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 : X0 (N ) → E defined over Q. There is a unique such map of minimal degree(up to composing with automorphisms of E), and its degree mE 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 mE satisfies certain divisibility
properties, especially, perhaps, by the prime 2. More precisely, we remind the reader thatthe class number of an imaginary quadratic extension K of Q is odd if and only if theabsolute discriminant DK 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 mE . 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 mE is
odd only if the conductor N of E is a prime congruent to 3 mod 8. In fact we prove thefollowing 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

1.2 Example. The following examples of elliptic curves with odd modular degree should
serve to illustrate conditions (3a), (3b) and (3c). The curve X0 (15) has modular degree
one and a rational two torsion point, and thus satisfies condition (3a). Another example
is given by the curvey 2 + xy = x3 − x2 − 58x − 105(2537E in Cremona’s tables) of conductor 43 · 59 with modular degree 445 and torsion
subgroup Z/4Z. The curves X0 (11) and X0 (19) both have modular degree one and satisfy
condition (3b). An example of larger conductor is given by
y 2 + y = x3 + x2 − 4x − 10of conductor 24859 and modular degree 3979. Finally, there are exactly four curves of
odd modular degree with complex multiplication, namely X0 (27), X0 (32), X0 (49) (all of
modular degree one) andy 2 + y = x3 + 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 ofgenerality 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 beeven 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 2r |mE where r = rank(E(Q)).
1.6 Remark. In [32] the parity of mE was determined for a very particular explicit classof 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 X0 (17) (equivalently, E does not have a rational
2-torsion point). If mE is odd, then N ≡ 3 mod 8.
2

One technique for proving that an elliptic curve E has even modular degree is to showthat the map π factors through X0 (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 tohandle is when N is actually prime, and in this case we deduce Theorem 1.1 from thefollowing result, proved in section 3.
1.8 Theorem. Let N be prime, let T denote the Hecke algebra over Z acting on weighttwo cuspforms on Γ0 (N ), and let m be a maximal ideal of T such that T/m = F2 , andsuch that the associated semi-simple Galois representation ρ : Gal(Q/Q) → GL2 (F2 ) is
irreducible. If the completion Tm = Z2 , then
1. m is supersingular at 2 (i.e. T2 ∈ m), or equivalently, ρ|D2 is absolutely irreducible.
2. ρ is totally complex.
The relevance of this result to Theorem 1.1 is that, since N is prime in the context ofTheorem 1.8, a result of Ribet [38] shows (assuming, as we may, that E is optimal) thatthe modular degree of E is even if and only if 2 is a congruence prime for the newform oflevel N attached to E.
1.9 Remark. The equivalence between the supersingularity of m and the absolute irreducibility of ρ|D2 can be deduced (in the standard way) as follows: if V/Z2 denote a finitegroup scheme subquotient of J[m]/Z2 whose generic fibre (regarded as a Galois module)
is isomorphic to ρ|D2 , 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, thenit admits either a multiplicative or an étale subgroup scheme, either of which gives riseto an étale subgroup scheme of the generic fibre V/Q2 . Thus ρ|D2 admits an unramified
subrepresentation, and hence is not absolutely irreducible. Conversely, if ρ|D2 is absolutely
reducible, then (extending scalars to F4 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 F4 ⊗F2 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); seefor 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 ρ : Gal(Q/Q) → GL2 (Fp ) is a surjective modular representation, then theoremsof Wiles and Taylor–Wiles [37, 34] show that the universal minimal deformation ring R∅
attached to ρ is isomorphic to the universal minimal modular deformation ring T∅ (= Tm ,since N is prime). Since T∅ is a finite W (Fp ) = Zp algebra with residue field Fp , it isexactly equal to Zp if and only if it is an étale Zp -algebra. On the other hand, since3R∅ ∼= T∅ , this is equivalent to R∅ being étale over Zp , 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(Q/Q) → GL2 (Fp [x]/(x2 ))of ρ. Thus to prove that Tm 6= Zp it suffices to show that there exists a non-trivial minimal
deformation of ρ to GL2 (Fp [x]/(x2 )).
In spirit, the proof of Theorem 1.8 follows this strategy; in other words, we determine
whether or not Tm = Z2 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 failureof Tm to be Gorenstein and the consequent failure of multiplicity one [21], and the fact that
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 withoutsome 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 whenp = 2 and the image of ρ is dihedral. (The main application of [11] to date has been to
representations with image SL2 (F4 ) ' A5 .) 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 ρ
has non-solvable image, and in any case, they do not give the precise R = T statementsthat would be needed for applications to Theorem 1.8.
Thus, instead of appealing to any general modularity results, we show that Tm is bigger
than Z2 by explicitly constructing (in certain situations) non-trivial deformations of ρm toF2 [x]/(x2 ) that are demonstrably modular (and hence contribute to the reduced cotangent
space of Tm ). 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 Z2 -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 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 ρ arising from the 2-torsion on an
elliptic curve E: he uses the symmetric square map from ρ to Sym2 ρ to lift elements from4the 2-Selmer group of E to the tangent space to the deformation ring of ρ. He also showsthat 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 theexperimental work of Watkins on the parity of modular degrees, together with the resultsof 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 ofsuch 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 overk (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 A0 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
A0 , and which together with A0 generates A. Thus A sits in the split short exact sequenceof groups
0 → E(k) → A → A0 → 1.
(1)
(This is of course well known. The surjection A → A0 may also be regarded as the map
A = Aut(E) → Aut(Pic0 (E)) induced by the functoriality of the formation of Picard
varieties — the target being the group of automorphisms of Pic0 (E) as a group variety —once we identify E and Pic0 (E) as group varieties in the usual way.)
The short exact sequence (1) induces a short exact sequence
0 → W ∩ E(k) → W → W0 → 1,
where W0 denotes the projection of W onto A0 . The known structure of A0 shows that
W0 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 thetheorem under the additional assumption that E is an optimal quotient of X0 (N ).
Let W denote the group of automorphisms of X0 (N ) generated by the Atkin–Lehnerinvolutions; W is an elementary abelian 2-group of rank equal to the number of primes
5dividing N . Since E is an optimal quotient of X0 (N ), the action of W on X0 (N ) descends
to an action on E. If w ∈ W were to act trivially on E, then the quotient map X0 (N ) → E
would factor through X0 (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 showsthat E[2](Q) is non-trivial.
Suppose now that N is odd, so that X0 (N ) and E both have good reduction at 2. We
may then apply the argument of the preceding paragraph over F2 , 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 −wN . If E is of odd analytic rank,
and if fE denotes the normalized newform of level N attached to E, then wN fE = fE , andso the automorphism of E induced by wN has trivial image in A0 . Thus wN acts on E via
translation by an element P ∈ E(Q). Since wN interchanges the cusps 0 and ∞ on X0 (N ),we see that P = π(0) − π(∞) (where π : X0 (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(fE , 1) = 0. Since thisL-value can be computed (up to a non-zero factor) by integrating fE from 0 to ∞ in theupper half-plane, we conclude that P = O, and thus that wN acts trivially on E. Hence π
factors through the quotient X0 (N )/wN of X0 (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 fE be the associated Hecke eigenform of level Γ0 (N ) and weight 2.
From a theorem of Ribet [38], 2|mE if and only if fE satisfies a congruence mod 2 withanother cuspidal eigenform of level N . The set of cuspidal eigenforms (in characteristiczero) congruent to f is indexed by Hom(Tm ⊗ Q2 , Q2 ). Thus fE satisfies no non-trivial
congruences if and only if Tm ⊗ Q2 = Q2 , or equivalently if and only if Tm = Z2 .

Of use to us will be the following theorem of Grothendieck on Abelian varieties withsemistable reduction [18, Exposé IX, Prop. 3.5]:
3.2 Theorem (Grothendieck). Let A be an Abelian variety over Q with semistablereduction at `. Let I` ⊂ Gal(Q/Q) denote a choice of inertia group at `. Then theaction of I` on the pn -division points of A for p 6= ` is rank two unipotent; i.e., as anendomorphism, for σ ∈ I` ,(σ − 1)2 A[pn ] = 0.
In particular, I` acts through its maximal pro-p quotient, which is procyclic.
6

Shimura proved that a modular form f of weight 2 and level Γ0 (N ) gives rise to amodular Abelian variety Af in such a way that the p-adic representations ρf attached tof arise from the torsion points of Af . 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 IN factor through a pro-cyclic 2-group. For representations ρ with image
inside GL2 (F2 ) ' S3 , this means in particular that the order of inertia at N is either 1
or 2.
Let us now consider a Galois representation ρ : Gal(Q/Q) → GL2 (F2 ), arising from a
cuspidal Hecke eigenform of level Γ0 (N ), whose image is not contained in a Borel subgroup.
(This is equivalent to ρ being irreducible, and also to the image of ρ having order divisible
by 3.) Let L be the fixed field of the kernel of ρ; the extension L/Q is unramified outside 2
and N . If L/Q is unramified at N , then ρ has Serre conductor 1, contradicting a theoremof Tate [33]. Thus by the discussion above we see that inertia at N factors through agroup of order 2, that L/Q is an S3 -extension, and (hence) that ρ is absolutely irreducible.
Let K/Q be a cubic subfield of L, and let F be the quadratic extension inside L. Sinceρ 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( ±N ) (as it isramified at N ). The extension L/F is unramified at the prime above N , since as explained
above, ρ|IN has order dividing two. Moreover, the extension L/K is ramified at 2 if andonly if ρ is supersingular, as follows from Remark 1.9.
3.3 Lemma. If 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( −N ), or N ≡ 5 mod 8 and F = Q( N ). Moreover if
F = Q( N ), then K and L are totally real.
We shall prove Theorem 1.8 by showing in the following subsections that if ρ satisfies
at least one of the following conditions:1. ρ is totally real;
2. ρ is unramified at 2;
3. ρ is ordinary, complex, and ramified at 2;
then Tm 6= Z2 .

3.2

ρ is totally real

The theory of modular deformations is not well understood when ρ is totally real. Thus ourarguments 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]).
7

3.4 Theorem. Let N be prime. Then J0 (N )(R) is connected.
If we let g denote the dimension of J := J0 (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[2m ]. Then J[2∞ ] is a 2-divisible group over Q admitting an action−→of T2 .
Since T2 is finite and flat over the complete local ring Z2 there exists a decomposition
Y
T2 =
Tm ,
where the product is taken over the maximal ideals m of T of residue characteristic two.
If g(m) denotes the rank of Tm over Z2 , thenXg(m) = rank(T2 /Z2 ) = g.
m
∞

∞

Q
If J[m ] := J[2 ] ⊗T2 Tm , then J[2∞ ] 'J[m∞ ] (compare [23] §7, p. 91). Fromlemma 7.7 of [23] we see that Tam J ⊗ Q2 is free of rank two over Tm ⊗ Q2 (where Tam J :=Hom(Q2 /Z2 , J[m∞ ](Q)) is the m-adic Tate module of J), and thus thatJ[m∞ ](C) ∼(2)= (Q2 /Z2 )2g(m) .
Let J[2]m := J[2] ⊗T2 Tm 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
dimZ/2Z (J[2]m (R)) = g(m)Proof.
The isomorphism (2) induces an isomorphism J[2]m (C) ∼= (Z/2Z)2g(m) . Letσ ∈ 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 dimZ/2Z (Jm [2](R)) > g(m)for some m, then sinceY
J[2](R) =J[2]m (R),and since (as was noted above) dimZ/2Z (J[2](R)) = g, we would deduce the inequality:
XX
g=dimZ/2Z (J[2]m (R)) >g(m) = g,mwhich is absurd.

Now let ρ be a totally real (absolutely) irreducible continuous modular representationof Gal(Q/Q) into GL2 (F2 ) of level Γ0 (N ), and let m be the corresponding maximal idealof T. The main result of [2] shows that the Gal(Q/Q)-representation J[m](Q) is a direct
sum of copies of ρ. Thus, since ρ is totally real, we find that
dimZ/2Z J[m](R) = dimZ/2Z J[m](C) ≥ dimZ/2Z ρ = 2.
Combining this inequality with the inclusion J[m](R) ⊆ J[2]m (R) and Lemma 3.5 we findthat g(m) ≥ 2, and thus that Tm 6= Z2 .
8

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 := Γ0 (N ) or Γ1 (N ), for some N ≥ 1, and a weight k, the spaceof cuspforms Sk (Γ, C) is unambiguously defined. For any subring R of C, we let Sk (Γ, R)
denote the R-submodule of Sk (Γ, C) consisting of cuspforms whose q-expansion coefficientslie in R. It is well know that the natural map R ⊗Z Sk (Γ, Z) → Sk (Γ, R) is an isomorphism,and so for an arbitrary ring R we define Sk (Γ, R) := R ⊗Z Sk (Γ, Z).
In the case when Γ := Γ1 (N ), the nebentypus action of (Z/N Z)× on Sk (Γ, C) preserves
Sk (Γ, Z), and hence induces a nebentypus action on Sk (Γ, R) for any R. If χ : (Z/N Z)× →C× is a character, then, letting Z[χ] denote the subring of C generated by the values ofχ, we let Sk (Γ, χ, Z[χ]) denote the subspace of Sk (Γ, Z[χ]) consisting of cuspforms withnebentypus character χ. If R is any Z[χ]-algebra, then we let Sk (Γ, χ, R) denote the imageof R ⊗Z[χ] Sk (Γ, χ, Z[χ]) under the tautological isomorphism R ⊗Z[χ] Sk (Γ, Z[χ]) ∼= Sk (Γ, R).
(This image is contained in the R-submodule of Sk (Γ, R) consisting of cuspforms on which(Z/N Z)× acts through the character χ. However, if R is not flat over Z[χ], then it neednot coincide with this submodule.)
The following lemma confirms that certain elementary manipulations with the spaces
Sk (Γ, R) are permissible in the context of the above definitions.
3.6 Lemma. If R is an F2 -algebra, then the following maps are well defined.
1. A map S1 (Γ1 (M ), χ, R) → S2 (Γ0 (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 U2 : S2 (Γ0 (2k+1 M ), R) → S2 (Γ0 (2k M ), R), for any M ≥ 1
and k ≥ 1.
Proof.
It suffices to treat the case R = F2 , since there is an isomorphism R ⊗F2
Sk (Γ, F2 ) ∼= Sk (Γ, R) by definition. For part (3), lift f ∈ S1 (Γ1 (M ), χ, F2 ) to characteristic
zero (possible, by definition), and then multiply by the Eisenstein series E1,χ . We obtain aform g of level Γ0 (M ) which we then reduce to obtain the desired map. Similarly, for part(4), lift f to characteristic zero. The operator U2 reduces the level (provided that k ≥ 1)
and preserves the integrality of q-expansions at ∞. Hence the reduction of g = U2 f lies in
S2 (Γ0 (2k N ), F2 ).

Let T denote the Z-algebra of endomorphisms of Sk (Γ, C) (or equivalently, of Sk (Γ, Z))
generated by the Hecke operators. The algebra T acts on Sk (Γ, Z), and hence acts on
Sk (Γ, R) for any ring R. Furthermore, the usual pairing hf, T i := a1 (T f ) (f ∈ Sk (Γ, R),
T ∈ T) induces an isomorphism Sk (Γ, R) ∼= Hom(T, R) [29, Thm. 2.2]. (Here Hom means
simply Hom of abelian groups.) A cuspform h ∈ Sk (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 f m defined over the finite field T/m, associated to which is a semi-simple Galois
representation ρm : Gal(Q/Q) → GL2 (T/m). If this Galois representation is absolutely
9irreducible, then Carayol (building on constructions of Deligne, Shimura, and Serre) hasconstructed a lifting of it to a continuous representation ρm : Gal(Q/Q) → GL2 (Tm ) [7,
Thm. 3].
Let R be a complete local ring, with maximal ideal n, such that R/n has positive
residue characteristic. If f ∈ Sk (Γ, R) is a Hecke eigenform, then f corresponds to a ringhomomorphism φ : T → R, which extends to a homomorphism Tm → 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 f m to the ring R.) If ρm is absolutely irreducible, then pushing forward ρm
via φ, we obtain an n-adically continuous representation Gal(Q/Q) → GL2 (R), which werefer to as the Galois representation associated to f .
Let m be a maximal ideal of T for which T/m = F2 . The completion Tm is naturally afinite flat Z2 -algebra, of some rank r > 0, say. The isomorphism Hom(T, F2 ) ∼= Sk (Γ, F2 )∞∼induces an isomorphism Hom(Tm , F2 ) = Sk (Γ, F2 )[m ]. (Here the target is the subspaceconsisting of cuspforms annihilated by some power of m.) Thus dimF2 (Sk (Γ, F2 )[m∞ ]) = r.
In particular, we obtain the following lemma.
3.7 Lemma. If there exist non-zero elements f , g ∈ S2 (Γ, F2 ) such that mf = m2 g = 0,
while mg 6= 0, then Tm 6= Z2 .
Proof. The existence of the elements f and g shows that dimF2 (Sk (Γ, F2 )[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, andif Γ = Γ1 (N ) with N ≥ 5, then there is a natural isomorphism SkKatz (Γ, R) ∼= Sk (Γ, R),
uniquely determined by its compatibility with the formation of q-expansions. In the casewhen Γ = Γ0 (N ), the comparison of SkKatz (Γ, R) and Sk (Γ, R) is more complicated, due tothe fact that the modular curve X0 (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

ρ is unramified at 2

We now return to the situation of Theorem 1.8. Suppose that ρ is unramified at 2. This
forces ρ to be ordinary. By the theory of companion forms [17] one expects that ρ arises
from a mod 2 form of level Γ1 (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 ρ is dihedral,as it is in our situation. Let f ∈ S1Katz (Γ1 (N ), F2 ) be this companion form for 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 S2Katz (Γ1 (N ), F2 ),and thus of S2 (Γ1 (N ), F2 ). (Note that it is no loss of generality to assume that N ≥ 5,
since X1 (N ) has genus zero for 1 ≤ N ≤ 4.) Wiese’s construction furthermore allows us tochoose f , and hence Af and g, so as to have trivial nebentypus. A lemma of Carayol [6,10§4.4]1 then assures us that Af and g in fact lie in S2 (Γ1 (N ), 1, F2 ) (where in this notation
1 denotes the trivial character of (Z/N Z)× ), that is, that Af and g lie in S2 (Γ0 (N ), F2 ).
∞X
If f has q-expansion f =an q n , then Af has the same q-expansion, while g has
q-expansion g =

∞
X

n=1

a2n q 2n ≡ V2 f , since an ∈ F2 . Since a1 = 1, we see that f and g are

n=1

linearly independent. Furthermore, one computes that (T` − a` )f = (T` − a` )g = 0 for
all odd `, that (T2 − a2 )f = 0 and (T2 − a2 )2 g = 0, and that (T2 − a2 )g 6= 0. Thus
mf = m2 g = 0, while mg 6= 0, and therefore, by Lemma 3.7, Tm 6= Z2 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 ρ is totally real, and this case of
Theorem 1.8 is already covered by section 3.2. If we assume that ρ is not totally real, thenwe may construct f directly as an element of S1 (Γ1 (N ), χ) for some character χ. (See thefirst paragraph of the proof of [36, Thm. 9].) One can then define Af ∈ S2 (Γ0 (N ), F2 ) byapplying the map of Lemma 3.6 (1) to f , as well as g = f 2 ∈ S2 (Γ0 (N ), F2 ). The argument
then proceeds just as above.

3.5

ρ is ordinary, complex, and ramified at 2

Suppose that ρ is ordinary, complex, and √ramified at 2. It follows that F/Q is complex
and ramified at 2, and thus that F = Q( −N ) for some N ≡ 1 mod 4. Moreover,√ theextension L/F is unramified everywhere. Since N ≡ 1 mod 4 it follows that H := L( −1)is also unramified everywhere over F . The field H is Galois over Q, and clearly
Gal(H/Q) ' S3 × Z/2Z.

(3)

We may embed S3 × Z/2Z into GL2 (F2 [x]/(x2 )) by fixing an identification of S3 withGL2 (F2 ), and mapping a generator of Z/2Z to the matrix


1+x
0
.
0
1+x
Composing the isomorphism (3) with this embedding yields a representation:
ρ : Gal(H/Q) ,→ GL2 (F2 [x]/(x2 )).
The representation ρ has trivial determinant (equivalently, determinant equal to the mod2 cyclotomic character). We also claim that ρ is finite flat at two. To check this, it suffices
factors through a group of order 2,
to prove this over Zur
2 . The representation ρ|Gal(Q2 /Qur
2 )1

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

11and one explicitly sees that it arises as the generic fibre of the group scheme (D ⊕ D)/Zur
2 ,where D is the non-trivial extension of Z/2Z by µ2 considered in [23] (Prop 4.2, p. 58).
Thus one expects ρ to arise from an F2 [x]/(x2 )-valued modular form of weight two andlevel N , corresponding to a surjective map of rings Tm → F2 [x]/(x2 ). This would follow
if we knew that Tm coincided with the minimal deformation ring associated to ρ. Rather
than proving this, however, we shall use weight one forms to explicitly construct a weight
two modular form giving rise to ρ.
Let χ4N be the character of conductor 4N associated to F . Consider two faithfulrepresentations
ψ1 : Gal(L/Q) ∼= S3 ,→ GL2 (C),

ψ2 : Gal(H/Q) ∼= S3 × Z/2Z ,→ GL2 (C).

Since F/Q is complex, these dihedral representations are odd and therefore give rise toweight one modular forms h1 , h2 in S1 (Γ1 (4N ), χ4N , C).
3.8 Lemma. The modular forms h1 , h2 are ordinary at 2, have coefficients in Z, and arecongruent modulo 2. Let(h2 − h1 )∈ Z[[q]].
g=
2
Then g mod 2 is non-zero, and the form h = h1 + xg ∈ S1 (Γ1 (4N ), χ4N , F2 [x]/(x2 )) is aneigenform for all the Hecke operators, including U2 . The GL2 (F2 [x]/(x2 ))-valued Galois
representation associated to h is isomorphic to ρ.
Proof. The modular forms are both ordinary at 2 because the representations ψ1 andψ2 have non-trivial subspaces on which inertia at two is trivial (since I2 acts through agroup of order 2). They both have coefficients in Z, since 2 cos(π/3) ∈ Z. The congruenceh1 ≡ h2 mod 2 follows from the fact that both are ordinary-at-2 Hecke eigenforms, andthat a(h1 , `) = a(h2 , `)χ4 (`) for all odd primes `, where χ4 is the character of conductor 4.
From this one also sees that g is non-trivial modulo two, and that h1 , h2 , g define forms in
S1 (Γ1 (4N ), χ4N , Z). Thus, by definition, h ∈ S1 (Γ1 (4N ), χ4N , F2 [x]/(x2 )). The claim thath is a Hecke eigenform follows formally (on q-expansions) from the fact that h1 and h2
are Hecke eigenforms that are congruent modulo 2. The discussion of Section 3.3, showsthe 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, wewould like to construct a corresponding weight two form of level N . Applying the map ofLemma 3.6 (1) (concretely, multiplying h by the Eisenstein series in M1 (Γ1 (4N ), χ4N )), we
see that there is a modular form h0 ∈ S2 (Γ0 (4N ), F2 [x]/(x2 )) having the same q-expansion
as h. Since h, and hence h0 , is an ordinary U2 eigenform, we may apply the U2 operator todeduce (using Lemma 3.6 (2)) that h0 ∈ S2 (Γ0 (2N ), F2 [x]/(x2 )). Applying Theorem 3.14
(proved in the following subsection) we then deduce that in fact h0 ∈ S2 (Γ0 (N ), F2 [x]/(x2 )),and (thus) that there is a modular form g 0 ∈ S2 (Γ0 (N ), F2 ) having the same q-expansion
as g.
12

As in the discussion of Section 3.3, let f m ∈ S2 (Γ0 (N ), F2 ) be the Hecke eigenform
associated to m. Then mf m = 0, while an easy calculation shows that mg ⊂ F2 f m \ {0},and so also that m2 g = 0. We conclude from Lemma 3.7 that Tm 6= Z2 .

3.6

Level-lowering for modular deformations

The goal of this section is to prove a level-lowering result for modular forms with coefficientsin 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 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 isthat 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 Γ0 (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
ρ : Gal(Q/Q) → GL2 (k)is (absolutely) irreducible, ordinary, and ramified at 2, then Tam J0 (2N ) (the m-adic Tate
module of J0 (2N )) is free of rank two over the completion Tm .
To be clear, the condition “ordinary at 2” means that the image of a decompositiongroup at 2 under ρ lies in a Borel subgroup of GL2 (k). Since k is of characteristic 2, we seethat (for an appropriate choice of basis) the restriction of ρ to an inertia group at 2 may
be written in the form
1 ∗
ρ|I2 =.
0 1
The assumption that ρ is ramified at 2 then implies that ∗ is not identically zero. Thus
the representation space of ρ has a unique line invariant under I2 , and so ρ is irreducible
if and only if it is absolutely irreducible.
3.10 Lemma. Let k be a finite field of characteristic 2. If ρ : Gal(Q2 /Q2 ) → GL2 (k) is
a continuous representation that is finite, ordinary, and ramified at 2, then ρ has a unique
finite flat prolongation over Z2 (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(Q2 /Q2 ) 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 ρ: étale
extended by étale; multiplicative extended by multiplicative; multiplicative extended byétale; 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
13

étale must split, by a consideration of the connected étale sequence). Thus, since ρ isassumed ramified, we see that any prolongation of ρ must be an extension of a rank oneétale 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 0 of ρ to a finite flat group scheme [27, Cor. 2.2.3]. Since étale (andhence multiplicative) group schemes are determined by their generic fibre, the result of
the preceding paragraph shows that the natural morphism M → M 0 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 Qp for some prime p, andlet O denote the ring of integers of K. If A is an abelian variety over K, then we let A/Odenote the connected component of the identity of the Néron model of A over Spec O. For
any power pr of p, the pr -torsion subgroup scheme A[pr ]/O of A/O is then a quasi-finite
flat group scheme over Spec O; we let F A[pr ]/O denote its maximal finite flat subgroupscheme, and A[pr ]0/O denote the maximal connected closed subgroup scheme of A[pr ]/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[pr ]/O is a p-divisible group, and A[p∞ ]0/O := lim A[pr ]0/O 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 Qp to be unrestricted, at the expense of imposing more restrictive
hypotheses on the reduction of the abelian varieties appearing in the exact sequence underconsideration.
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∞ ]0/O → B[p∞ ]0/O → C[p∞ ]0/O → 0
is a short exact sequence of p-divisible groups over Spec O. Equivalently, for any power prof p, the induced sequence
0 → A[pr ]0/O → B[pr ]0/O → C[pr ]0/O → 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[pr ]0/O 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∞ ]0/O → B[p∞ ]0/O is a closed embedding.
Let C 0 ⊂ B be an abelian subvariety chosen so that the induced map C 0 → C is an
isogeny. Then C 0 also has good reduction, and so C 0 [p∞ ]0/O → C[p∞ ]0/O is an epimorphism
14

of p-divisible groups over Spec O. Thus the induced map B[p∞ ]0/O → C[p∞ ]0/O is also an
epimorphism of p-divisible groups. A consideration of generic fibres shows that the kernelof this surjection coincides with the scheme-theoretic image of A[p∞ ]0/O in B[p∞ ]0/O , 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 thecase when “∆(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 J0 (2N ) → J0 (N ) × J0 (N )induced by Albanese functoriality applied to the two “degeneracy maps” from level 2N
to level N , then A is an abelian subvariety of J0 (2N ) having purely toric reduction at 2,
whilst the quotient B of J0 (2N ) by A has good reduction at 2. From Proposition 3.11 we
obtain (for any r ≥ 1) the short exact sequence
0 → A[2r ]0/Z2 → J0 (2N )[2r ]0/Z2 → B[2r ]0/Z2 → 0
of connected finite flat group schemes over Spec Z2 . By functoriality of the formation ofthis short exact sequence, and since A is a T-invariant subvariety of J0 (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 actionof T, and hence of Z2 ⊗Z T, via endomorphisms, then by the localization Mm we mean the
kernel G[ε0 ], where the idempotent ε0 ∈ Z2 ⊗Z T is defined as ε0 := 1 − ε, with ε ∈ Z2 ⊗Z T
being the idempotent corresponding to the direct factor Tm of Z2 ⊗ZL
T. Since ε and ε0
are orthogonal idempotents (by construction), the natural map G[ε] G[ε0 ] ∼= G is an
isomorphism. Thus passage to Gm 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[2r ]0m/Z2 → J0 (2N )[2r ]0m/Z2 → B[2r ]0m/Z2 → 0.

(4)

Passing to Q2 -valued points induces a short exact sequence of Gal(Q2 /Q2 )-modules
0 → A[2r ]0m (Q2 ) → J0 (2N )[2r ]0m (Q2 ) → B[2r ]0m (Q2 ) → 0,

(5)which is a subexact sequence of the short exact sequence of Gal(Q2 /Q2 )-modules
0 → A[2r ]m (Q2 ) → J0 (2N )[2r ]m (Q2 ) → B[2r ]m (Q2 ) → 0.

(6)

Let A[2r ]m (Q2 )χ (respectively J0 (2N )[2r ]m (Q2 )χ , respectively B[2r ]m (Q2 )χ ) denote themaximal Gal(Q2 /Q2 )-subrepresentation of A[2r ]m (Q2 ) (respectively J0 (2N )[2r ]m (Q2 ), respectively B[2r ]m (Q2 )) on which the inertia group acts through the 2-adic cyclotomic character χ. The short exact sequence (6) induces an exact sequence
0 → A[2r ]m (Q2 )χ → J0 (2N )[2r ]m (Q2 )χ → B[2r ]m (Q2 )χ .

15

(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 Tm [Gal(Q2 /Q2 )]-modules
underlying the corresponding exact sequence of Tm [Gal(Q/Q)]-modules
0 → A[2r ]m (Q) → J0 (2N )[2r ]m (Q) → B[2r ]m (Q) → 0.
Since ρ is assumed irreducible as a k[Gal(Q/Q)]-representation, each of the modules appearing in this exact sequence is a successive extension of copies of ρ. The same is thustrue of each of the modules appearing in the exact sequence (6).
Since A has purely toric reduction, it is clear that A[2r ]0m/Z2 is of multiplicative type,and soA[2r ]0m (Q2 ) ⊂ A[2r ]m (Q2 )χ .
(8)
Fix a filtration 0 = W0 ⊂ W1 ⊂ · · · ⊂ Wn = A[2r ]m (Q2 ) of A[2r ]m (Q2 ) for which thesuccessive quotients Wi+1 /Wi are isomorphic to ρ. Since A has purely toric reduction thequotient A[2r ]/Q2 /A[2r ]0/Q2 is Cartier dual to Â[2r ]0/Q2 (where Â is the dual abelian varietyto A), and so A[2r ]m (Q2 )/A[2r ]0m (Q2 ) is an unramified Gal(Q2 /Q2 )-representation. Sinceρ is assumed ramified at 2, this implies that\\
Wi A[2r ]0m (Q2 ) ( Wi+1 A[2r ]0m (Q2 )for each i ≥ 0. Furthermore,
Wi+1 6⊂ A[2r ]m (Q2 )χ + Wi
for each i ≥ 0, because χ mod 2 is trivial. Since Wi+1 /Wi ∼= ρ is two dimensional over kfor each i ≥ 0, we conclude by induction on i that\\
Wi A[2r ]0m (Q2 ) = Wi A[2r ]m (Q2 )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[2r ]/Z2 = B[2r ]/Z2 . As noted
above, any Jordan–Hölder filtration of the localization B[2r ]m (Q) as a T[Gal(Q/Q)]module has all its associated graded pieces isomorphic to ρ. Taking scheme-theoretic closures of such a filtration in B[2r ]/Z2 , we obtain a filtration of the localization B[2r ]m/Z2 byfinite flat closed subgroup schemes, whose associated graded pieces are prolongations of ρ.
Now Lemma 3.10 shows that the connected component of any such finite flat prolongation
is multiplicative. Thus B[2r ]0m/Z2 is indeed multiplicative, whilst B[2r ]m (Q2 )/B[2r ]0m (Q2 )is an unramified Gal(Q2 /Q2 )-module. Arguing as in the preceding paragraph gives therequired equality
B[2r ]0m (Q2 ) = B[2r ]m (Q2 )χ .
Since any extension of multiplicative type groups is again of multiplicative type, we
see that J0 (2N )[2r ]0m/Z2 is also of multiplicative type, and that the exact sequence (5) is16

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 J0 (2N )[2]0m (Q2 ) is the maximal
unramified Gal(Q2 /Q2 )-subrepresentation of J0 (2N )[2]m (Q2 ) (since χ mod 2 is trivial).
Recall that there is a natural isomorphism Tan(J0 (2N )[2]0/F2 ) ∼= Tan(J0 (2N )/F2 ) (indeed,this is true with J0 (2N )/F2 replaced by any group scheme over F2 ), and also a natural
isomorphism Tan(J0 (2N )[2]0/F2 ) ∼= J0 (2N )[2]0 (Q2 ) ⊗F2 F2 (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 J0 (2N )[2]0m (Q2 ) is a cyclic Tm -module, and thus that the maximal
unramified Gal(Q2 /Q2 )-subrepresentation of J0 (2N )[2]m (Q2 ) is a cyclic Tm -module.
Let ρm : Gal(Q/Q) → GL2 (Tm ) denote the Galois representation associated to m by [7,Thm. 3]. Carayol has proved [7, Thm. 4] that there is an isomorphism Tam J0 (2N ) ∼= J ⊗Tmρm for some ideal J in Tm , and thus an isomorphism J0 (2N )[2]m (Q2 ) ∼= (J/2J) ⊗Tm ρm .
We conclude that J/2J is a cyclic Tm -module, and hence that J is a principal ideal in Tm .
The discussion of [7, 3.3.2] shows that in fact J ∼= Tm and that Tam J0 (2N ) is free of rank
two over Tm , as claimed.

3.13 Corollary. In the situation of Theorem 3.9, the completion Tm is a Gorenstein Z2 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(Q/Q) →GL2 (A) that is modular of level Γ0 (2N ) for some odd natural number N , in the sense that itarises from a Hecke eigenform h ∈ S2 (Γ0 (2N ), A). Let ρ denote the residual representation
attached to ρ (so ρ arises from the Hecke eigenform h ∈ S2 (Γ0 (2N ), k) obtained by reducing
h modulo the maximal ideal of A).
3.14 Theorem. If ρ : Gal(Q/Q) → GL2 (A) is a modular Galois representation of levelΓ0 (2N ) as above, such that
1. ρ is (absolutely) irreducible,
2. ρ is ordinary and ramified at 2, and3. 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 Tm of T17

at some maximal ideal m of residue characteristic 2, and the residual representation ρ isthe residual Galois representation attached to the maximal ideal m. We let ρm denote the
Galois representation
ρm : Gal(Q/Q) → GL2 (Tm )attached to m by [7, Thm. 3]. The Galois representation ρ attached to h coincides withthe pushforward of ρm via φ.
Replacing A by the image of φ, we may and do assume from now on that φ is surjective. We let I ⊂ Tm denote the kernel of φ. Since A is Artinian, we may chooser ≥ 1 so that 2r ∈ I. Theorem 3.9 shows that the m-adic Tate module Tam J0 (2N ) is
isomorphic as a Gal(Q/Q)-representation to ρm ; thus J0 (2N )[2r ]m (Q) is isomorphic to
the reduction mod 2r of ρm . Since Tm is a Gorenstein Z2 -algebra, by corollary 3.13,
we see that Tm /2r Tm is a Gorenstein Z/2r Z-algebra, and thus that there is an isomorphism (Tm /2r Tm )[I] ∼= HomZ/2r (Tm /I, Z/2r ) of Tm /I = A-modules. In particular,
J0 (2N )[I](Q/Q) ⊂ J0 (2N )[2r ]m (Q) is a faithful A-module, isomorphic as an A[Gal(Q/Q)]module to HomZ/2r (Tm /I, Z/2r ) ⊗A ρ. To simplify notation, we will write(9)
V := J0 (2N )[I](Q) ∼= HomZ/2r (Tm /I, Z/2r ) ⊗A ρ.
By assumption, ρ prolongs to a finite flat group scheme M over Spec Z2 . If we fix
a Jordan–Hölder filtration of ρ as an A[Gal(Q/Q)]-module, then the associated gradedpieces are each isomorphic to 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 schemeV∼= HomZ/2r (Tm /I, Z/2r ) ⊗A M
over Z2 . 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 schemeét
M by a multiplicative A-module scheme M0 , 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 J0 (2N ) over Spec Z2 . For a scheme over Z2 ,use the subscript “s” to denote its special fibre over F2 . The special fibre Js admits thefollowing filtration by T-invariant closed subgroups:
0 ⊂ T ⊂ Js0 ⊂ Js ,where T is the maximal torus contained in Js , and Js0 is the connected component of
the identity of Js . The quotient Js0 /T is an abelian variety on which T acts through its
quotient Told (where Told denotes the quotient of T that acts faithfully on the space of
2-old forms of level 2N ). The connected component group Φ := Js /Js0 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]).
18

3.15 Lemma. The Zariski closure of V in J is a finite flat A-module scheme over Z2(which is thus isomorphic to V).
Proof.
Since V 0 is a multiplicative type group scheme, inertia at 2 acts on V 0 (Q2 )through the cyclotomic character. It follows from Lemma 3.12 that V 0 is contained in
the generic fibre of J0 (2N )[2r ]0m/Z2 , 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 thatthe embedding of V 0 in J0 (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 thelemma.

Lemma 3.15 allows us to regard V as a closed T-submodule scheme of J , and thus toregard Vs as a closed T-submodule scheme of Js . Since ρ is irreducible and Φ is Eisenstein,
we T
see that Vs is in fact contained in Js0 . On the other hand, since T is a torus, we see thatVs T ⊂ Vs0 . Thus Vsét appears as a subquotient of Js0 /T , and in particular the T-action
on V ét factors through the quotient Told of T. Since Vsét is a faithful A-module scheme, we
see that the map φ : T → A factors through Told , 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 resultson finite flat models due to Raynaud [27].

4

N a proper prime power

There are only finitely many elliptic curves of conductor 2k for all k, and we may explicitly
determine which have odd modular degree. Therefore we assume that E has conductor N ,where N = pk with k ≥ 2 and p ≥ 3. Let χ be the unique quadratic character of conductor
p. Let E 0 be the elliptic curve E twisted by χ. The curve E 0 also has conductor N , andmoreover, the associated modular forms fE and fE 0 are congruent modulo 2, since twistingby quadratic characters preserves E[2]. Since N is odd, any non-trivial congruence modulo
2 between fE and other forms in S2 (Γ0 (N )) forces the modular degree mE to be even [38].
Thus we are done unless fE = fE 0 = fE ⊗ χ. It follows (for example, by [28]) that the
Galois representation associated to fE is induced from an imaginary quadratic field, andE has complex multiplication by this field. Alternatively, the equality fE = fE 0 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 = 1632 , 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
19conditions that E must satisfy in order to have odd modular degree. In this section we say alittle about the related question of whether or not 2 is a congruence prime for the associatedmodular form fE , when E satisfies either of conditions (3a) or (3b) of the theorem.
For curves E with a rational two torsion point, the modular form fE automaticallysatisfies a mod two congruence with an Eisenstein series, and so detecting whether fsatisfies a congruence with a cuspform is a more subtle phenomenon than in the nonEisenstein 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 Tm = Z2 for such representations. The specificdetermination of when Tm = Z2 , however, was already achieved (for N prime and ρ
reducible) by Merel in [25]:
5.1 Theorem. Let N ≡ 1 mod 8 be prime, and let Tm be the localization at the Eisensteinprime at 2. Then Tm 6= Z2 if and only if N = u2 + 16v 2 and v ≡ (N − 1)/8 mod 2.
If E is a Neumann–Setzer curve, then N = u2 + 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 6≡ 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 ρ is irreducible and supersingular, andthat 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(Q/Q) → GL2 (F2 [x]/(x2 )) lifting ρ. For representations ρ that are complex and ramified at 2, but ordinary, we constructed such a ρdirectly 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 beconstructed so directly.

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

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