 2.2]. (Here Hom means simply Hom of abelian groups.) A cuspform h ∈ S_{k}(Γ, R) is an eigenform for all the Hecke operators precisely when the associated homomorphism T → R is a ring homomorphism.

If 𝔪 is a maximal ideal in T, then the quotient map T → T/𝔪 corresponds to an eigenform \overline{f}_{𝔪} defined over the finite field T/𝔪, associated to which is a semi-simple Galois representation \overline{ρ}_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T/𝔪). If this Galois representation is absolutely irreducible, then Carayol (building on constructions of Deligne, Shimura, and Serre) has constructed a lifting of it to a continuous representation ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T_{𝔪}) [7, Thm. 3].

Let R be a complete local ring, with maximal ideal 𝔫, such that R/𝔫 has positive residue characteristic. If f ∈ S_{k}(Γ, R) is a Hecke eigenform, then f corresponds to a ring homomorphism ϕ : T → R, which extends to a homomorphism T_{𝔪} → R, for some maximal ideal 𝔪 (the preimage of 𝔫 under ϕ). (The eigenform f can be thought of as a lifting of the eigenform \overline{f}_{𝔪} to the ring R.) If \overline{ρ}_{𝔪} is absolutely irreducible, then pushing forward ρ_{𝔪} via ϕ, we obtain an 𝔫-adically continuous representation Gal(\overline{Q}/Q) → GL_{2}(R), which we refer to as the Galois representation associated to f.

Let 𝔪 be a maximal ideal of T for which T/𝔪 = F_{2}. The completion T_{𝔪} is naturally a finite flat Z_{2}-algebra, of some rank r > 0, say. The isomorphism Hom(T, F_{2}) ≅ S_{k}(Γ, F_{2}) induces an isomorphism Hom(T_{𝔪}, F_{2}) ≅ S_{k}(Γ, F_{2})[𝔪^{∞}]. (Here the target is the subspace consisting of cuspforms annihilated by some power of 𝔪.) Thus dim_{F_{2}}(S_{k}(Γ, F_{2})[𝔪^{∞}]) = r. In particular, we obtain the following lemma.

3.7 Lemma. If there exist non-zero elements f, g ∈ S_{2}(Γ, F_{2}) such that 𝔪f = 𝔪^{2}g = 0, while 𝔪g ≠ 0, then T_{𝔪} ≠ Z_{2}.

Proof. The existence of the elements f and g shows that dim_{F_{2}}(S_{k}(Γ, F_{2})[𝔪^{∞}]) > 1. □

If the level N of Γ is invertible in R, then we may also define the space of Katz modular forms over level Γ and weight k over R, as certain rules on elliptic curves with Γ-level structure defined over R-schemes, following the prescription of [19]. If k ≥ 2, and if Γ = Γ_{1}(N) with N ≥ 5, then there is a natural isomorphism S_{k}^{Katz}(Γ, R) ≅ S_{k}(Γ, R), uniquely determined by its compatibility with the formation of q-expansions. In the case when Γ = Γ_{0}(N), the comparison of S_{k}^{Katz}(Γ, R) and S_{k}(Γ, R) is more complicated, due to the fact that the modular curve X_{0}(N) need not be a fine moduli space. (In the case when N is prime and k = 2, a detailed study of their relationship is given in [23, §II.4]. We will not need to apply any of the results of this study.)

3.4 \overline{ρ} is unramified at 2

We now return to the situation of Theorem 1.8. Suppose that \overline{ρ} is unramified at 2. This forces \overline{ρ} to be ordinary. By the theory of companion forms [17] one expects that \overline{ρ} arises from a mod 2 form of level Γ_{1}(N) and weight 1. Although the results of [17] do not apply in this case, Wiese [36] explicitly constructs such forms when the image of \overline{ρ} is dihedral, as it is in our situation. Let f ∈ S_{1}^{Katz}(Γ_{1}(N), F_{2}) be this companion form for \overline{ρ}. Let A be the Hasse invariant modulo 2, which is a (Katz) modular form (not a cuspform) of level one with q expansion given by 1. Then Af and g = f^{2} are elements of S_{2}^{Katz}(Γ_{1}(N), F_{2}), and thus of S_{2}(Γ_{1}(N), F_{2}). (Note that it is no loss of generality to assume that N ≥ 5, since X_{1}(N) has genus zero for 1 ≤ N ≤ 4.) Wiese’s construction furthermore allows us to choose f, and hence Af and g, so as to have trivial nebentypus. A lemma of Carayol [6, §4.4]^{1} then assures us that Af and g in fact lie in S_{2}(Γ_{1}(N), 1, F_{2}) (where in this notation 1 denotes the trivial character of (Z/NZ)^{×}), that is, that Af and g lie in S_{2}(Γ_{0}(N), F_{2}).

If f has q-expansion f = ∑_{n=1}^{∞} a_{n}q^{n}, then Af has the same q-expansion, while g has

q-expansion g = ∑_{n=1}^{∞} a_{n}^{2}q^{2n} ≡ V_{2}f, since a_{n} ∈ F_{2}. Since a_{1} = 1, we see that f and g are

linearly independent. Furthermore, one computes that (T_{ℓ} − a_{ℓ})f = (T_{ℓ} − a_{ℓ})g = 0 for all odd ℓ, that (T_{2} − a_{2})f = 0 and (T_{2} − a_{2})^{2}g = 0, and that (T_{2} − a_{2})g ≠ 0. Thus 𝔪f = 𝔪^{2}g = 0, while 𝔪g ≠ 0, and therefore, by Lemma 3.7, T_{𝔪} ≠ Z_{2} and we are done.

In fact one can avoid the appeal to Carayol’s lemma in the above argument. The only difficult point of Wiese’s construction is the case when \overline{ρ} is totally real, and this case of Theorem 1.8 is already covered by section 3.2. If we assume that \overline{ρ} is not totally real, then we may construct f directly as an element of S_{1}(Γ_{1}(N), χ) for some character χ. (See the first paragraph of the proof of [36, Thm. 9].) One can then define Af ∈ S_{2}(Γ_{0}(N), F_{2}) by applying the map of Lemma 3.6 (1) to f, as well as g = f^{2} ∈ S_{2}(Γ_{0}(N), F_{2}). The argument then proceeds just as above.

3.5 \overline{ρ} is ordinary, complex, and ramified at 2

Suppose that \overline{ρ} is ordinary, complex, and ramified at 2. It follows that F/Q is complex and ramified at 2, and thus that F = Q(\sqrt{−N}) for some N ≡ 1 mod 4. Moreover, the extension L/F is unramified everywhere. Since N ≡ 1 mod 4 it follows that H := L(\sqrt{−1}) is also unramified everywhere over F. The field H is Galois over Q, and clearly

Gal(H/Q) ≃ S_{3} × Z/2Z.  (3)

We may embed S_{3} × Z/2Z into GL_{2}(F_{2}[x]/(x^{2})) by fixing an identification of S_{3} with GL_{2}(F_{2}), and mapping a generator of Z/2Z to the matrix

(\begin{matrix}1 + x & 0 \\ 0 & 1 + x \end{matrix}) .

Composing the isomorphism (3) with this embedding yields a representation:

ρ : Gal(H/Q) ↪ GL_{2}(F_{2}[x]/(x^{2})).

The representation ρ has trivial determinant (equivalently, determinant equal to the mod 2 cyclotomic character). We also claim that ρ is finite flat at two. To check this, it suffices to prove this over Z_{2}^{ur}. The representation ρ|_{Gal(\overline{Q}_{2}/Q_{2}^{ur})} factors through a group of order 2, and one explicitly sees that it arises as the generic fibre of the group scheme (D ⊕ D)/Z_{2}^{ur}, where D is the non-trivial extension of Z/2Z by μ_{2} considered in [23] (Prop 4.2, p. 58). Thus one expects ρ to arise from an F_{2}[x]/(x^{2})-valued modular form of weight two and level N, corresponding to a surjective map of rings T_{𝔪} → F_{2}[x]/(x^{2}). This would follow if we knew that T_{𝔪} coincided with the minimal deformation ring associated to \overline{ρ}. Rather than proving this, however, we shall use weight one forms to explicitly construct a weight two modular form giving rise to ρ.

1The calculations of the following paragraph show that f and g are eigenforms for the Hecke operators T_{ℓ} for every prime ℓ not dividing 2N, with eigenvalue equal to the trace of the image under \overline{ρ} of Frobenius at ℓ. Since \overline{ρ} is not induced from a character of Gal(\overline{Q}/Q(i)) (we saw above that it is rather induced from a cubic character of Gal(\overline{Q}/Q(\sqrt{±N}))), Carayol’s lemma applies.

Let χ_{4N} be the character of conductor 4N associated to F. Consider two faithful representations

ψ_{1} : Gal(L/Q) ≅ S_{3} ↪ GL_{2}(C), ψ_{2} : Gal(H/Q) ≅ S_{3} × Z/2Z ↪ GL_{2}(C).

Since F/Q is complex, these dihedral representations are odd and therefore give rise to weight one modular forms h_{1}, h_{2} in S_{1}(Γ_{1}(4N), χ_{4N}, C).

3.8 Lemma. The modular forms h_{1}, h_{2} are ordinary at 2, have coefficients in Z, and are congruent modulo 2. Let

g = \frac{(h_{2} − h_{1})}{2} ∈ Z[[q]].

Then g mod 2 is non-zero, and the form h = h_{1} + xg ∈ S_{1}(Γ_{1}(4N), χ_{4N}, F_{2}[x]/(x^{2})) is an eigenform for all the Hecke operators, including U_{2}. The GL_{2}(F_{2}[x]/(x^{2}))-valued Galois representation associated to h is isomorphic to ρ.

Proof. The modular forms are both ordinary at 2 because the representations ψ_{1} and ψ_{2} have non-trivial subspaces on which inertia at two is trivial (since I_{2} acts through a group of order 2). They both have coefficients in Z, since 2 cos(π/3) ∈ Z. The congruence h_{1} ≡ h_{2} mod 2 follows from the fact that both are ordinary-at-2 Hecke eigenforms, and that a(h_{1}, ℓ) = a(h_{2}, ℓ)χ_{4}(ℓ) for all odd primes ℓ, where χ_{4} is the character of conductor 4. From this one also sees that g is non-trivial modulo two, and that h_{1}, h_{2}, g define forms in S_{1}(Γ_{1}(4N), χ_{4N}, Z). Thus, by definition, h ∈ S_{1}(Γ_{1}(4N), χ_{4N}, F_{2}[x]/(x^{2})). The claim that h is a Hecke eigenform follows formally (on q-expansions) from the fact that h_{1} and h_{2} are Hecke eigenforms that are congruent modulo 2. The discussion of Section 3.3, shows the existence of a Galois representation associated to h, which by comparing traces of Frobenius one easily sees is isomorphic to ρ. □

Now that we have constructed the weight one form h of level 4N giving rise to ρ, we would like to construct a corresponding weight two form of level N. Applying the map of Lemma 3.6 (1) (concretely, multiplying h by the Eisenstein series in M_{1}(Γ_{1}(4N), χ_{4N})), we see that there is a modular form h′ ∈ S_{2}(Γ_{0}(4N), F_{2}[x]/(x^{2})) having the same q-expansion as h. Since h, and hence h′, is an ordinary U_{2} eigenform, we may apply the U_{2} operator to deduce (using Lemma 3.6 (2)) that h′ ∈ S_{2}(Γ_{0}(2N), F_{2}[x]/(x^{2})). Applying Theorem 3.14 (proved in the following subsection) we then deduce that in fact h′ ∈ S_{2}(Γ_{0}(N), F_{2}[x]/(x^{2})), and (thus) that there is a modular form g′ ∈ S_{2}(Γ_{0}(N), F_{2}) having the same q-expansion as g.

As in the discussion of Section 3.3, let \overline{f}_{𝔪} ∈ S_{2}(Γ_{0}(N), F_{2}) be the Hecke eigenform associated to 𝔪. Then 𝔪\overline{f}_{𝔪} = 0, while an easy calculation shows that 𝔪g ⊂ F_{2}\overline{f}_{𝔪} \ \{0\}, and so also that 𝔪^{2}g = 0. We conclude from Lemma 3.7 that T_{𝔪} ≠ Z_{2}.

3.6 Level-lowering for modular deformations

The goal of this section is to prove a level-lowering result for modular forms with coefficients in Artinian rings that strengthens the case p = 2 of [13, Thm. 2.8] (which in turn extends a level lowering result proved by Mazur [30, Thm. 6.1] in the odd prime case).

We first establish a version of the multiplicity one theorem [37, Thm. 2.1] for p = 2. Under the additional assumption that \overline{ρ} is not finite at 2, this theorem was proved in [4, §2] (as was the corresponding result for odd level). Thus the key point in our theorem is that \overline{ρ} is allowed to be finite at 2, even though the level is taken to be even.

3.9 Theorem. Let N be an odd natural number, and let T denote the full Z-algebra of Hecke operators acting on weight two cuspforms of level Γ_{0}(2N). If 𝔪 is a maximal ideal in T whose residue field k is of characteristic 2, and for which the associated residual Galois representation

\overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(k)

is (absolutely) irreducible, ordinary, and ramified at 2, then Ta_{𝔪}J_{0}(2N) (the 𝔪-adic Tate module of J_{0}(2N)) is free of rank two over the completion T_{𝔪}.

To be clear, the condition “ordinary at 2” means that the image of a decomposition group at 2 under \overline{ρ} lies in a Borel subgroup of GL_{2}(k). Since k is of characteristic 2, we see that (for an appropriate choice of basis) the restriction of \overline{ρ} to an inertia group at 2 may be written in the form

\overline{ρ}_{|I_{2}} = (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}) .

The assumption that \overline{ρ} is ramified at 2 then implies that ∗ is not identically zero. Thus the representation space of \overline{ρ} has a unique line invariant under I_{2}, and so \overline{ρ} is irreducible if and only if it is absolutely irreducible.

3.10 Lemma. Let k be a finite field of characteristic 2. If \overline{ρ} : Gal(\overline{Q}_{2}/Q_{2}) → GL_{2}(k) is a continuous representation that is finite, ordinary, and ramified at 2, then \overline{ρ} has a unique finite flat prolongation over Z_{2} (up to unique isomorphism). Furthermore, this prolongation is an extension of a rank one étale k-vector space scheme by a rank one multiplicative kvector space scheme.

Proof. Any finite flat group scheme that prolongs an unramified continuous representation of Gal(\overline{Q}_{2}/Q_{2}) on a one-dimensional k-vector space is either étale or multiplicative. Thus there are a priori four possible structures for a finite flat prolongation of \overline{ρ}: étale extended by étale; multiplicative extended by multiplicative; multiplicative extended by étale; or étale extended by multiplicative. However, all but the last possibility necessarily gives rise to an unramified generic fibre (note that any extension of multiplicative by étale must split, by a consideration of the connected étale sequence). Thus, since \overline{ρ} is assumed ramified, we see that any prolongation of \overline{ρ} must be an extension of a rank one étale k-vector space scheme scheme by a rank one multiplicative k-vector space scheme.

To see that such a prolongation is unique, consider the maximal and minimal prolongations M and M′ of \overline{ρ} to a finite flat group scheme [27, Cor. 2.2.3]. Since étale (and hence multiplicative) group schemes are determined by their generic fibre, the result of the preceding paragraph shows that the natural morphism M → M′ necessarily induces an isomorphism on the connected components of the identity, and on the corresponding groups of connected components (i.e., the corresponding maximal étale quotients). By the 5-lemma, this morphism is thus an isomorphism, and the lemma follows. □

We now show that certain results of Mazur [24] cited in the proof of [37, Thm. 2.1] extend to the case p = 2. We put ourselves in the context of [24, §1], and use the notation introduced there. Namely, we let K denote a finite extension of Q_{p} for some prime p, and let O denote the ring of integers of K. If A is an abelian variety over K, then we let A_{/O} denote the connected component of the identity of the Néron model of A over Spec O. For any power p^{r} of p, the p^{r}-torsion subgroup scheme A[p^{r}]_{/O} of A_{/O} is then a quasi-finite flat group scheme over Spec O; we let F A[p^{r}]_{/O} denote its maximal finite flat subgroup scheme, and A[p^{r}]_{/O}^{0} denote the maximal connected closed subgroup scheme of A[p^{r}]_{/O}. Since we took A_{/O} to be the connected component of the Néron model of A, the inductive limit F A[p^{∞}]_{/O} := lim_{⟶} F A[p^{r}]_{/O} is a p-divisible group, and A[p^{∞}]_{/O}^{0} := lim_{⟶} A[p^{r}]_{/O}^{0} is the maximal connected p-divisible subgroup of F A[p^{∞}]_{/O}.

The following proposition is a variation on [24, Prop. 1.3], in which we allow the ramification of K over Q_{p} to be unrestricted, at the expense of imposing more restrictive hypotheses on the reduction of the abelian varieties appearing in the exact sequence under consideration.

3.11 Proposition. Let 0 → A → B → C → 0 be an exact sequence of abelian varieties over K such that A has purely toric reduction, whilst C has good reduction. Then the induced sequence of p-divisible groups

0 → A[p^{∞}]_{/O}^{0} → B[p^{∞}]_{/O}^{0} → C[p^{∞}]_{/O}^{0} → 0

is a short exact sequence of p-divisible groups over Spec O. Equivalently, for any power p^{r} of p, the induced sequence

0 → A[p^{r}]_{/O}^{0} → B[p^{r}]_{/O}^{0} → C[p^{r}]_{/O}^{0} → 0

is a short exact sequence of finite flat group schemes over Spec O.

Proof. Since A has purely toric reduction, the group scheme A[p^{r}]_{/O}^{0} is of multiplicative type for each r. Thus it necessarily maps isomorphically onto its scheme theoretic image in B_{/O}, and thus the induced map A[p^{∞}]_{/O}^{0} → B[p^{∞}]_{/O}^{0} is a closed embedding.

Let C′ ⊂ B be an abelian subvariety chosen so that the induced map C′ → C is an isogeny. Then C′ also has good reduction, and so C′[p^{∞}]_{/O}^{0} → C[p^{∞}]_{/O}^{0} is an epimorphism of p-divisible groups over Spec O. Thus the induced map B[p^{∞}]_{/O}^{0} → C[p^{∞}]_{/O}^{0} is also an epimorphism of p-divisible groups. A consideration of generic fibres shows that the kernel of this surjection coincides with the scheme-theoretic image of A[p^{∞}]_{/O}^{0} in B[p^{∞}]_{/O}^{0}, and so the proposition is proved. □

Proof of Theorem 3.9. We closely follow the method of proof of [37, Thm. 2.1 (ii)] in the case when “Δ_{(p)} is trivial mod 𝔪” (in the terminology of that proof; see [37, pp. 485–488]). If we let A denote the connected part of the kernel of the map J_{0}(2N) → J_{0}(N) × J_{0}(N) induced by Albanese functoriality applied to the two “degeneracy maps” from level 2N to level N, then A is an abelian subvariety of J_{0}(2N) having purely toric reduction at 2, whilst the quotient B of J_{0}(2N) by A has good reduction at 2. From Proposition 3.11 we obtain (for any r ≥ 1) the short exact sequence

0 → A[2^{r}]_{/Z_{2}}^{0} → J_{0}(2N)[2^{r}]_{/Z_{2}}^{0} → B[2^{r}]_{/Z_{2}}^{0} → 0

of connected finite flat group schemes over Spec Z_{2}. By functoriality of the formation of this short exact sequence, and since A is a T-invariant subvariety of J_{0}(2N), we see that this is in fact a short exact sequence of T-module schemes.

If G is a 2-power torsion commutative group scheme over some base S, with an action of T, and hence of Z_{2} ⊗_{Z} T, via endomorphisms, then by the localization M_{𝔪} we mean the kernel G[ε′], where the idempotent ε′ ∈ Z_{2} ⊗_{Z} T is defined as ε′ := 1 − ε, with ε ∈ Z_{2} ⊗_{Z} T being the idempotent corresponding to the direct factor T_{𝔪} of Z_{2} ⊗_{Z} T. Since ε and ε′ are orthogonal idempotents (by construction), the natural map G[ε] ⨁ G[ε′] ≅ G is an isomorphism. Thus passage to G_{𝔪} is an exact functor on the category of 2-power torsion T-module schemes over S.

In particular, localizing the above short exact sequence at 𝔪 induces the corresponding short exact sequence

0 → A[2^{r}]_{𝔪/Z_{2}}^{0} → J_{0}(2N)[2^{r}]_{𝔪/Z_{2}}^{0} → B[2^{r}]_{𝔪/Z_{2}}^{0} → 0.  (4)

Passing to \overline{Q}_{2}-valued points induces a short exact sequence of Gal(\overline{Q}_{2}/Q_{2})-modules

0 → A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) → J_{0}(2N)[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) → B[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) → 0,  (5)

which is a subexact sequence of the short exact sequence of Gal(\overline{Q}_{2}/Q_{2})-modules

0 → A[2^{r}]_{𝔪}(\overline{Q}_{2}) → J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}_{2}) → B[2^{r}]_{𝔪}(\overline{Q}_{2}) → 0.  (6)

Let A[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ} (respectively J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}, respectively B[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}) denote the maximal Gal(\overline{Q}_{2}/Q_{2})-subrepresentation of A[2^{r}]_{𝔪}(\overline{Q}_{2}) (respectively J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}_{2}), respectively B[2^{r}]_{𝔪}(\overline{Q}_{2})) on which the inertia group acts through the 2-adic cyclotomic character χ. The short exact sequence (6) induces an exact sequence

0 → A[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ} → J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ} → B[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}.  (7)

3.12 Lemma. Each of the groups schemes appearing in the exact sequence (4) is of multiplicative type, and the exact sequences (5) and (7) coincide (as subsequences of (6)).

Proof. We first remark that (6) is the exact sequence of T_{𝔪}[Gal(\overline{Q}_{2}/Q_{2})]-modules underlying the corresponding exact sequence of T_{𝔪}[Gal(\overline{Q}/Q)]-modules

0 → A[2^{r}]_{𝔪}(\overline{Q}) → J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}) → B[2^{r}]_{𝔪}(\overline{Q}) → 0.

Since \overline{ρ} is assumed irreducible as a k[Gal(\overline{Q}/Q)]-representation, each of the modules appearing in this exact sequence is a successive extension of copies of \overline{ρ}. The same is thus true of each of the modules appearing in the exact sequence (6).

Since A has purely toric reduction, it is clear that A[2^{r}]_{𝔪/Z_{2}}^{0} is of multiplicative type, and so

A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) ⊂ A[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}.  (8)

Fix a filtration 0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{n} = A[2^{r}]_{𝔪}(\overline{Q}_{2}) of A[2^{r}]_{𝔪}(\overline{Q}_{2}) for which the successive quotients W_{i+1}/W_{i} are isomorphic to \overline{ρ}. Since A has purely toric reduction the quotient A[2^{r}]_{/Q_{2}}/A[2^{r}]_{/Q_{2}}^{0} is Cartier dual to Â[2^{r}]_{/Q_{2}}^{0} (where Â is the dual abelian variety to A), and so A[2^{r}]_{𝔪}(\overline{Q}_{2})/A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) is an unramified Gal(\overline{Q}_{2}/Q_{2})-representation. Since \overline{ρ} is assumed ramified at 2, this implies that

W_{i} ⋂ A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) ⊊ W_{i+1} ⋂ A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2})

for each i ≥ 0. Furthermore,

W_{i+1} ⊄ A[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ} + W_{i}

for each i ≥ 0, because χ mod 2 is trivial. Since W_{i+1}/W_{i} ≅ \overline{ρ} is two dimensional over k for each i ≥ 0, we conclude by induction on i that

W_{i} ⋂ A[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) = W_{i} ⋂ A[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}

for each i ≥ 0. Taking i = n then shows that the inclusion (8) is in fact an equality.

Since B has good reduction at 2, we have equality F B[2^{r}]_{/Z_{2}} = B[2^{r}]_{/Z_{2}}. As noted above, any Jordan–Hölder filtration of the localization B[2^{r}]_{𝔪}(\overline{Q}) as a T[Gal(\overline{Q}/Q)]- module has all its associated graded pieces isomorphic to \overline{ρ}. Taking scheme-theoretic closures of such a filtration in B[2^{r}]_{/Z_{2}}, we obtain a filtration of the localization B[2^{r}]_{𝔪/Z_{2}} by finite flat closed subgroup schemes, whose associated graded pieces are prolongations of \overline{ρ}. Now Lemma 3.10 shows that the connected component of any such finite flat prolongation is multiplicative. Thus B[2^{r}]_{𝔪/Z_{2}}^{0} is indeed multiplicative, whilst B[2^{r}]_{𝔪}(\overline{Q}_{2})/B[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) is an unramified Gal(\overline{Q}_{2}/Q_{2})-module. Arguing as in the preceding paragraph gives the required equality

B[2^{r}]_{𝔪}^{0}(\overline{Q}_{2}) = B[2^{r}]_{𝔪}(\overline{Q}_{2})^{χ}.

Since any extension of multiplicative type groups is again of multiplicative type, we see that J_{0}(2N)[2^{r}]_{𝔪/Z_{2}}^{0} is also of multiplicative type, and that the exact sequence (5) is a subsequence of the exact sequence (7). We have furthermore shown that first and third non-trivial terms of these two sequences coincide. This implies that these exact sequences do indeed coincide. □

Specializing Lemma 3.12 to the case r = 1 shows that J_{0}(2N)[2]_{𝔪}^{0}(\overline{Q}_{2}) is the maximal unramified Gal(\overline{Q}_{2}/Q_{2})-subrepresentation of J_{0}(2N)[2]_{𝔪}(\overline{Q}_{2}) (since χ mod 2 is trivial). Recall that there is a natural isomorphism Tan(J_{0}(2N)[2]_{/\overline{F}_{2}}^{0}) ≅ Tan(J_{0}(2N)_{/\overline{F}_{2}}) (indeed, this is true with J_{0}(2N)_{/\overline{F}_{2}} replaced by any group scheme over \overline{F}_{2}), and also a natural isomorphism Tan(J_{0}(2N)[2]_{/\overline{F}_{2}}^{0}) ≅ J_{0}(2N)[2]^{0}(\overline{Q}_{2}) ⊗_{F_{2}} \overline{F}_{2} (as follows from the discussion on [37, p. 488]). Localizing at 𝔪, and taking into account [37, lem. 2.2], which is valid for p = 2, we find that J_{0}(2N)[2]_{𝔪}^{0}(\overline{Q}_{2}) is a cyclic T_{𝔪}-module, and thus that the maximal unramified Gal(\overline{Q}_{2}/Q_{2})-subrepresentation of J_{0}(2N)[2]_{𝔪}(\overline{Q}_{2}) is a cyclic T_{𝔪}-module.

Let ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T_{𝔪}) denote the Galois representation associated to 𝔪 by [7, Thm. 3]. Carayol has proved [7, Thm. 4] that there is an isomorphism Ta_{𝔪}J_{0}(2N) ≅ J ⊗_{T_{𝔪}} ρ_{𝔪} for some ideal J in T_{𝔪}, and thus an isomorphism J_{0}(2N)[2]_{𝔪}(\overline{Q}_{2}) ≅ (J/2J) ⊗_{T_{𝔪}} ρ_{𝔪}. We conclude that J/2J is a cyclic T_{𝔪}-module, and hence that J is a principal ideal in T_{𝔪}. The discussion of [7, 3.3.2] shows that in fact J ≅ T_{𝔪} and that Ta_{𝔪}J_{0}(2N) is free of rank two over T_{𝔪}, as claimed. □

3.13 Corollary. In the situation of Theorem 3.9, the completion T_{𝔪} is a Gorenstein Z_{2}- algebra.

Proof. This follows from the theorem together with the self-duality of the 𝔪-adic Tate module under the Weil pairing. □

We now prove our level lowering result. Let A be an Artinian ring with finite residue field k of characteristic 2, and suppose given a continuous representation ρ : Gal(\overline{Q}/Q) → GL_{2}(A) that is modular of level Γ_{0}(2N) for some odd natural number N, in the sense that it arises from a Hecke eigenform h ∈ S_{2}(Γ_{0}(2N), A). Let \overline{ρ} denote the residual representation attached to ρ (so \overline{ρ} arises from the Hecke eigenform \overline{h} ∈ S_{2}(Γ_{0}(2N), k) obtained by reducing h modulo the maximal ideal of A).

3.14 Theorem. If ρ : Gal(\overline{Q}/Q) → GL_{2}(A) is a modular Galois representation of level Γ_{0}(2N) as above, such that

1. \overline{ρ} is (absolutely) irreducible,

2. \overline{ρ} is ordinary and ramified at 2, and

3. ρ is finite flat at 2,

then ρ arises from an A-valued Hecke eigenform of level N.

Proof. The Hecke eigenform h corresponds to a ring homomorphism ϕ : T → A. Since A is local of residue characteristic 2, the map ϕ factors through the completion T_{𝔪} of T at some maximal ideal 𝔪 of residue characteristic 2, and the residual representation \overline{ρ} is the residual Galois representation attached to the maximal ideal 𝔪. We let ρ_{𝔪} denote the Galois representation

ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T_{𝔪})

attached to 𝔪 by [7, Thm. 3]. The Galois representation ρ attached to h coincides with the pushforward of ρ_{𝔪} via ϕ.

Replacing A by the image of ϕ, we may and do assume from now on that ϕ is surjective. We let I ⊂ T_{𝔪} denote the kernel of ϕ. Since A is Artinian, we may choose r ≥ 1 so that 2^{r} ∈ I. Theorem 3.9 shows that the 𝔪-adic Tate module Ta_{𝔪}J_{0}(2N) is isomorphic as a Gal(\overline{Q}/Q)-representation to ρ_{𝔪}; thus J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}) is isomorphic to the reduction mod 2^{r} of ρ_{𝔪}. Since T_{𝔪} is a Gorenstein Z_{2}-algebra, by corollary 3.13, we see that T_{𝔪}/2^{r}T_{𝔪} is a Gorenstein Z/2^{r}Z-algebra, and thus that there is an isomorphism (T_{𝔪}/2^{r}T_{𝔪})[I] ≅ Hom_{Z/2^{r}}(T_{𝔪}/I, Z/2^{r}) of T_{𝔪}/I = A-modules. In particular, J_{0}(2N)[I](\overline{Q}/Q) ⊂ J_{0}(2N)[2^{r}]_{𝔪}(\overline{Q}) is a faithful A-module, isomorphic as an A[Gal(\overline{Q}/Q)]- module to Hom_{Z/2^{r}}(T_{𝔪}/I, Z/2^{r}) ⊗_{A} ρ. To simplify notation, we will write

V := J_{0}(2N)[I](\overline{Q}) ≅ Hom_{Z/2^{r}}(T_{𝔪}/I, Z/2^{r}) ⊗_{A} ρ.  (9)

By assumption, ρ prolongs to a finite flat group scheme M over Spec Z_{2}. If we fix a Jordan–Hölder filtration of ρ as an A[Gal(\overline{Q}/Q)]-module, then the associated graded pieces are each isomorphic to \overline{ρ}, and so Lemma 3.10 and [5, Prop. 2.5] together imply that M is uniquely determined by ρ, whilst [5, Lem. 2.4] then implies that M is naturally an A-module scheme. From (9) we see that V also prolongs to a finite flat A-module scheme

V ≅ Hom_{Z/2^{r}}(T_{𝔪}/I, Z/2^{r}) ⊗_{A} M

over Z_{2}. Again, Lemma 3.10 and [5, Prop. 2.5] show that V is the unique finite flat prolongation of V.

Lemma 3.10 furthermore implies that M is the extension of an étale A-module scheme M^{ét} by a multiplicative A-module scheme M^{0}, each of which is free of rank one as an A-module scheme. Thus V is also an extension of an étale A-module scheme V^{ét} by multiplicative A-module scheme V^{0}, each of which is faithful as an A-module scheme. Let V^{ét} and V^{0} denote the generic fibres of these schemes.

We write J to denote the Néron model of J_{0}(2N) over Spec Z_{2}. For a scheme over Z_{2}, use the subscript “s” to denote its special fibre over F_{2}. The special fibre J_{s} admits the following filtration by T-invariant closed subgroups:

0 ⊂ T ⊂ J_{s}^{0} ⊂ J_{s},

where T is the maximal torus contained in J_{s}, and J_{s}^{0} is the connected component of the identity of J_{s}. The quotient J_{s}^{0}/T is an abelian variety on which T acts through its quotient T_{old} (where T_{old} denotes the quotient of T that acts faithfully on the space of 2-old forms of level 2N). The connected component group Φ := J_{s}/J_{s}^{0} is Eisenstein [30, Thm. 3.12].

The following lemma provides an analogue in our situation of [30, lem. 6.2] (and generalizes one step of the argument in the proof of [13, Thm. 2.8]).

3.15 Lemma. The Zariski closure of V in J is a finite flat A-module scheme over Z_{2} (which is thus isomorphic to V). Proof. Since V^{0} is a multiplicative type group scheme, inertia at 2 acts on V^{0}(\overline{Q}_{2}) through the cyclotomic character. It follows from Lemma 3.12 that V^{0} is contained in the generic fibre of J_{0}(2N)[2^{r}]_{𝔪/Z_{2}}^{0}, and thus that the Zariski closure of V^{0} in J is indeed finite flat, and in fact of multiplicative type. Thus it coincides with V^{0}, and so we see that the embedding of V^{0} in J_{0}(2N) prolongs to an embedding of V^{0} in J. Since the quotient V^{ét} = V/V^{0} is étale, Lemma 5.9.2 of [18, Exposé IX] serves to complete the proof of the lemma. □

Lemma 3.15 allows us to regard V as a closed T-submodule scheme of J, and thus to regard V_{s} as a closed T-submodule scheme of J_{s}. Since \overline{ρ} is irreducible and Φ is Eisenstein, we see that V_{s} is in fact contained in J_{s}^{0}. On the other hand, since T is a torus, we see that V_{s} ⋂ T ⊂ V_{s}^{0}. Thus V_{s}^{ét} appears as a subquotient of J_{s}^{0}/T, and in particular the T-action on V^{ét} factors through the quotient T_{old} of T. Since V_{s}^{ét} is a faithful A-module scheme, we see that the map ϕ : T → A factors through T_{old}, completing the proof of the theorem. □

We remark that the obvious analogue of Theorem 3.14 in the case of odd residue characteristic is also true. The proof is similar but easier, relying on the uniqueness results on finite flat models due to Raynaud [27].

4 N a proper prime power

There are only finitely many elliptic curves of conductor 2^{k} for all k, and we may explicitly determine which have odd modular degree. Therefore we assume that E has conductor N, where N = p^{k} with k ≥ 2 and p ≥ 3. Let χ be the unique quadratic character of conductor p. Let E′ be the elliptic curve E twisted by χ. The curve E′ also has conductor N, and moreover, the associated modular forms f_{E} and f_{E′} are congruent modulo 2, since twisting by quadratic characters preserves E[2]. Since N is odd, any non-trivial congruence modulo 2 between f_{E} and other forms in S_{2}(Γ_{0}(N)) forces the modular degree m_{E} to be even [38]. Thus we are done unless f_{E} = f_{E′} = f_{E} ⊗ χ. It follows (for example, by [28]) that the Galois representation associated to f_{E} is induced from an imaginary quadratic field, and E has complex multiplication by this field. Alternatively, the equality f_{E} = f_{E′} implies that E is isogenous to its twist, and one may deduce this way that E has CM. If E has CM and prime power conductor, then E is one of finitely many well known elliptic curves, for which we can directly determine the modular degree by consulting current databases (for N = 163^{2}, we use the elliptic curve database of Stein–Watkins, described in [31]).

5 Further remarks

Certainly not every E satisfying the conditions of Theorem 1.1 will actually have odd modular degree, and one could try to refine this result by deducing additional necessary conditions that E must satisfy in order to have odd modular degree. In this section we say a little about the related question of whether or not 2 is a congruence prime for the associated modular form f_{E}, when E satisfies either of conditions (3a) or (3b) of the theorem.

For curves E with a rational two torsion point, the modular form f_{E} automatically satisfies a mod two congruence with an Eisenstein series, and so detecting whether f satisfies a congruence with a cuspform is a more subtle phenomenon than in the non-Eisenstein situation. One approach might be to relate the Hecke algebra to an appropriate universal deformation ring (if the latter exists). If N is prime, this can be done [5], and this enables one to determine when T_{𝔪} = Z_{2} for such representations. The specific determination of when T_{𝔪} = Z_{2}, however, was already achieved (for N prime and \overline{ρ} reducible) by Merel in [25]:

5.1 Theorem. Let N ≡ 1 mod 8 be prime, and let T_{𝔪} be the localization at the Eisenstein prime at 2. Then T_{𝔪} ≠ Z_{2} if and only if N = u^{2} + 16v^{2} and v ≡ (N − 1)/8 mod 2.

If E is a Neumann–Setzer curve, then N = u^{2} + 64 for some u ∈ Z. The result of Merel above then clearly implies that the optimal Neumann–Setzer curve E has odd modular degree if and only if N ≢ 1 mod 16. (An alternative proof of this fact, relying on the results of [23], is given in [32], Thm. 2.1.) If E has composite conductor, then one might try to generalize the results of [25] or [5] to this setting.

Suppose now that E has prime conductor, that \overline{ρ} is irreducible and supersingular, and that Q(E[2]) is totally complex. If one had an R = T result of the type discussed in the introduction, then to obtain further necessary conditions for E to have odd modular degree, it would suffice to establish sufficient conditions for the existence of an appropriate non-trivial minimal deformation ρ : Gal(\overline{Q}/Q) → GL_{2}(F_{2}[x]/(x^{2})) lifting \overline{ρ}. For representations \overline{ρ} that are complex and ramified at 2, but ordinary, we constructed such a ρ directly in subsection 3.5 by considering quadratic genus fields. When E is supersingular, such deformations ρ (when they exist) may be more subtle and can not necessarily be constructed so directly.

References

[1] A. Agashe, The Birch and Swinnerton-Dyer formula for modular abelian varieties of analytic rank zero, Berkeley Ph.D. Thesis, 2000.

[2] N. Boston, H. Lenstra, K. Ribet, Quotients of group rings arising from twodimensional representations, C. R. Acad. Sci. Paris, Série I 312 (1991), 323–328.

[3] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.

[4] K. Buzzard, On level-lowering for mod 2 representations, Math. Res. Lett. 7 (2000), no. 1, 95–110.

[5] F. Calegari, M. Emerton, On the ramification of Hecke algebras at Eisenstein primes, Invent. Math. 160 (2005), no. 1, 97–144.

[6] H. Carayol, Sur les représentations galoisiennes mod l attachées aux formes modulaires, Duke Math. J. 59 (1989), no. 3, 785–801.

[7] H. Carayol, Formes modulaires et répresentations Galoisiennes á valeurs dans un anneau local complet, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (B. Mazur, G. Stevens, eds.), Contemp. Math. 165 (1994), 213–235.

[8] J. Cremona, M. Watkins, data available at

  http://www.maths.nott.ac.uk/personal/jec/ftp/data

[9] E. de Shalit, Hecke rings and universal deformation rings, in Modular forms and Fermat’s last theorem (Boston, MA, 1995), 421–445, Springer.

[10] F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), no. 1, 137–166.

[11] M. Dickinson, On the modularity of certain 2-adic Galois representations, Duke Math. J. 109 (2001), no. 2, 319–382.

[12] N. Dummigan On a conjecture of Watkins, J. Thor. Nombres Bordeaux 18 (2006), no. 2, 345–355.

[13] B. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594.

[14] M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), no. 2, 307–327.

[15] M. Flach, On the degree of modular parametrizations, Séminaire de Théorie des Nombres, Paris, 1991–92, 23–36, Progr. Math., 116, Birkhäuser Boston, Boston, MA, 1993.

[16] J. Fontaine, Il n’y a pas de variété abélienne sur Z, Invent. Math. 81, 1985, no. 3, 515–538.

[17] B. Gross, A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math. J. 61 (1990), no. 2, 445–517.

[18] A. Grothendieck, Groups de monodromie en géométrie algébrique I, (SGA 7), Séminaire de Géométrie Algébrique du Bois-Marie, 1967-1969, Lecture Notes in Math., vol. 288 , Springer, Berlin and New York, 1972, 313–523.

[19] N. Katz, p-adic properties of modular schemes and modular forms, Modular functions of one variable, III, (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 350 , Springer, Berlin and New York, 1973, 69–190.

[20] C. Khare, J. P. Wintenberger, Serre’s modularity conjecture: the odd conductor case (II), preprint, 2006.

[21] L. Kilford, Some non-Gorenstein Hecke algebras attached to spaces of modular forms, J. Number Theory 97 (2002), no. 1, 157–164.

[22] W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.

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

[24] B. Mazur Rational isogenies of prime degree, Invent. Math. 44 (1978), no. 2, 129–162.

[25] L. Merel, L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de J_{0}(p), J. Reine Angew. Math. 477 (1996), 71–115.

[26] F. Oort, J. Tate, Group schemes of prime order. Ann. Sci. Ècole Norm. Sup. (4) 3 (1970), 1–21.

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

[28] K. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., vol. 601, Springer, Berlin and New York, 1977, 17–51.

[29] K. Ribet, Mod p Hecke operators and congruences between modular forms, Invent. Math. 71, (1983), no. 1, 193–205.

[30] K. Ribet, On modular representations of Gal(\overline{Q}/Q) arising from modular forms, Invent. Math. 100, (1990), no. 2, 431–476.

[31] W. Stein, M. Watkins, A Database of Elliptic Curves—First Report, Algorithmic number theory (Sydney, 2002), 267–275, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002.

[32] W. Stein, M. Watkins, Modular parametrizations of Neumann–Setzer elliptic curves, IMRN, no. 27 (2004), 1395–1405.

[33] J. Tate, The non-existence of certain Galois extensions of Q unramified outside 2, in Arithmetic Geometry (N. Childress, J. Jones, eds.), Contemp. Math. 174 (1994), 153–156.

[34] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), no. 3, 553–572.

[35] M. Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 (2002), no. 4, 487–502.

[36] G. Wiese, Dihedral Galois representations and Katz modular forms, Doc. Math. 9 (2004), 123–133.

[37] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math. 141 (1995), no. 3, 443–551.

[38] D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), no. 3, 372–384.

Email addresses: fcale@math.harvard.edu
emerton@math.northwestern.edu