On the Ramification of Hecke Algebras at Eisenstein Primes

Frank Calegari Matthew Emerton

January 7, 2005

1 Introduction

Fix a prime p, and a modular residual representation \overline{ρ} : G_{Q} → GL_{2}(\overline{F}_{p}). Suppose f is a normalized cuspidal Hecke eigenform of some level N and weight k that gives rise to \overline{ρ}, and let K_{f} denote the extension of Q_{p} generated by the q-expansion coefficients a_{n}(f) of f. The field K_{f} is a finite extension of Q_{p}. What can one say about the extension K_{f}/Q_{p}? Buzzard [1] has made the following conjecture: if N is fixed, and k is allowed to vary, then the degree [K_{f} : Q_{p}] is bounded independently of k.

Little progress has been made on this conjecture so far; indeed, very little seems to have been proven at all regarding the degrees [K_{f} : Q_{p}]. The goal of this paper is to consider a question somewhat orthogonal to that of Buzzard, namely, to fix the weight and vary the level. Moreover, we only consider certain reducible representations \overline{ρ} that arise in Mazur’s study of the Eisenstein Ideal [7]. Our results suggest that the degrees [K_{f} : Q_{p}] are, in fact, arithmetically significant.

Suppose that N ≥ 5 is prime, and that p is a prime which exactly divides the numerator of (N −1)/12. Mazur ([7], Prop. 9.6, p. 96 and Prop. 19.1, p. 140) has shown that there is a weight two normalized cuspidal Hecke eigenform defined over \overline{Q}_{p}, unique up to conjugation by G_{Q_{p}} (the Galois group of \overline{Q}_{p} over Q_{p}), satisfying the congruence

a_{ℓ}(f) ≡ 1 + ℓ mod 𝔭  (1)

(where 𝔭 is the maximal ideal in the ring of integers of K_{f}, and ℓ ranges over primes distinct from N). It follows moreover from [7] (Prop. 19.1, p. 140) that K_{f} is a totally ramified extension of Q_{p}, and thus that the degree [K_{f} : Q_{p}] is equal to the (absolute) ramification degree of K_{f}. Denote this ramification degree by e_{p}.

In this paper we prove the following theorem, in the case when p = 2.

Theorem 1.1 Suppose that p = 2 and that N ≡ 9 mod 16, and let f be a weight two eigenform on Γ_{0}(N) satisfying the congruence (1). If 2^{m} is the largest power of 2 dividing the class number of the field Q(\sqrt{−N}), then e_{2} = 2^{m−1} − 1.

When p is odd, we establish the following less definitive result.

Theorem 1.2 Suppose that p is an odd prime exactly dividing the numerator of (N−1)/12. Let f be a weight two eigenform on Γ_{0}(N) satisfying the congruence (1).

(i) Suppose that p = 3. (Our hypothesis on N thus becomes N ≡ 10 or 19 mod 27). Then e_{3} = 1 if and only if the 3-part of the class group of Q(\sqrt{−3}, N^{1/3}) is cyclic.

(ii) Suppose that p ≥ 5. (Our hypothesis on N thus becomes p∥N − 1). Then e_{p} = 1 if the p-part of the class group of Q(N^{1/p}) is cyclic.

The question of computing e_{p} has been addressed previously, in the paper [9] of Merel. In this work, Merel establishes a necessary and sufficient criterion for e_{p} = 1. Merel’s criterion for e_{p} = 1 is not expressed in terms of class groups; rather, it is expressed in terms of whether or not the congruence class modulo N of a certain explicit expression is a pth power.

When p = 2, Merel, using classical results from algebraic number theory, was able to reinterpret his explicit criterion for e_{2} = 1 so as to prove that e_{2} = 1 if and only if m = 2. (It is known that m ≥ 2 if and only if N ≡ 1 mod 8; see Proposition 4.1 below.) Theorem 1.1 strengthens this result, by relating the value of e_{2} in all cases to the order of the 2-part of the class group of Q(\sqrt{−N}).

When p is odd, Merel was not able to reinterpret his explicit criterion in algebraic number theoretic terms. However, combining Merel’s result with Theorem 1.2 (and the analogue of this theorem for more general primes N, i.e. those for which p divides N − 1, but not necessarily exactly) yields the following result.

Theorem 1.3 Let N ≥ 5 be prime.

(i) Let N ≡ 1 mod 9. The 3-part of the class group of Q(\sqrt{−3}, N^{1/3}) is cyclic if and only if (\frac{N−1}{3})! is not a cube modulo N. Equivalently, if we let N = ππ̄ denote the factorization of N in Q(\sqrt{−3}), then the 3-part of the class group of Q(N^{1/3}, \sqrt{−3}) is cyclic if and only if the 9th power residue symbol (\frac{π}{π̄} )_{9} is non-trivial. ^{1}

Furthermore, if these equivalent conditions hold, then the 3-part of the class group of Q(N^{1/3}) (which a fortiori is cyclic of order divisible by three) has order exactly three.

(ii) Let p ≥ 5, and let N ≡ 1 mod p. If the p-part of the class group of Q(N^{1/p}) is cyclic then

∏_{ℓ=1}^{(N−1)/2} ℓ^{ℓ}

is not a pth power modulo N.

1The claimed equivalence follows from the formula ((\frac{N−1}{3})!)^{3} ≡ π mod π̄, which was pointed out to us by Noam Elkies. René Schoof has told us that one can prove part (i) of Theorem 1.3 using class field theory. It is not apparent, however, that (ii) can be proved in this way.

The proofs of Theorems 1.1 and 1.2 depend on arguments using deformations of Galois representations. Briefly, if T denotes the completion at its p-Eisenstein ideal of the Hecke algebra acting on weight two modular forms on Γ_{0}(N), then we identify T with the universal deformation ring for a certain deformation problem. The theorems are then proved by an explicit analysis of this deformation problem over Artinian F_{p}-algebras.

It may be of independent interest to note that our identification of T as a universal deformation ring also allows us to recover all the results of Mazur proved in the reference [7] regarding the structure of T and the Eisenstein ideal: for example, that T is monogenic over Z_{p} (and hence Gorenstein); that the Eisenstein ideal is principal, and is generated by T_{ℓ} − (1 + ℓ) if and only if ℓ ≠ N is a good prime; and also that T_{N} = 1 in T.

Let us now give a more detailed explanation of our method. For the moment, we relax our condition on N, assuming simply that N and p are distinct primes. We begin by defining a continuous representation \overline{ρ} : G_{Q} → GL_{2}(F_{p}). If p is odd, we let

\overline{ρ} = (\begin{matrix}\overline{χ}_{p} & 0 \\ 0 & 1 \end{matrix}) ,

where \overline{χ}_{p} is the mod p reduction of the cyclotomic character. If p is even, we let

\overline{ρ} = (\begin{matrix}1 & ϕ \\ 0 & 1 \end{matrix}) ,

where ϕ : G_{Q} → F_{2} is the unique F_{2}-valued homomorphism inducing an isomorphism Gal(Q(\sqrt{−1})/Q) ≅ F_{2}.

Let \overline{V} denote the two dimensional vector space on which \overline{ρ} acts, and fix a line \overline{L} in \overline{V} that is not invariant under G_{Q} (equivalently, G_{Q_{p}}).

Fix once and for all a choice of inertia group I_{N} at N. If A is an Artinian local ring with residue field F_{p}, consider the set of triples (V, L, ρ), where V is a free A-module, L is a direct summand of V that is free of rank one over A, and ρ is a continuous homomorphism G_{Q} → GL(V), satisfying the following conditions:

Def1. The triple (V, L, ρ) is a deformation of (\overline{V} , \overline{L}, \overline{ρ}).

Def2. The representation ρ is unramified away from p and N, and is finite at p (i.e. V, regarded as a G_{Q_{p}}-module, arises as the generic fibre of a finite flat group scheme over Z_{p}).

Def3. The inertia subgroup at N acts trivially on the submodule L of V.

Def4. The determinant of ρ is equal to the composition of the cyclotomic character χ_{p} : G_{Q} → Z_{p}^{×} with the natural map Z_{p}^{×} → A^{×}.

If we let Def(A) denote the collection of such triples modulo strict equivalence ([8] §8, p. 257), then Def defines a deformation functor on the category of Artinian local rings A.

Note that the representation \overline{ρ} is reducible, and is either the direct sum of two characters (if p is odd) or an extension of the trivial character by itself (if p = 2). Nevertheless, one has the following result.

Proposition 1.4 The deformation functor Def is pro-representable by a complete Noetherian local Z_{p}-algebra R.

The proposition follows directly from that fact that the only endomorphisms of the triple (\overline{V} , \overline{L}, \overline{ρ}) are the scalars. (See for example [13], Prop. 1.2.) (The authors learned the idea of introducing a locally invariant line to rigidify an otherwise unrepresentable deformation problem from Mark Dickinson, who has applied it to analyse the deformation theory of residually irreducible representations that are ordinary, but not p-distinguished, locally at p.)

Having defined a universal deformation ring, we now introduce the corresponding Hecke algebra. As indicated above, we let T denote the completion at its p-Eisenstein ideal of the Z-algebra of Hecke operators acting on the space of all modular forms (i.e. the cuspforms together with the Eisenstein series) of level Γ_{0}(N) and weight two. (The p-Eisenstein ideal is the maximal ideal in the Hecke algebra generated by the elements T_{ℓ} − (1 + ℓ) (ℓ ≠ N), T_{N} − 1, and p).

The following result relates R and T.

Theorem 1.5 If ρ^{univ} denotes the universal deformation of \overline{ρ} over the universal deformation ring R, then there is an isomorphism of Z_{p}-algebras R ≅ T, uniquely determined by the requirement that the trace of Frobenius at ℓ under ρ^{univ} (for primes ℓ ≠ p, N) maps to the Hecke operator T_{ℓ} ∈ T.

Let us now return to the setting of Theorems 1.1 and 1.2. Thus we suppose again that p exactly divides the numerator of (N − 1)/12, and let f be as in the statements of the theorems. If O denotes the ring of integers in K_{f}, and 𝔭 its maximal ideal, then the results of [7] imply (taking into account the congruence satisfied by N) that the Hecke algebra T admits the following description:

T = {(a, b) ∈ Z_{p} × O | a mod p = b mod 𝔭}.

From this description of T, one easily computes that T/p is isomorphic to F_{p}[X]/X^{e_{p}+1}. Theorem 1.5 thus yields the following charaterization of e_{p}.

Corollary 1.6 The natural number e_{p} is the largest integer e for which we may find a triple (V, L, ρ) in Def(F_{p}[X]/X^{e+1}) such that the induced map R → F_{p}[X]/X^{e+1} is surjective.

Theorems 1.1 and 1.2 are a consequence of this corollary, together with an explicit analysis of the deformations of (\overline{V} , \overline{L}, \overline{ρ}) over Artinian local rings of the form F_{p}[X]/X^{n}.

If p^{2} divides the numerator of (N − 1)/12, then the residually Eisenstein cusp forms of level N need not be mutually conjugate. However, one still has an isomorphism of the form T/p = F_{p}[x]/x^{g_{p}+1}, where g_{p} + 1 denotes the rank of T over Z_{p}. (Thus g_{p} is the rank over Z_{p} of the cuspidal quotient of T.) In particular, the cuspidal Hecke algebra localized at the Eisenstein prime is isomorphic to Z_{p} if and only if g_{p} = 1. In this way our analysis of deformations over F_{p}[X]/X^{n} suffices to prove Theorem 1.3. More generally, our paper can be seen as providing a partial answer to Mazur’s question ([7], p. 140): “Is there anything general that can be said … about g_{p}?”.

The organization of the paper is as follows. In Section 2 we develop some results about group schemes that will be required in our study of the deformation functor Def. In Section 3 we prove Theorem 1.5, using the numerical criterion of Wiles [17] (subsequently strengthened by Lenstra [6]). As in [16], we use the class field theory of cyclotomic fields to obtain the required upper bound for the size of an appropriate Galois cohomology group; the numerical criterion is then established by comparing this upper bound with the congruence modulus of the weight two Eisenstein series on Γ_{0}(N) (which is known by [7] to equal the numerator of (N − 1)/12). Finally in Sections 4 (respectively 5) we perform the analysis necessary to deduce Theorem 1.1 (respectively 1.2 and 1.3) from Corollary 1.6.

Let us close this introduction by emphasising that the only result of [7] required for the proof of Theorem 1.5 is the computation of the congruence modulus between the Eisenstein and cuspidal locus in the Hecke algebra of weight two and level N. (Namely, that this congruence modulus is equal to the numerator of (N −1)/12.) As remarked upon above, we are then able to deduce all the results of [7] regarding T and its quotient T^{0} from Theorem 1.5. The necessary arguments are presented at the end of Section 3.

Acknowledgments. The authors would like to thank Brian Conrad for his close reading of an earlier version of this paper. His many remarks not only improved the exposition, but also saved us from a blunder or two. The authors are also grateful for the comments of the anonymous referee; these too were helpful in improving the exposition of the paper.

2 Some group scheme-theoretic calculations

Let us fix a prime p, and a natural number n. We begin with some generalities on finite flat group schemes. All group schemes to be considered here and below will be assumed commutative, whether or not this is explicitly noted.

For any scheme S we let Gr(S) denote the category of (commutative, in light of the convention signalled above) finite flat group schemes over the base S. Passing from an object of Gr(S) to the corresponding fppf sheaf that it represents embeds Gr(S) as a full additive subcategory of the abelian category of abelian sheaves on the fppf site of S. We let Sh(S) denote this latter category, and in this way we regard Gr(S) as a full subcategory of Sh(S). A key point is that Gr(S) is closed under extensions in Sh(S) (see Lemma 2.3) thus it is an exact category in the sense of Quillen [12].

We suppose from now on that S = Spec O with O a Dedekind domain whose field of fractions K is of characteristic zero. We let η denote Spec K, fix an algebraic closure \overline{K} of K, and write G_{K} := Gal(\overline{K}/K). Since K is of characteristic zero, passing to \overline{K}- valued points induces an equivalence between the category Gr(η) and the category of finite discrete G_{K}-modules (and we will freely identify an object of Gr(η) with the corresponding G_{K}-module). In particular, this category is abelian. Since any object of Gr(S) is equal to the scheme theoretic closure of its generic fibre, restriction from Gr(S) to Gr(η) is a faithful functor (although typically not fully faithful).

If M_{/K} is a G_{K}-module, we will refer to an object M of Gr(S) whose generic fibre is isomorphic to M_{/K} as a finite flat prolongation of M_{/K} over S. The collection of such prolongations form a category in an evident way (morphisms being morphisms in Gr(S) that restrict to the identity on the generic fibre). Note that there is at most one morphism between any two prolongations of M_{/K} (since restriction to the generic fibre is faithful). In particular, if M_{/K} admits a prolongation that is unique up to isomorphism, then it is unique up to unique isomorphism.

We now describe some simple but crucial aspects of the homological algebra of the exact category Gr(S).

Lemma 2.1 If S is a Dedekind scheme of generic characteristic zero, then the category Gr(S) admits kernels, cokernels, images, and coimages. Furthermore, the formation of each of these is compatible with passage to the generic fibre.

Proof. Let f : G → H be a morphism in the category Gr(S). Since for a Dedekind domain, flat coincides with torsion free, we see that the scheme-theoretic closure of the kernel of f_{/K} is a finite flat subgroup scheme of G, while the scheme-theoretic closure of the image of f_{/K} in H is a finite flat subgroup scheme of H. One checks that these are the kernel and image respectively of f in the category Gr(S). The quotient of G by the kernel of f is then a coimage of f in the category Gr(S), while the quotient of H by the image of f is then a cokernel of f in the category Gr(S). □

Note that the constructions of the preceding lemma typically do not coincide with the corresponding constructions in the larger category Sh(S). For example, take S = Spec Z, and let f : (Z/2)_{/Z} → (μ_{2})_{/Z} be the map that induces the identity on generic fibres. Then f has zero kernel, zero cokernel, coimage equal to (Z/2)_{/Z}, and image equal to (μ_{2})_{/Z}.

Suppose now that G is an object of Gr(S) with endomorphisms by a ring A, and suppose that M is a finitely presented right A-module. We may define the object M ⊗_{A} G of Gr(S) in the following way: Choose a presentation A^{r} ⟶^{ϕ} A^{s} ⟶ M → 0 of M, and define M ⊗_{A} G to be the cokernel of the induced map G^{r} ⟶^{ϕ⊗id_{G}} G^{s}.

Lemma 2.2 The object M ⊗_{A} G of Gr(S) is well-defined, up to natural isomorphism, independent of the choice of presentation of M.

Proof. We leave the easy proof to the reader. □

Since the formation of cokernels is compatible with passage to the generic fibre, we see that there is a natural isomorphism of G_{K}-modules (M ⊗_{A} G)_{/K} ≅ M ⊗_{A} (G_{/K}). We also record here the following lemma used implicitly throughout the rest of the text:

Lemma 2.3 The category Gr(S) is closed under extensions in Sh(S).

Proof. This follows from [10], Cor. 17.5, III.17-7.

The following result is useful for obtaining finite flat A-module schemes. The authors thank Brian Conrad for providing the proof.

Lemma 2.4 Suppose that M_{/K} is a finite discrete G_{K}-module that has a unique (up to isomorphism) finite flat prolongation M over S. If M_{/K} admits an A-module structure (compatible with its G_{K}-module structure), then this extends uniquely to an A-module scheme structure on M.

Proof. Since restriction to the generic fibre is a faithful functor on Gr(S), it suffices to show that for each a ∈ A, the corresponding endomorphism of M_{/K} extends to an endomorphism of M. These extensions will then necessarily be unique, and induce an A-module scheme structure on M.

If a ∈ A, let (Γ_{a})_{/K} ⊂ (M × M)_{/K} denote the graph of the endomorphism of M_{/K} induced by a. This is then a Galois submodule of (M × M)_{/K} that maps isomorphically to M_{/K} under the first projection. The Zariski closure of (Γ_{a})_{/K} in M × M is thus a closed finite flat subgroup scheme Γ_{a} of M ×M prolonging (Γ_{a})_{/K}. Our assumption that M_{/K} has a unique prolongation over S up to isomorphism implies that the first projection from Γ_{a} to M is again an isomorphism, and hence that Γ_{a} is (as the notation suggests) the graph of an endomorphism of M, which extends multiplication by a on M_{/K}. □

There is one more homological algebra result that we will need.

Proposition 2.5 Let V_{/K} be a finite discrete G_{K}-module, and suppose that V_{/K} has a unique (up to isomorphism) prolongation to an object V of Gr(S). If M_{/K} is any finite discrete G_{K}-module that admits a filtration by G_{K}-submodules whose subquotients are isomorphic to V_{/K}, and that admits a prolongation to a finite flat group scheme M over S, then this prolongation is unique up to isomorphism. (The above discussion shows that this isomorphism is then also necessarily unique.) Furthermore, any composition series of M_{/K} with successive quotients isomorphic to V_{/K} prolongs to a composition series for M consisting of closed finite flat subgroup schemes whose subquotients are isomorphic to V.

Proof. Let M and M′ be two choices of a finite flat group scheme over S prolonging M_{/K}. The results of [14] show that we may find a prolongation of M_{/K} that maps (in the category of such prolongations) to each of M and M′. Thus we may assume we are given a map M → M′ that induces the identity on generic fibres. By assumption we may find an embedding V_{/K} ⊂ M_{/K}. Passing to scheme theoretic closures in each of M and M′, and taking into our assumption on the uniqueness of V (up to isomorphism), this prolongs to an embedding of V into each of M and M′, so that the map M → M′ restricts to the identity map between these two copies of V. Replacing M_{/K} by M_{/K}/V_{/K}, M by M/V, and M′ by M′/V, and arguing by induction on the order of M, the proposition follows from the 5-lemma (applied, for example, in the category of sheaves on the fppf site over S). □

For any prime power p^{n}, we let Gr(p^{n}, S) denote the full, exact subcategory of Gr(S) consisting of finite flat group schemes of exponent p^{n}. We are primarily interested in finite flat group schemes of exponent p^{n} (for various p and n) that are extensions of Z/p^{n} by μ_{p^{n},}, and so we work in the categories Gr(p^{n}, S) from now on. We write Ext_{S,p^{n}}^{1}(–, –) to denote the first Yoneda Ext bifunctor on the exact category Gr(p^{n}, S). (Note in particular, then, that by stipulation, for any objects G and H of Gr(p^{n}, S), elements of Ext_{S,p^{n}}^{1}(G, H) correspond to extensions of the group scheme G by H that are of exponent p^{n}.) Our base scheme S will typically be either an open subset of Spec Z, or else Spec Z_{p}.

Lemma 2.6 The natural map Ext_{Z_{p},p^{n}}^{1}(Z/p^{n}, μ_{p^{n}}) → Ext_{Q_{p},p^{n}}^{1}(Z/p^{n}, μ_{p^{n}}), induced by restricting to the generic fibre, is injective.

Proof. Kummer theory identifies the map in the statement of the lemma with the obviously injective map Z_{p}^{×}/(Z_{p}^{×})^{p^{n}} → Q_{p}^{×}/(Q_{p}^{×})^{p^{n}}. □

If p = 2, we let V_{n}^{min} denote the extension of Z/2^{n} by μ_{2^{n}} in the category Gr(2^{n}, Q) corresponding by Kummer theory to the element −1 ∈ Q^{×}/(Q^{×})^{2^{n}}. If p is odd, we let V_{n}^{min} denote the direct sum Z/p^{n} ⨁ μ_{p^{n}} in the category Gr(p^{n}, Q). We may (and do) regard V_{n}^{min} as an object of the category of G_{Q}-modules annihilated by p^{n}.

More explicitly, let χ_{p} denote the p-adic cyclotomic character. Then if p = 2, the G_{Q}-module V_{n}^{min} corresponds to the representation

ρ_{n}^{min} : G_{Q} → GL_{2}(Z/2^{n})

given by

σ ↦ (\begin{matrix}χ_{2}(σ) & (χ_{2}(σ) − 1)/2 \\ 0 & 1 \end{matrix}) mod 2^{n},

whilst if p is odd, the G_{Q}-module V_{n}^{min} corresponds to the representation

ρ_{n}^{min} : G_{Q} → GL_{2}(Z/p^{n})

given by

σ ↦ (\begin{matrix}χ_{p}(σ) & 0 \\ 0 & 1 \end{matrix}) mod p^{n}.

(Here we have denoted by σ an element of G_{Q}.)

Proposition 2.7 For any natural number M, the G_{Q}-module V_{n}^{min} has a unique prolongation to an object of Gr(p^{n}, Z[1/M]).

Proof. The Galois module V_{n}^{min} is unramified away from p, and so V_{n}^{min} has a unique prolongation to a finite étale group scheme over Z[1/Mp]. It thus suffices to show that V_{n}^{min}, regarded as a G_{Q_{p}}-module, has a unique prolongation to an object of Gr(p^{n}, Z_{p}). If p is odd, then this is a direct consequence of [2], Thm. 2. Thus we assume for the remainder of the proof that p = 2. In this case, V_{n}^{min} is defined to be the extension of Z/2^{n} by μ_{2^{n}} corresponding to −1 ∈ Q_{2}^{×}. Since −1 in fact lies in Z_{2}^{×}, V_{n}^{min} does prolong to a finite flat group scheme over Z_{2}. We must show that this prolongation is unique.

We begin with the case n = 1. Suppose that G is a finite flat group scheme over Z_{2} having (V_{1}^{min})_{/Q_{2}} as its associated Galois representation. The scheme-theoretic closure of the fixed line in V_{1}^{min} yields an order two finite flat subgroup scheme H of G. Both H and G/H are thus finite flat group schemes of order two. The results of [11] show that Z/2 and μ_{2} are the only group schemes of order 2 over Z_{2}. Thus G is an extension of either Z/2 or μ_{2} by either Z/2 or μ_{2}. Since neither G nor its Cartier dual are unramified (since V_{1}^{min} is self-dual and ramified at 2), we see that both Z/2 and μ_{2} must appear. Since V_{1}^{min} is a non-trivial G_{Q_{2}}-module, a consideration of the connected-étale exact sequence attached to G shows that in fact G is an extension of Z/2 by μ_{2}. The fact that G is determined uniquely by V_{1}^{min} now follows from Lemma 2.6. The uniqueness in the case of arbitrary n follows from the result of the preceding paragraph, together with Proposition 2.5. □

Lemma 2.8 Let D_{n} denote the (uniquely determined, by Proposition 2.7) prolongation of V_{n}^{min} to an object of Gr(p^{n}, Z). We have Ext_{Z,p^{n}}^{1}(Z/p^{n}, D_{n}) = 0.

Proof. Writing D_{n} as an extension of Z/p^{n} by μ_{p^{n}}, we obtain the exact sequence of Yoneda Ext groups

Ext_{Z,p^{n}}^{1}(Z/p^{n}, μ_{p^{n}}) → Ext_{Z,p^{n}}^{1}(Z/p^{n}, D_{n}) → Ext_{Z,p^{n}}^{1}(Z/p^{n}, Z/p^{n}).

The third of these groups always vanishes, since Z has no non-trivial finite étale covers. The first group is isomorphic to Z^{×}/Z^{×p}, so if p is odd it vanishes, and if p = 2 it has order two, with the non-trivial element corresponding by Kummer theory to −1 ∈ Z^{×}. Since D_{n} is itself classified by this same element when p = 2, we see that the first arrow vanishes in all cases, and thus so does the middle group. □

Corollary 2.9 Suppose that A is an Artinian local ring with maximal ideal 𝔭 and residue field F_{p}, that V is a free A-module of rank two, and that ρ : G_{Q_{p}} → GL(V) is a deformation of (V_{1}^{min})_{/Q_{p}} that is finite flat at p. Then there is a unique up to unique isomorphism finite flat group scheme M over Z_{p} whose generic fibre equals V. Furthermore, the A-action on V prolongs to an A-action on M, the connected-étale sequence of M realizes M as the extension of an étale finite flat A-module scheme of A-rank one by a multiplicative finite flat A-module scheme of A-rank one, and M admits a filtration by closed finite flat sub-A-module schemes with successive quotients isomorphic to D_{1}.

Proof. If we choose a Jordan-Hölder filtration of A as a module over itself, then this induces a filtration of V with successive quotients isomorphic to V_{1}^{min}. Thus we are in the situation of Proposition 2.5, and the uniqueness of M follows, as does the existence of the required filtration of M. (Note that this uniqueness result, together with Lemma 2.4, shows that the A-actions on V and on each of the steps of its filtration extend respectively to an A-action on M and on the steps of its filtration.) Finally, let 0 → M^{0} → M → M^{ét} → 0 be the connected-étale sequence of M. The functorial nature of its construction implies that it is an exact sequence of closed finite flat A-submodule schemes of M. Thus the exact sequence 0 → M_{/Q_{p}}^{0} → V → M_{/Q_{p}}^{ét} → 0 obtained by restricting to Q_{p} yields a two-step filtration of V by A-submodules. The formation of this filtration is clearly functorial in A. Thus if we tensor M with A/𝔭 over A, and take into account that A/𝔭 ⊗_{A} M = D_{1}, we find that A/𝔭 ⊗_{A} M^{0} = D_{1}^{0} = μ_{p} and that A/𝔭 ⊗_{A} M^{ét} = D_{1}^{ét} = Z/p. Thus each of M_{/Q_{p}}^{0} and M_{/Q_{p}}^{ét} are cyclic A-modules. Since M_{/Q_{p}} is free of rank two over A, they must both be free A-modules of rank one, as claimed. We also see that M^{0} is multiplicative, as claimed. □

Proposition 2.10 Let A be an Artinian local Z/p^{n}-algebra with maximal ideal 𝔭 and residue field F_{p}. If V is a free A-module of rank two and ρ : G_{Q_{p}} → GL(V) is a deformation of (V_{1}^{min})_{/Q_{p}} that is finite at p, then the coinvariants of V with respect to the inertia group I_{p} are free of rank one over A.

Proof. The preceding corollary shows that V admits a two-step filtration, with rank one free quotients, corresponding to the connected-étale sequence of the prolongation of V to a group scheme over Z_{p}. In particular, the inertial coinvariants V_{I_{p}} admit a surjection onto a free A-module of rank one. On the other hand, if 𝔭 is the maximal ideal of A, then (V_{I_{p}})/𝔭 = (V/𝔭)_{I_{p}} = (V_{1}^{min})_{I_{p}}. This latter space is directly checked to be one dimensional over F_{p}, implying that V_{I_{p}} is a cyclic A-module. Altogether, we find that V_{I_{p}} is free of rank one over A, as claimed. □

Proposition 2.11 Let A be an Artinian local Z/p^{n}-algebra with maximal ideal 𝔭 and residue field F_{p}. If V is a free A-module of rank two and ρ : G_{Q} → GL(V) is a deformation of V_{1}^{min} that is unramified away from p and finite at p, then there is an A-linear isomorphism of G_{K}-modules V ≅ A ⊗_{Z/p^{n}} V_{n}^{min}. (Note that we ignore for the moment the question of whether this isomorphism can be taken to be an isomorphism of deformations of V_{1}^{min}.)

Proof. Corollary 2.9 shows that V prolongs to a finite flat A-module scheme M over Spec Z, which admits a filtration by closed finite flat subgroup schemes, with successive quotients isomorphic to D_{1}.

If p is odd, then Proposition I.4.5 of [7] shows that M is the product of a constant finite flat closed subgroup scheme and a μ-type finite flat closed subgroup scheme. (Recall that a finite flat group scheme is said to be of μ-type if it is Cartier dual to a constant group scheme). Furthermore, these subgroup schemes are unique (and hence this direct product decomposition of M is unique), since when p is odd, there are non-zero morphisms from a constant étale group scheme to a μ-type group scheme (over Q, and hence over Z).

Each of these subgroups is thus an A-submodule scheme of M, and we easily conclude that V ≅ A ⊗_{Z/p^{n}} V_{n}^{min}.

If p = 2, then Propositions I.2.1 and I.3.1 of [7] show that M is the extension of a constant group scheme by a μ-type group. Again, each of these groups is seen to be an A-module scheme, and we easily conclude that M is in fact an extension of the constant A-module scheme A by the μ-type A-module scheme A ⊗_{Z/2^{n}} μ_{2^{n}}. The group of all such extensions is classified by

H^{1}(Spec Z, A ⊗_{Z/2^{n}} μ_{2^{n}}) ≅ A ⊗_{Z/2^{n}} Z^{×}/(Z^{×})^{2^{n}} ≅ A/2 ⊗_{F_{2}} {±1}.

We thus see that this cohomology group is free of rank one over A/2. Since V_{1}^{min} corresponds by Kummer theory to the non-trivial element of {±1}, we see that the elements of this cohomology group corresponding to deformations of V_{1}^{min} form a principal homogeneous space under (A/2)^{×}. The action of A^{×} on H^{1}(Spec Z, A ⊗_{Z/2^{n}} μ_{2^{n}}) corresponds simply to “changing the basis” of A ⊗_{Z/2^{n}} μ_{2^{n}}. Thus it does not change the isomorphism class of the finite flat group scheme underlying a given extension, and so we see that there is is a unique finite flat group scheme over Z that deforms V_{1}^{min} over A (which must then be A ⊗_{Z/2^{n}} V_{n}^{min}). □

We leave it to the reader to verify the following lemma.

Lemma 2.12 If A is an Artinian local Z/p^{n}-algebra with maximal ideal 𝔭, then the ring of Galois equivariant endomorphisms of A ⊗_{Z/p^{n}} V^{min} admits the following description:

(i) If p = 2, then End_{A[G_{Q}]}(A ⊗_{Z/2^{n}} V_{n}^{min}) = \{(\begin{matrix}a & b \\ 0 & a − 2b \end{matrix}) | a, b ∈ A\}.

(ii) If p is odd, then End_{A[G_{Q}]}(A ⊗_{Z/p^{n}} V_{n}^{min}) = \{(\begin{matrix}a & 0 \\ 0 & d \end{matrix}) | a, d ∈ A\}.

3 Proving that R = T

We let Def denote the deformation problem described in the introduction. We begin by describing some equivalent formulations of condition 4 in the definition of Def.

Lemma 3.1 If A is an Artinian local ring with residue field F_{p}, and if (V, L, ρ) is a triple satisfying conditions Def1, Def2, and Def3 in the definition of Def, then the following conditions are equivalent:

Def4. The determinant of ρ is equal to the cyclotomic character χ_{p}.

Def4a. The determinant of ρ is unramified away from p.

Def4b. The inertia subgroup at N acts trivially on the quotient V/L.

Proof. It is obvious that Def4 implies Def4a and Def4b. By assumption ρ is unramified away from p and N, and I_{N} (inertia at N) acts trivially on L, from which it also follows that Def4a and Def4b are equivalent. If Def4a holds, then (by the Kronecker-Weber theorem) the determinant of ρ is determined by its action on inertia at p. Corollary 2.9 shows that the finite flat group scheme over Z_{p} that prolongs V is the extension of an étale A-module scheme of rank one over A by a multiplicative A-module scheme of rank one over A. Consequently, the determinant of ρ, restricted to inertia at p, is equal to χ_{p}, and so Def4a implies Def4. □

Our proof of Theorem 1.5 employs the technique introduced in [17]: namely, we first consider a minimal deformation ρ^{min} of (\overline{V} , \overline{L}, \overline{ρ}) over Z_{p}, and then verify the numerical criterion of [17].

Let us define the minimal deformation problem Def^{min}, as the subfunctor of Def consisting of those deformations of (\overline{V} , \overline{L}, \overline{ρ}) that are unramified away from p. Let us also define (Def^{min})′ to be the functor that classifies all deformations of (\overline{V} , \overline{ρ}) that are unramified away from p and finite at p. Forgetting the I_{N}-fixed line L gives a natural transformation Def^{min} → (Def^{min})′.

Let us now define the triple (V^{min}, L^{min}, ρ^{min}). We take V^{min} = Z_{p} ⨁ Z_{p}. If p = 2, then we let ρ^{min} denote the representation

σ ↦ (\begin{matrix}χ_{2}(σ) & (χ_{2}(σ) − 1)/2 \\ 0 & 1 \end{matrix})

(here σ denotes an element of G_{Q}), while if p is odd, we let ρ^{min} denote the direct sum of χ_{p} (the p-adic cyclotomic character) and 1 (the trivial character). In each case, the pair (V^{min}, ρ^{min}) is certainly a lifting of (\overline{V} , \overline{ρ}). We take L^{min} to be any free of rank one Z_{p}-submodule of V^{min} lifting the line \overline{L} in \overline{V}.

Note that for any natural number n, we have V^{min}/p^{n} = V_{n}^{min} (the Galois module introduced in the preceding section).

Proposition 3.2 The natural transformation Def^{min} → (Def^{min})′ is an isomorphism of functors. Moreover, the deformation functor Def^{min} is pro-represented by (V^{min}, L^{min}, ρ^{min}) in Def^{min}(Z_{p}).

Proof. Let A be an Artinian local Z_{p}-algebra, and let (V, ρ) be an object of (Def^{min})′(A). Proposition 2.11 shows that there is an isomorphism V ≅ A ⊗_{Z_{p}} V^{min}. The explicit description of the endomorphisms of A ⊗_{Z_{p}} V^{min} provided by Lemma 2.12 shows that we may furthermore choose this isomorphism so that it is strict. Thus we see that (Def^{min})′ is pro-represented by Z_{p}, with (V^{min}, ρ^{min}) as universal object.

Now suppose that (V, L, ρ) is an object of Def^{min}(A). Using Lemma 2.12 again, we see that we may choose the strict endomorphism V ≅ A ⊗_{Z_{p}} V^{min} of the preceding paragraph in such a way that L is identified with A ⊗_{Z_{p}} L^{min}. Thus Z_{p} also pro-represents Def^{min}, with universal object (V^{min}, L^{min}, ρ^{min}). This establishes the proposition. □

Note that the preceding lemma implies in particular that the class of (V^{min}, L^{min}, ρ^{min}) in Def(Z_{p}) is independent of the choice of L^{min} (provided that it lifts \overline{L}).

Let R denote the universal deformation ring that pro-represents the functor Def, and let (V^{univ}, L^{univ}, ρ^{univ}) denote the universal deformation of (\overline{V} , \overline{L}, \overline{ρ}) over R. Corresponding to (V^{min}, L^{min}, ρ^{min}) there is a homomorphism R → Z_{p} of Z_{p}-algebras. We let I denote the kernel of this homomorphism. The following more explicit description of I will be useful.

Proposition 3.3 If S is any finite set of primes containing p and N, then I is generated by the set

{1 + ℓ − Trace(ρ^{univ}(Frob_{ℓ})) | ℓ ∉ S}.

Proof. Let I_{S} denote the ideal generated by the stated set. Clearly I_{S} ⊂ I. We will show that the Galois representation G_{Q} → GL_{2}(R/I_{S}) obtained by reducing ρ^{univ} modulo I_{S} is unramified at N. It will follow from Proposition 3.2 that I ⊂ I_{S}, and the proposition will be proved. The argument is a variation of that used to prove Prop. 2.1 of [16].

Suppose first that p is odd. Let us choose a basis for V^{univ}, and write

ρ^{univ}(σ) = (\begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix}) ,

for σ ∈ G_{Q}. We may assume that if c ∈ G_{Q} denotes complex conjugation, then

ρ^{univ}(c) = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) .

We find that

a(σ) = \frac{1}{2} (Trace(ρ^{univ}(σ)) − Trace(ρ^{univ}(cσ))) ,

and that

d(σ) = \frac{1}{2} (Trace(ρ^{univ}(σ)) + Trace(ρ^{univ}(cσ))) .

Since by construction Trace(ρ^{univ}(σ)) ≡ 1 + χ_{p}(σ) mod I_{S}, we find that

a(σ) ≡ χ_{p}(σ) mod I_{S},

whilst

d(σ) ≡ 1 mod I_{S}.

In particular, if σ is an element of the inertia group I_{N}, then

ρ^{univ}(σ) ≡ (\begin{matrix}1 & b(σ) \\ c(σ) & 1 \end{matrix}) mod I_{S}.

The universal I_{N}-fixed line is spanned by a vector of the form (1, x), where x ∈ R^{×}. We conclude that if σ ∈ I_{N} then

(1 + b(σ)x, c(σ) + x) ≡ (1, x) mod I_{S},

and thus that

b(σ) ≡ c(σ) ≡ 0 mod I_{S}.

This implies that ρ^{univ} mod I_{S} is unramified at N, as required.

Consider now the case p = 2. Again, we write

ρ^{univ}(σ) = (\begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix}) ,

for σ ∈ G_{Q}. We may assume that if c ∈ G_{Q} denotes complex conjugation, then

ρ^{univ}(c) = (\begin{matrix}−1 & −1 \\ 0 & 1 \end{matrix}) .

We may also assume that the universal I_{N}-fixed line is spanned by the vector (0, 1). By considering Trace(ρ^{univ}(cσ)), for σ ∈ G_{Q}, we find that

−a(σ) − c(σ) + d(σ) ≡ 1 − χ_{2}(σ) mod I_{S}.

If σ ∈ I_{N}, then since σ fixes (0, 1), we find that

b(σ) = 0, d(σ) = 1.

The preceding equations, the fact that det ρ^{univ} = χ_{2}, and the fact that χ_{2}(σ) = 1 for σ ∈ I_{N}, imply that also

a(σ) ≡ 1, c(σ) ≡ 0 mod I_{S}.

Altogether, we conclude that ρ^{univ} mod I_{S} is unramified at N, as required. □

The preceding result has the following important corollary.

Corollary 3.4 If S is any finite set of primes containing p and N, then the complete local Z_{p}-algebra R is topologically generated by the elements Trace(ρ^{univ}(Frob_{ℓ})), for ℓ ∉ S.

Proof. This follows immediately from the description of I provided by Proposition 3.3, the fact that R is I-adically complete, and the fact that R/I ≅ Z_{p}. □

We now compute the order of I/I^{2}, which is one of the two ingredients we will eventually use in our verification of the Wiles-Lenstra numerical criterion.

Theorem 3.5 The order of I/I^{2} (which is a power of p) divides (N^{2} − 1)/24.

Proof. As usual, the first step of the argument involves identifying the Pontrjagin dual of I/I^{2} with a certain (inductive limit of) Ext groups. We begin by describing the relevant extensions.

Let n be a natural number, and let (V_{n}^{min}, L_{n}^{min}, ρ_{n}^{min}) denote the reduction modulo p^{n} of (V^{min}, L^{min}, ρ^{min}). We consider extensions of Galois modules

0 → (V_{n}^{min}, L_{n}^{min}) → (E, F) → (V_{n}^{min}, L_{n}^{min}) → 0;

here the notation indicates that E is a Z/p^{n}[G_{Q}]-module that extends V_{n}^{min} by itself, and that F is a submodule of E (not assumed to be Galois invariant) that provides an extension of L_{n}^{min} by itself.

We let A_{n} denote the additive category of such extensions for which E is annihilated by p^{n}, is unramified away from p and N, is finite at p, and for which both F and E/F are fixed (element-wise) by the inertia group I_{N}. Morphisms between two objects (E_{1}, F_{1}) and (E_{2}, F_{2}) of the category A_{n} are given by isomorphisms of Galois modules E_{1} → E_{2} that take F_{1} to F_{2}, and that make the diagram

\xymatrix{0 \ar[r] & V_{n}^{min} \ar[r] \ar[d]^{_{id_{V{min \atop n}}}} & E_{1} \ar[r] \ar[d] & V_{n}^{min} \ar[r] \ar[d]^{_{id_{V{min \atop n}}}} & 0 \\ 0 \ar[r] & V_{n}^{min} \ar[r] & E_{2} \ar[r] & V_{n}^{min} \ar[r] & 0}

commute. The direct sum is given by the usual Baer sum of extensions.

We let A_{n} denote the Grothendieck group of isomorphism classes of objects of A_{n}. There are natural Galois equivariant maps (V_{n+1}^{min}, L_{n+1}^{min}) → (V_{n}^{min}, L_{n}^{min}) and (V_{n}^{min}, L_{n}^{min}) → (V_{n+1}^{min}, L_{n+1}^{min}), the first being given by reduction modulo p^{n}, and the second by regarding the sources as the p^{n}-torsion in the target. Pulling back by the first of these maps, and pushing forward by the second, we obtain a functor F_{n} : A_{n} → A_{n+1}. Thus F_{n} induces a homomorphism A_{n} → A_{n+1} of Grothendieck groups. The usual identification of the relative tangent space to a deformation functor with an appropriate Ext-group in an appropriate category of Galois modules shows that

Hom(I/I^{2}, Q_{p}/Z_{p}) ≅ lim_{⟶} A_{n}.  (2)

(Note that we are using the equivalence of conditions Def4 and Def4b in the definition of Def provided by Lemma 3.1.)

We will prove the lemma by showing that the right hand side of this isomorphism has order dividing (N^{2} − 1)/24. In fact, we will not work directly with the rather complicated Ext groups A_{n}. Rather, we fill construct an injection of inductive systems {A_{n}}_{n≥1} → {B_{n}}_{n≥1}, with each B_{n} being a simpler Ext group, and investigate the limit lim_{⟶} B_{n} instead.

If (E, F) is an object of A_{n}, then since Z/p^{n} (with the trivial G_{Q}-action) is a quotient of V_{n}^{min}, whilst μ_{p^{n}} (with its natural G_{Q}-action) is a submodule of V_{n}^{min}, the extension E determines an extension E′ of G_{Q} modules

0 → Z/p^{n} → E′ → μ_{p^{n}} → 0.  (3)

Let B_{n} denote the group of isomorphism classes of extensions of G_{Q}-modules of the form (3) that are unramified away from p and N, and that prolong over Z_{p} to an extension of the finite flat group scheme μ_{p^{n}} by the finite flat group scheme Z/p^{n}. The natural maps μ_{p^{n+1}} → μ_{p^{n}} and Z/p^{n} → Z/p^{n+1}, given respectively by raising to the pth power and by multiplication by p, induce a map B_{n} → B_{n+1}. Thus the B_{n} form an inductive system. The passage from E to E′ gives rise to a homomorphism of inductive sequences

{A_{n}}_{n≥1} → {B_{n}}_{n≥1},  (4)

which we will next show is injective.

Lemma 3.6 If (E, F) is an object of A_{n} for which E is a trivial extension, then the pair (E, F) is also a trivial extension.

Proof. Let us remind the reader that if E is the trivial extension of V_{n}^{min} by itself, then

the automorphisms of E (as an object of A_{n}) are of the form (\begin{matrix}Id & A \\ 0 & Id \end{matrix}) , where A is an

element of End_{G_{Q}}(V_{n}^{min}). This being said, the lemma is easily checked using Lemma 2.12.

Alternatively, we may appeal to Proposition 3.2. Since E is assumed to be a trivial extension, it is in particular unramified at N, and thus corresponds to a deformation for the subproblem Def^{min} of Def. The triviality of E implies that this deformation is trivial, when regarded as an deformation for the problem (Def^{min})′. Since Def^{min} maps isomorphically to (Def^{min})′, we obtain the assertion of the lemma. □

Lemma 3.7 If (E, F) is an object of A_{n} for which the corresponding extension E′ in B_{n} is trivial, then E is also a trivial extension.

Proof. We begin by pointing out the category A_{n} has a natural involution, given by passing to Cartier duals. Indeed, since V_{n}^{min} is Cartier self-dual by construction, if (E, F) is an object of A_{n}, then the Cartier dual E^{∗} is itself an extension of V_{n}^{min} by itself in a natural way. We define F^{∗} to be the annihilator of F in E^{∗}; our assumptions then make it clear that (E^{∗}, F^{∗}) is again an object of A_{n}. (Note that V_{n}^{min} is identified with its Cartier dual (V_{n}^{min})^{∗} via the alternating pairing ∧ : V_{n}^{min} ×V_{n}^{min} → det(V_{n}^{min}) ≅ χ_{p} mod p^{n}, which implies that (L_{n}^{min})^{∗} = L_{n}^{min}.) It is clear that the extension (E^{∗})′ of μ_{p^{n}} by Z/p^{n} arising from E^{∗} is obtained by taking the Cartier dual of the extension E′ arising from E. Thus if E′ is trivial, so is (E^{∗})′.

We now prove the lemma. Let D_{n} denote the (unique, by Proposition 2.7) prolongation of V_{n}^{min} to a finite flat group scheme over Z[1/N]. Proposition 2.5 shows that E has a unique prolongation to a finite flat group scheme E over Z[1/N], that provides an extension of D_{n} by itself. We let D_{n}^{(1)} denote the copy of D_{n} that appears as a submodule of E, and let D_{n}^{(2)} denote the copy of D_{n} that appears as a quotient. Also, we let μ_{p^{n}}^{(i)} (respectively (Z/p^{n})^{(i)}) denote the copy of μ_{p^{n}} (respectively Z/p^{n}) that appears as a subgroup scheme (respectively a quotient group scheme) of D_{n}^{(i)}, for i = 1, 2. The extension (3) corresponding to E thus prolongs to an extension E′ of μ_{p^{n}} by Z/p^{n} as finite flat groups schemes over Z[1/N].

We begin by observing that our hypothesis that E′ is a trivial extension implies that E′ is also a trivial extension. Indeed, since E′ is étale over Z[1/Np], the splitting of the extension of Galois modules E′ implies the splitting of the corresponding extension of group schemes E′ over Z[1/Np]. Also, a consideration of the connected-étale sequence shows that E′_{|Z_{p}} is a split extension (for any E′!). Thus E′ splits over Z[1/N], as claimed.

The quotient E/μ_{p^{n}}^{(1)} is an extension of D_{n}^{(2)} by (Z/p^{n})^{(1)}. Thus it yields a class e ∈ Ext_{Z[1/N],p^{n}}(D_{n}^{(2)}, (Z/p^{n})^{(1)}). This latter group sits in the exact sequence

Ext_{Z[1/N],p^{n}}((Z/p^{n})^{(2)}, (Z/p^{n})^{(1)}) → Ext_{Z[1/N],p^{n}}(D_{n}^{(2)}, (Z/p^{n})^{(1)})
→ Ext_{Z[1/N],p^{n}}(μ_{p^{n}}^{(2)}, (Z/p^{n})^{(1)}).

(5)

The image of e under the second arrow of (5) classifies the extension E′, and thus vanishes. Thus there is a class e′ ∈ Ext_{Z[1/N],p^{n}}((Z/p^{n})^{(2)}, (Z/p^{n})^{(1)}) that maps to e under the first arrow of (5). We can construct such an extension class e′ concretely as follows: Since the image of e vanishes, we may choose a lift of μ_{p^{n}}^{(2)} to a subgroup scheme μ of E/μ_{p^{n}}^{(1)}. The quotient (E/μ_{p^{n}}^{(1)})/μ is then an extension of (Z/p^{n})^{(2)} by (Z/p^{n})^{(1)}, which gives a realization of a class e′ mapping to e. Our assumption on the submodule F of E implies that it maps surjectively onto (E/μ_{p^{n}}^{(1)})/μ (the generic fibre of (E/μ_{p^{n}}^{(1)})/μ), and thus that the action of inertia at N on (E/μ_{p^{n}}^{(1)})/μ is trivial. Thus (E/μ_{p^{n}}^{(1)})/μ has a prolongation to a finite flat group scheme over Z, yielding an extension of Z/p^{n} by itself. There are no such non-trivial extensions that are finite flat over Z, and thus the extension class e′ is trivial. Hence the extension class e is also trivial, and so E/μ_{p^{n}}^{(1)} is a split extension of D_{n}^{(2)} by (Z/p^{n})^{(1)}.

If Ẽ denotes the preimage in E of the subgroup μ_{p^{n}}^{(2)} ⊂ D_{n}^{(2)}, then we find that Ẽ is Cartier dual to E^{∗}/μ_{p^{n}}^{(1)}. (Where we are momentarily applying the notation μ_{p^{n}}^{(1)} not to E, but to its Cartier dual E^{∗}.) The observations at the beginning of the argument show that the reasoning of the preceding paragraph applies equally well to E^{∗}, and thus that E^{∗}/μ_{p^{n}}^{(1)} is a split extension of D_{n}^{(2)} by (Z/p^{n})^{(1)}. Consequently, passing back from E^{∗} to E, we conclude that Ẽ is a split extension of μ_{p^{n}}^{(2)} by D_{n}^{(1)}. Consider the exact sequence

Ext_{Z[1/N],p^{n}}((Z/p^{n})^{(2)}, D_{n}^{(1)}) → Ext_{Z[1/N],p^{n}}(D_{n}^{(2)}, D_{n}^{(1)})
→ Ext_{Z[1/N],p^{n}}(μ_{p^{n}}^{(2)}, D_{n}^{(1)})

(6)

If e′′ denotes the class of E in the middle group, then we have just seen that its image under the second arrow of (6) vanishes. Thus we may find a class e′′′ ∈ Ext_{Z[1/N],p^{n}}((Z/p^{n})^{(2)}, D_{n}^{(1)}) mapping to e′′ under the first arrow of (6). We can construct such a class e′′′ concretely as follows: Since the image of e′′ vanishes, we may lift μ_{p^{n}}^{(2)} to a subgroup scheme μ′ of Ẽ. The quotient E/μ′ then provides an extension of (Z/p^{n})^{(2)} by D_{n}^{(1)} whose extension class e′′′ maps to e′′. Our assumption on F implies that its image in E/μ′ (the generic fibre of E/μ′) projects isomorphically onto (Z/p^{n})^{(2)}, and thus that inertia at N acts trivially on E/μ′, and so E/μ′ has a prolongation to a finite flat group scheme over Z that extends Z/p^{n} by D_{n}. Lemma 2.8 shows that any such extension is split, and thus that e′′′ vanishes. Consequently e′′ also vanishes, and so E is a split extension, as claimed. □

The preceding two results show that the map (4) is injective for each n, as claimed. Passing to the direct limit in n yields an injective map

lim_{⟶} A_{n} → lim_{⟶} B_{n}.  (7)

Lemma 3.8 The group lim_{⟶} B_{n} is finite, of order at most the p-power part of (N^{2} − 1)/24.

Proof. It suffices to show that B_{n} is of order at most the p-power part of (N^{2} − 1)/24 for sufficiently large values of n. Let Σ = {p, N, ∞}, and let G_{Σ} denote the Galois group of the maximal extension of Q in \overline{Q} unramified away from the elements of Σ. Extensions of the form (3) are classified by the Galois cohomology group H^{1}(G_{Σ}, μ_{p^{n}}^{⊗−1}). If such an extension prolongs to an extension of finite flat groups over Z_{p}, then it is in fact trivial locally at p, since the connected group scheme μ_{p^{n}} cannot have a non-trivial extension over the étale group scheme Z/p^{n}. Thus B_{n} is equal to the kernel of the natural map

H^{1}(G_{Σ}, μ_{p^{n}}^{⊗−1}) → H^{1}(G_{Q_{p}}, μ_{p^{n}}^{⊗−1}).

Let K_{n} denote the extension of Q obtained by adjoining all p^{n}th roots of unity in \overline{Q}. Let H denote the normal subgroup of G_{Σ} which fixes K_{n}; the quotient G_{Σ}/H is naturally isomorphic to (Z/p^{n})^{×}. The prime p is totally ramified in K_{n}. Thus, if π denotes the unique prime of K_{n} lying over p, the quotient G_{Q_{p}}/G_{K_{n,π}} also maps isomorphically to (Z/p^{n})^{×}. The inflation-restriction exact sequence gives a diagram

[figure: 0; H^{1}((Z/p^{n})^{×}, μ_{p^{n}}^{⊗−1}); H^{1}(G_{Σ}, μ_{p^{n}}^{⊗−1}); H^{1}(H, μ_{p^{n}}^{⊗−1})^{(Z/p^{n})^{×}}; 0; H^{1}((Z/p^{n})^{×}, μ_{p^{n}}^{⊗−1}); H^{1}(G_{Q_{p}}, μ_{p^{n}}^{⊗−1}); H^{1}(G_{K_{n,π}}, μ_{p^{n}}^{⊗−1})^{(Z/p^{n})^{×}}.]

Taking into account the discussion of the preceding paragraph, this diagram in turn induces an injection

B_{n} ↪ ker (H^{1}(H, μ_{p^{n}}^{⊗−1})^{(Z/p^{n})^{×}} → H^{1}(G_{K_{n,π}}, μ_{p^{n}}^{⊗−1})^{(Z/p^{n})^{×}}).

Since H acts trivially on μ_{p^{n}}^{⊗−1}, there is an isomorphism

H^{1}(H, μ_{p^{n}}^{⊗−1})^{(Z/p^{n})^{×}} ≅ Hom_{(Z/p^{n})^{×}}(H, μ_{p^{n}}^{⊗−1}).

Thus B_{n} injects into the subgroup of Hom_{(Z/p^{n})^{×}}(H, μ_{p^{n}}^{⊗−1}) consisting of homomorphisms that are trivial on G_{K_{n,π}}.

Any element of Hom_{(Z/p^{n})^{×}}(H, μ_{p^{n}}^{⊗−1}) that is trivial on G_{K_{n,π}} factors through the Galois group Gal(L_{n}/K_{n}), where L_{n} is the extension of K_{n} defined in the statement of the following lemma. Lemma 3.8 is now seen to follow from the conclusion of that lemma. □

Lemma 3.9 Let L_{n} denote the maximal abelian extension of K_{n} of exponent dividing p^{n} that is unramified away from N, in which the prime lying over p splits completely, and on whose Galois group (Z/p^{n})^{×} = Gal(K_{n}/Q) acts via χ_{p}^{−1}. Then L_{n} is a cyclic extension of K_{n}, and the degree [L_{n} : K_{n}] divides the p-power part of (N^{2} − 1)/24.

Proof. In the following proof, for an abelian group G, let G/{m} = G/G^{m} denote the quotient of G by all mth powers in G. For integers a, b let G/{(a, b)} = G/{m}, where m is the greatest common divisor of a and b. Let ζ be a choice of primitive p^{n}th root of unity. If O_{n} denotes the ring of integers in K_{n}, then 1 − ζ generates the unique prime ideal of O_{n} lying above p. Let ((O_{n}/N)^{×}/{p^{n}})_{(−1)} denote the maximal quotient of (O_{n}/N)^{×}/{p^{n}} on which Gal(K_{n}/Q) acts via χ_{p}^{−1}. Since Herbrand’s criterion shows that the χ_{p}^{−1}-eigenspace in the p-part of the class group of K_{n} vanishes, global class field theory shows that the Galois group of L_{n}/K_{n} is equal to the cokernel of the composite

O_{n}[(1 − ζ)^{−1}]^{×} → (O_{n}/N)^{×} → ((O_{n}/N)^{×}/{p^{n}})_{(−1)}.

Fix a prime 𝔫 of K_{n} lying over N. We first claim that the injection (O_{n}/𝔫)^{×} ↪ (O_{n}/N)^{×} (coming from the Chinese remainder theorem) induces an isomorphism

(O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)} ≅ ((O_{n}/N)^{×}/{p^{n}})_{(−1)}.  (8)

To see this, we first recall that χ_{p} induces an isomorphism Gal(K_{n}/Q) ≅ (Z/p^{n})^{×}; we will write σ_{a} to denote the Galois element corresponding to a ∈ (Z/p^{n})^{×} via χ_{p}. The group Gal(K_{n}/Q) acts transitively on the primes of K_{n} lying over N, and the stabilizer of any one of these primes (and so in particular of 𝔫) is identified by χ_{p} with the cyclic subgroup ⟨N⟩ generated by N of (Z/p^{n})^{×}. Thus if {a_{1}, …, a_{r}} is a set of coset representatives for ⟨N⟩ in Z/p^{n} (labelled so that a_{1} represents the identity coset), then (taking into account that {a_{1}^{−1}, …, a_{r}^{−1}} also gives a set of coset representatives) the Chinese remainder theorem provides an isomorphism

∏_{i=1}^{r}(O_{n}/σ_{a_{i}^{−1}}(𝔫))^{×} ≅ (O_{n}/N)^{×}.  (9)

If x, y ∈ (O_{n}/N)^{×}/p^{n}, write x ∼ y if x and y have the same image in ((O_{n}/N)^{×}/{p^{n}})_{(−1)}. Using the isomorphism (9) to write x = (x_{1}, …, x_{r}), with x_{i} ∈ (O_{n}/σ_{a_{i}})^{×}/{p^{n}}, we see that

x = (x_{1}, …, x_{r}) ∼ (∏_{i=1}^{r} σ_{a_{i}^{−1}}(x_{i})^{a_{i}^{−1}}, 1, …, 1).

Taking into account the fact that σ_{N} acts on (O_{n}/N)^{×} as the Frobenius automorphism, i.e. via raising to Nth powers, we see that (8) is indeed an isomorphism, and that the inverse isomorphism is given by the map

image of x = (x_{1}, …, x_{r}) in ((O_{n}/N)/p^{n})_{(−1)}

↦ image of ∏_{i=1}^{r} σ_{a_{i}}(x_{i})^{a_{i}} in (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)}. (10)

Of course, this map is independent of the particular choice of coset representatives {a_{i}}.

Since (O_{n}/𝔫)^{×} is a cyclic group, the isomorphism (8) shows that ((O_{n}/N)^{×}/{p^{n}})_{(−1)} has order bounded by the p-part of N^{2} − 1. Thus if p ≥ 5 the lemma is proved.

We now perform a more refined analysis, which will prove the lemma in the remaining cases (i.e. p = 2 or 3). The formula (10) shows that under the isomorphism (8), the subgroup of

((O_{n}/N)^{×}/{p^{n}})_{(−1)}

generated by (1 − ζ) corresponds to the subgroup of (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)} generated by

∏_{a∈(Z/p^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a}.

(Here and below, in expressions such as this, we will suppress the particular choice of coset representatives for elements of (Z/p^{n})^{×}/⟨N⟩; the product is well-defined as an element of (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)}, independently of this choice.)

Suppose first that p is odd, and write N = ω(N)N_{1} in Z_{p}, where ω(N) is the Teichmüller lift and N_{1} is a 1-unit. Let c denote the order of (Z/p)/⟨ω(N)⟩; note that c is prime to p. If p^{f} denotes the exact power of p dividing N^{2} − 1, and p^{f′} denotes the exact power of p dividing N_{1} − 1, then f′ ≥ f, with equality if p = 3. Let us assume that n ≥ f′, so that (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)} is cyclic of order p^{f}, generated by the image of ζ, or of −ζ.

Since 2c is prime to p, the subgroup of (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)} generated by

∏_{a∈(Z/p^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a}

coincides with the subgroup generated by

( ∏_{a∈(Z/p^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a})^{2c} = ( ∏_{a∈(Z/p^{n})^{×}/⟨N_{1}⟩}(1 − ζ^{a})^{a})^{2}

= ∏_{a∈(Z/p^{n})^{×}/⟨N_{1}⟩}(1 − ζ^{a})^{a}(1 − ζ^{−a})^{−a} = ∏_{a∈(Z/p^{f′})^{×}}(−ζ)^{a^{2}}.

(The above expressions are all well-defined as elements of, and the equalities all hold in, the quotient (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)}.)

If p ≥ 5, then since there are quadratic residues distinct from 1 in (Z/p)^{×}, we compute that ∑_{a∈(Z/p^{f′})^{×}} a^{2} ≡ 0 mod p^{f′}, and so ∏_{a∈(Z/p^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a} generates the trivial subgroup of the group (O_{n}/𝔫)^{×}/{(p^{n}, N^{2} − 1)}. In this case, our “refined analysis” adds no further restrictions to the degree of L_{n} over K_{n}. However, if p = 3, then 1 is the only quadratic residue in (Z/3)^{×}, and one computes that the power of 3 dividing ∑_{a∈(Z/p^{f′})^{×}} a^{2} is exactly 3^{f′−1} = 3^{f−1}. Thus we find that the degree [L_{n} : K_{n}] is bounded above by 3^{f−1}. This is the exact power of 3 dividing (N^{2} − 1)/24, and thus we have proved the lemma in the case p = 3.

Suppose now that p = 2. Write N = ±1 · N_{1}, where N_{1} ≡ 1 mod 4. Let 2^{f} be the exact power of 2 dividing N^{2} − 1, and let 2^{f′} be the exact power of 2 dividing N_{1} − 1. Note that f = f′ + 1. Also, assume that n ≥ f. In particular, n ≥ 2, and so −ζ is also a primitive 2^{n}th root of unity. The quotient (O_{n}/𝔫)^{×}/{(2^{n}, N^{2} − 1)} is then cyclic of order 2^{f}, generated by ζ, or by −ζ.

We may rewrite ∏_{a∈(Z/2^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a} in the form

∏_{a∈(Z/2^{n})^{×}/⟨N⟩}(1 − ζ^{a})^{a} = ∏_{a∈(1+4Z/2^{n})/⟨N_{1}⟩}(1 − ζ^{a})^{a}(1 − ζ^{−a})^{−a} = ∏_{a∈(1+4Z/2^{n})/(1+2^{f′}Z/2^{n})}(−ζ)^{a^{2}}.

(The above expressions are all well-defined as elements of, and the equalities all hold in, the quotient (O_{n}/𝔫)^{×}/{(2^{n}, N^{2} − 1)}.) One computes that the largest power of 2 dividing ∑_{a∈(1+4Z/2^{n})/(1+2^{f′}Z/2^{n})} a^{2} is 2^{f′−2} = 2^{f−3}. Thus the degree [L_{n} : K_{n}] is bounded above by 2^{f−3}. This is the exact power of 2 dividing (N^{2} − 1)/24, and so we have proved the lemma in the case p = 2. □

Conclusion of proof of Theorem 3.5: The Theorem follows from the isomorphism (2), the injectivity of (7), and Lemma 3.8. □

The reduced Zariski tangent space of the deformation ring R can be computed via a calculation similar to that used to prove Theorem 3.5. We state the result here, but postpone the details of the calculation to the following sections. (See Proposition 4.11 for the case p = 2, and Proposition 5.5 for the case of odd p.)

Proposition 3.10 If 𝔭 denotes the maximal ideal of R, then the reduced Zariski tangent space 𝔭/(𝔭^{2}, p) of R is of dimension at most one over F_{p}. More precisely, 𝔭/(𝔭^{2}, p) vanishes unless p divides the numerator of (N − 1)/12, in which case it has dimension one over F_{p}.

Having introduced the deformation ring R, we now turn to constructing the corresponding Hecke ring T. We consider the space M_{2}(N) of all modular forms of weight two on Γ_{0}(N) defined over \overline{Q}_{p}, and the commutative Z_{p}-algebra H of endomorphisms of M_{2}(N) generated by the Hecke operators T_{n}. We define the p-Eisenstein maximal ideal of the algebra H to be the ideal generated by the elements T_{n} − σ^{∗}(n) (where σ^{∗}(n) = ∑_{{0<d|n \atop (d,N)=1}} d

for any positive integer n) together with the prime p, and let T denote the completion of H at its p-Eisenstein maximal ideal. Then T is a reduced Z_{p}-algebra. We let J denote the kernel of the surjection T → Z_{p} describing the action of T on the Eisenstein series E_{2}^{∗}, where

E_{2}^{∗} = 1 − N − 24 ∑_{n=1}^{∞} q^{n}σ^{∗}(n).

Let T^{0} denote the quotient of T that acts faithfully on cuspforms, and let J^{0} denote the image of J in T^{0}. (This is the localization at p of the famous Eisenstein ideal of [7].)

Lemma 3.11 The order of T^{0}/J^{0} (which is a power of p) is equal to the p-power part of the numerator of (N − 1)/12.

Proof. This is Proposition II.9.7 of [7]. □

Proposition 3.12 There is an object (V, L, ρ) of Def(T), uniquely determined by the property that Trace(ρ(Frob_{ℓ})) = T_{ℓ}, for ℓ ≠ p, N. Furthermore, the diagram

[figure: R; T; R/I; Z_{p}; T/J]

is commutative.

Proof. Since, by Corollary 3.4, the universal deformation ring R is topologically generated by the traces Trace(ρ^{univ}(Frob_{ℓ})), there is at most one object (V, L, ρ) of Def(T) satisfying the condition Trace(ρ(Frob_{ℓ})) = T_{ℓ} for ℓ ≠ p, N. This gives the uniqueness statement of the proposition. In order to construct the required object (V, L, ρ), we proceed in several steps.

Lemma 3.13 Let \overline{V}′ be a two dimensional discrete G_{Q}-module over a finite extension k of F_{p}. Suppose that \overline{V}′ is finite at p, unramified away from p and N, contains an I_{N}-fixed line that is not G_{Q}-stable, and has semi-simplification isomorphic to the semi-simplification of k ⊗_{F_{p}} \overline{V} . Then \overline{V}′ ≅ k ⊗_{F_{p}} \overline{V} .

Proof. Since \overline{V}′ and k ⊗_{F_{p}} \overline{V} have isomorphic semi-simplifications, we see that \overline{V}′ is an extension of one of k ⊗_{F_{p}} μ_{p} or k ⊗_{F_{p}} Z/p (thought of as étale groups schemes over Q, or equivalently as G_{Q}-representations) by the other. Both these one dimensional representations are unramified at N, and \overline{V}′ contains one or the other as a G_{Q}-submodule. It also contains an I_{N}-fixed line which is not a G_{Q}-submodule. Thus \overline{V}′ is in fact spanned by I_{N}-fixed lines, and so is unramified at N. By assumption it is finite at p, and so it has a prolongation to a finite flat group scheme over Z.

If p is odd, then \overline{V} must prolong to an extension of one of k ⊗_{F_{p}} μ_{p} or k ⊗_{F_{p}} Z/p by the other as a group scheme over Spec Z (since by [2], Thm. 2, p-power order group schemes over Z are determined by their associated Galois representations). There are no such nontrivial extensions ([7], Ch. I for k = F_{p}, from which the result can easily be deduced), and thus \overline{V}′ ≅ k ⊗_{F_{p}} \overline{V} . In the case that p = 2, note first that since both k ⊗_{F_{2}} μ_{2} and k ⊗_{F_{2}} Z/2 yield the trivial character of G_{Q}, the module \overline{V} cannot be the direct sum of these two characters; if it were, every line (including the I_{N}-fixed line appearing in the statement of the lemma) would be G_{Q}-stable. Taking this into account, it is easily seen (again using the results of [7], Ch. I) that \overline{V}′ ≅ k ⊗_{F_{2}} \overline{V}. □

Lemma 3.14 Let K be a finite extension of Q_{p}, with ring of integers O. Let k denote the residue field of O, and let O′ denote the order in O consisting of elements whose image in k lies in the prime subfield F_{p} of k. Suppose one is given a two dimensional K-vector space W, and a continuous representation G_{Q} → GL(W) that is finite at p (in the sense that one, or equivalently any, G_{Q}-invariant O-lattice in W is finite at p), semistable at N (in the sense that W contains an I_{N}-fixed line), unramified away from p and N, such that the semi-simple residual representation attached to W is isomorphic to the direct sum of the trivial character and the mod p cyclotomic character.

If W is irreducible, then we may find a free O′-module of rank two V, equipped with a continuous representation ρ : G_{Q} → GL(V), and containing an I_{N}-fixed line L, such that the triple (V, L, ρ) deforms (\overline{V} , \overline{L}, \overline{ρ}), and such that K ⊗_{O′} V ≅ W as G_{Q}-modules.

Proof. Choose any G_{Q}-stable lattice V′ in W, and let L′ denote the intersection of V′ with the I_{N}-fixed line in W. Since W is irreducible, the line L′ is not G_{Q}-stable. Thus we may find a non-negative integer n such that L′/p^{n} is G_{Q}-stable in V′/p^{n}, but such that L′/p^{n+1} is not G_{Q}-stable in V′/p^{n+1}. If we define V′′ to be the preimage in V′ of L′/p^{n}, then we see that L′/p, when regarded as a subspace of V′′/p, is not G_{Q}-stable. Lemma 3.13 implies that V′′/p ≅ k ⊗_{F_{p}} \overline{V}. Using the description of the automorphisms of k ⊗_{F_{p}} \overline{V} afforded by Lemma 2.12, we deduce easily that in fact there is an isomorphism of pairs (V′′/p, L′/p) ≅ k ⊗_{F_{p}} (\overline{V} , \overline{L}). If we choose a basis for (V′′, L′) over O that reduces to an F_{p} basis for (\overline{V} , \overline{L}), then the O′-span of this basis gives rise to the required pair (V, L). □

If T̃ denotes the normalization of T, then we may write T̃ = ∏_{i=1}^{d} O_{i}, where each O_{i} is a discrete valuation ring, of finite index over Z_{p}. The rings O_{i} are in bijection with the conjugacy classes of normalized eigenforms f_{i} in M_{2}(N) that satisfy the congruence f_{i} ≡ E_{2}^{∗} mod 𝔭_{i} (where 𝔭_{i} denotes the maximal ideal of O_{i}); as before, E_{2}^{∗} denotes the weight two Eisenstein series on Γ_{0}(N). The ring O_{i} is the ring of integers in the subfield of \overline{Q}_{p} generated by the Fourier coefficients of f_{i}. The injection T → T̃ = ∏_{i=1}^{d} O_{i} is characterised by the property T_{n} ↦ (a_{n}(f_{i}))_{i=1,...,d}. Note that E_{2}^{∗} is one such form f_{i}. We may choose the labeling so that E_{2}^{∗} = f_{1}; then O_{1} = Z_{p} = T/J.

As in the statement of Lemma 3.14, for each i = 1, …, d, define O′_{i} to be the order in O_{i} obtained as the preimage under the map to the residue field of the prime subfield F_{p}. By construction O′_{i} is a complete Noetherian local ring with residue field F_{p}. Also, the natural map T → O_{i} factors through O′_{i}.

Lemma 3.15 For each i = 1, …, d, we may construct an object (V_{i}, L_{i}, ρ_{i}) ∈ Def(O′_{i}) with the property that Trace(ρ_{i}(Frob_{ℓ})) is equal to the image of T_{ℓ} in O′_{i}, for each ℓ ≠ p, N.

Proof. If i = 1, so that O′_{i} = Z_{p}, we take (V_{1}, L_{1}, ρ_{1}) to be the triple (V^{min}, L^{min}, ρ^{min}). Suppose now that i ≥ 2, so that O_{i} corresponds to a cuspform f_{i}. If we consider the usual irreducible Galois representation into GL_{2}(Q_{p} ⊗_{Z_{p}} O_{i}) attached to f_{i}, and apply Lemma 3.14, then we again obtain the required triple. □

Conclusion of proof of Proposition 3.12: Each of the triples (V_{i}, L_{i}, ρ_{i}) constructed in the previous lemma corresponds to a homomorphism ϕ_{i} : R → O′_{i}. The product of all these yields a homomorphism ϕ : R → ∏_{i=1}^{d} O′_{i}. Since R is topologically generated by the elements Trace(ρ^{univ}(Frob_{ℓ})) (ℓ ≠ p, N), we see that ϕ factors through T. The map ϕ in turn corresponds to a triple (V, L, ρ) ∈ Def(T), satisfying the requirements of the proposition. By construction, the diagram appearing in the statement of the proposition commutes. □

Let T′ denote the image in T of the map constructed in Proposition 3.12. Our ultimate goal is to prove that this map is an isomorphism, and so in particular that T′ = T. However, we will proceed in stages.

Write J′ = T′ ⋂ J, let (T′)^{0} denote the image of T′ in T^{0}, and let (J′)^{0} denote the image of J′ in (T′)^{0}. We have the morphism of short exact sequences

\xymatrix{0 \ar[r] & T′ \ar[r] \ar[d] & Z_{p} ⨁(T′)^{0} \ar[r]^{(x,y)↦x−y} \ar[d] & (T′)^{0}/(J′)^{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & T \ar[r] & Z_{p} ⨁ T^{0} \ar[r]^{(x,y)↦x−y} & T^{0}/J^{0} \ar[r] & 0.}

Applying the snake lemma we obtain the following exact sequence:

0 ⟶ ker((T′)^{0}/(J′)^{0} → T^{0}/J^{0}) ⟶ coker(T′ → T)
⟶ coker((T′)^{0} → T^{0}) ⟶ coker((T′)^{0}/(J′)^{0} → T^{0}/J^{0}) ⟶ 0.

We also have the following tautological exact sequence:

0 ⟶ ker((T′)^{0}/(J′)^{0} → T^{0}/J^{0}) ⟶ (T′)^{0}/(J′)^{0}
⟶ T^{0}/J^{0} ⟶ coker((T′)^{0}/(J′)^{0} → T^{0}/J^{0}) ⟶ 0.

Thus we find that

#(T′)^{0}/(J′)^{0} − #(T^{0}/J^{0}) = # coker(T′ → T) − # coker((T′)^{0} → T^{0}).  (11)

Since T → T^{0} is surjective, we conclude that the right hand side of (11) is non-negative, and thus that the order of (T′)^{0}/(J′)^{0} is at least equal to that of T^{0}/J^{0}. By Lemma 3.11, the order of this latter group has order equal to the p-power part of the numerator of (N − 1)/12. Thus the order of (T′)^{0}/(J′)^{0} is at least equal to this number.

Suppose now that N ≢ −1 mod 2p. The p-power part of (N^{2} − 1)/24 is then equal to the p-power part of the numerator of (N − 1)/12. Theorem 3.5 thus shows that the numerical criterion of [17] (as strengthened in [6]) applies to prove that the surjection R → T′ of the preceding proposition is an isomorphism of local complete intersections. Furthermore, we conclude that in fact (T′)^{0}/(J′)^{0} has order exactly equal to the power of p dividing the numerator of (N − 1)/12, that is, to the order of T^{0}/J^{0}. The equation (11) then shows that T′ = T if and only if (T′)^{0} = T^{0}.

Lemma 3.16 The inclusion T′ → T is an isomorphism.

Proof. It follows from Corollary 3.4, together with the construction of the map R → T of Proposition 3.12, that T′ contains T_{ℓ} for all ℓ ≠ N, p. Proposition 2.10 shows that ρ^{univ} has a rank one space of I_{p}-coinvariants, on which Frob_{p} then acts as multiplication by a unit. It follows from the construction of R → T, and the known structure of Galois representations attached to weight two modular forms, that the image of this unit in T is equal to the Hecke operator T_{p}. Thus T′ contains T_{p}.

It remains to show that T_{N} lies in T′. By the remark preceding the statement of the lemma, it in fact suffices to show that T_{N} lies in (T′)^{0}. The surjection R → (T′)^{0} induces an object (V^{0}, L^{0}, ρ^{0}) ∈ Def((T′)^{0}). The concrete construction of the map R → T (and hence the map R → T^{0}) shows that this representation is built out of Galois representations attached to weight two cuspforms on Γ_{0}(N), which are (so to speak) genuinely semi-stable at N. In particular, the line L^{0} is not only fixed by I_{N}, but is stable under the decomposition group at N. Standard properties of Galois representations attached to weight two cusp forms show that the eigenvalue of Frob_{N} on this line is furthermore equal to T_{N}. Thus T_{N} ∈ (T′)^{0}, and so we see that (T′)^{0} = T^{0}, as required. □

The preceding lemma completes the proof of Theorem 1.5 in the case when N ≢ −1 mod 2p. If, on the other hand, we have N ≡ −1 mod 2p, then Proposition 3.10 shows that the Zariski tangent space of R is trivial. In this case, the map R → Z_{p} is an isomorphism. Also, Lemma 3.11 then implies that T^{0} = 0, and hence that T = Z_{p}. Thus the map R → T is certainly an isomorphism in this case, and we have completely proved Theorem 1.5 of the introduction.

Let us make two remarks:

(A) An alternative approach to proving Proposition 3.12 is as follows. The results of [7], Section II.16, show that if V^{0} denotes the p-Eisenstein part of the p-adic Tate module of J_{0}(N), then V^{0} is free of rank two over T^{0}, and the G_{Q}-action on V^{0} yields a deformation ρ^{0} of \overline{ρ} over T^{0}. The I_{N}-invariants in this representation form a rank one free submodule L^{0} of this representation. The discussion of [7], Section II.11 shows that both the cuspidal and Shimura subgroup map isomorphically onto the connected component group of the fibre over N of the Néron model of J_{0}(N), and this in turn implies that (V^{0}, L^{0}, ρ^{0}) provides an object of Def(T^{0}). Thus we obtain a corresponding map R → T^{0}. Taking the product of this with the map R → R/I = Z_{p}, we obtain the required map R → T of Proposition 3.12. Finally, the explicit description of T^{0} provided by [7], Cor. II.16.2 assures us that the map R → T is surjective.

We have chosen to present the alternative argument above both because it is more elementary (the only ingredient required from [7], Ch. II, is the computation of the order of T^{0}/J^{0}), and because we are then able to recover the results of [7], Sections II.16, II.17, as we explain below.

(B) In the proof of Lemma 3.16, we have struggled slightly to prove that T_{N} in fact lies in T′. This is somewhat amusing, since actually T_{N} = 1 in T! This follows from [7], Prop. II.17.10. When p is odd, the argument is straightforward: namely, since T_{N}^{2} = 1 for general reasons (the Galois representations attached to modular forms of weight two on Γ_{0}(N) are semi-stable at N and Cartier self-dual), it suffices to note that T_{N} ≡ 1 modulo the maximal ideal of T. When p = 2, Mazur’s proof of this result depends on his detailed analysis of the 2-Eisenstein torsion in J_{0}(N). We present an alternative proof below, using the deformation theoretic techniques of this paper.

We close this section by explaining how Theorem 1.5 allows us to recover the main results of Section II of [7].

Corollary 3.17 The Z_{p}-algebra T (and consequently also its quotient T^{0}) is generated by a single element over Z_{p}. In particular, both T and T^{0} are local complete intersections, and hence Gorenstein.

Proof. Theorem 1.5 shows that it suffices to verify the analogous statement for the deformation ring R. Proposition 3.10 shows that if 𝔭 denotes the maximal ideal of R, then 𝔭/(𝔭^{2}, p) has dimension at most one over F_{p}, and the corollary follows. □

The fact that T^{0} is monogenic over Z_{p} was originally proved by Mazur ([7], Cor. 16.2). Since T is reduced, finite flat, and monogenic over Z_{p}, and is equipped with a map T → T/J ≅ Z_{p}, we see that we may write T ≅ Z_{p}[X]/Xf(X), where X generates the ideal J in T, the monic polynomial f(X) ∈ Z_{p}[X] satisfies f(X) ≡ X^{g_{p}} mod p, and there is an isomorphism T^{0} ≅ Z_{p}[X]/f(X). (Here we follow [7] in letting g_{p} denote the rank of T^{0} over Z_{p}.) The image of X in Z_{p}[X]/f(X) generates the ideal J^{0} in T^{0}.

In [7], Prop. II.18.10, Mazur treats the questions of exhibiting explicit generators of J^{0} (or equivalently, explicit choices for the element “X” of the preceding paragraph). We recall his result here, and give a deformation-theoretic proof.

Proposition 3.18 Suppose that p divides the numerator of (N − 1)/12. Let ℓ be a prime different from N. Say that ℓ is good (with respect to the pair (p, N)) if (i) one of ℓ or p is odd, ℓ is not a pth power modulo N, and (ℓ − 1)/2 ≢ 0 mod p (note that this last expression makes sense, since either ℓ is odd, in which case (ℓ − 1)/2 is an integer, or else p is odd, in which case 1/2 is well-defined modulo p); or (ii) ℓ = p = 2 and −4 is not an 8th power modulo N. ^{2} Then T_{ℓ} − (1 + ℓ) generates the ideal J^{0} if and only if ℓ is a good prime.

(\frac{ℓ − 1}{2}) φ(\overline{ℓ}) = φ(x).

2This definition originally appeared in [7], p. 124. However, condition (ii) is misstated there. In particular, on page 139 Mazur writes that (for ℓ = 2 and \overline{ℓ} = x^{2} mod N)

In reality this equality is only valid up to an element of H^{+}/IH^{+} killed by 2 (recall there is a noncanonical isomorphism H^{+}/IH^{+} ≃ Z/nZ). This ambiguity can not be avoided by replacing x by −x, since φ(−1) = 0. If 4|n, however, then this equality suffices to determine when ϵ^{+}(2) generates the 2-primary subgroup of H^{+}/IH^{+}; thus our criterion agrees with Mazur’s when N ≡ 1 mod 16.

Proof. Let R ≅ T → F_{p}[X]/X^{2} be a map that classifies a (unique up to scaling, by Proposition 3.10) non-trivial element in the reduced Zariski tangent space of R. If ℓ is distinct from p, then we must show that T_{ℓ} − (1 + ℓ) = Trace(ρ^{univ}(Frob_{ℓ})) − (1 + ℓ) has non-zero image under this map if and only if ℓ is a good prime. If α_{p} ∈ R ≅ T denotes the scalar by which Frob_{p} acts on the rank one quotient module of I_{p}-coinvariants of V^{univ}, then T_{p} = α_{p}, and so we must also show that α_{p} − (1 + p) has non-zero image under this map if and only if p is a good prime. Both cases follow from Proposition 4.11 in the case when p = 2, and from Proposition 5.5 in the case of odd p. □

As was remarked upon above, the next result (and the final result of this section) is also originally due to Mazur.

Proposition 3.19 In T we have the equality T_{N} = 1.

Proof. As we recalled above, this result is straightforward when p is odd. Thus we assume that p = 2. The T_{N}-eigenvalue of E_{2}^{∗} is equal to 1. Thus, in order to show that T_{N} = 1, it suffices to show that for each cuspform f_{i} (i = 2, …, d – we are using the notation introduced during the proof of Proposition 3.12), the image of T_{N} in O_{i} is equal to 1. If N ≢ 1 mod 8, then there are no cuspforms to consider (by Proposition 3.10 and Theorem 1.5, for example, or by Proposition II.9.7 of [7]), and hence there is nothing to prove. Thus we assume for the remainder of the argument that N ≡ 1 mod 8.

Fix a cuspform f_{i}, and let S denote the local ring

S = {(a, b) ∈ Z/4 × O_{i}/2𝔭_{i} | a mod 2 = b mod 𝔭_{i}}.

The objects (V_{2}^{min}, L_{2}^{min}, ρ_{2}^{min}) ∈ Def(Z/4) and the object in Def(O′_{i}/2𝔭_{i}) obtained by reducing modulo 2𝔭_{i} the object (V_{i}, L_{i}, ρ_{i}) ∈ Def(O′_{i}) (the latter was constructed in the course of proving Proposition 3.12) glue to yield an object (V, L, ρ) ∈ Def(S). Since N ≡ 1 mod 8, we see that G_{Q_{N}} acts trivially on V_{2}^{min}. Since (V_{i}, L_{i}, ρ_{i}) is constructed from the Galois representation attached to the cuspform f_{i}, we see that G_{Q_{N}} stabilises L_{i}, and Frob_{N} acts as multiplication by T_{N} on L_{i}. Thus the line L is stabilized by G_{Q_{N}} (in addition to being fixed by I_{N}), and Frob_{N} acts as multiplication by the image of T_{N} in S. If the image of T_{N} in O_{i} is equal to −1, then we see that the image of T_{N} in S is equal to (1, −1). Now (1, −1) ≢ (1, 1) mod 2S. Thus the object (V/2, L/2, ρ/2) ∈ Def(S/2) obtained by reducing (V, L, ρ) modulo 2 has the property that L is stable, but not trivial, under the action of G_{Q_{N}}. On the other hand, Theorem 4.4, together with Lemma 4.2, shows that there are no such elements of Def(S/2). This contradiction proves the proposition. □

4 Explicit deformation theory: p = 2

Let us begin by fixing an odd prime N, and recalling some class field theory of the field K = Q(\sqrt{(−1)^{(N+1)/2}N}). We let H denote the 2-power part of the strict class group Cl(O_{K}) of the ring of integers O_{K} of K, and let E denote the corresponding cyclic 2-power extension of K, which is unramified at all finite primes. Genus theory shows that H is cyclic, and non-trivial. Thus E is a non-trivial cyclic 2-power extension of K; its unique quadratic subextension is equal to K(\sqrt{−1}). We let π_{K} denote the unique prime of K lying over 2; its image in H generates the two-torsion subgroup H[2] of H, if N ≢ −1 mod 8.

The following result is classical, but we will recall a proof for the benefit of the reader.

Proposition 4.1 The order of H is divisible by four if and only if N ≡ 1 mod 8. The order of H is divisible by eight if and only if furthermore −4 is an 8th power modulo N.

Proof. If N ≡ −1 mod 4, then K is a real quadratic field. If E^{+} denotes the 2-Hilbert class field of K (so E^{+} is the maximal totally real subextension of E), then we see that E is equal to the compositum of E^{+} and K(\sqrt{−1}). Since E is cyclic over K, we deduce that E^{+} must in fact be trivial. Thus in this case H is of order two.

Suppose now that N ≡ 1 mod 4, and that E contains a degree four sub-extension. Since E/K is cyclic, this sub-extension is unique, and hence Galois over Q. It must contain Q(\sqrt{−1}), and one sees easily that it is in fact a biquadratic extension of Q(\sqrt{−1}), unramified away from N. Since it is Galois over Q, it must be of the form Q(\sqrt{−1}, \sqrt{ν}, \sqrt{ν̄}), where ν is an element of Z[\sqrt{−1}] (and ν̄ is its conjugate) satisfying νν̄ = N.

However, for the extension Q(\sqrt{−1}, \sqrt{ν}, \sqrt{ν̄})/Q(\sqrt{−1}) to actually be unramified at 2, it must be that ν ≡ 1 mod 4. The element ν can be chosen in this manner if and only if N ≡ 1 mod 8. Thus we see that E has a degree four subfield if and only if N satisfies this congruence.

Finally, let us consider the question of whether the order of H is divisible by eight. This is the case if and only if the two-torsion subgroup H[2] of H has trivial image in H/4; equivalently, if and only if π_{K} has trivial image in H/4. This holds, in turn, if and only if π_{K} splits completely in Q(\sqrt{−1}, \sqrt{ν}, \sqrt{ν̄}). Clearly, this is true if and only if the ideal (1+i) splits completely in this field, regarded as an extension of Q(\sqrt{−1}). This holds, in turn, if and only if (1 + i) is a quadratic residue modulo ν (or equivalently module ν̄). Raising to 4th powers, and taking into account the isomorphism Z/N ≅ Z[\sqrt{−1}]/ν, we see that this is equivalent to −4 being an 8th power modulo N. □

The following lemma is used in the proof of Proposition 3.19.

Lemma 4.2 The inertia group I_{N} and the decomposition group G_{Q_{N}} have the same image in Gal(E/Q).

Proof. There is a unique prime lying above N in K, and it is principal. Thus this prime splits completely in the 2-Hilbert class field E of K, and so I_{N} and G_{Q_{N}} both have trivial image in Gal(E/K). Since N is ramified in K/Q, the lemma follows. □

Let H′ denote the 2-power part of the strict ray class group of K of conductor π_{K}^{2}, and let H′′ denote the 2-power part of the strict ray class group of K of conductor π_{K}^{3}. (Here “strict” means that in the case when K is real quadratic, we allow ramification at infinity.) We let E′ and E′′ denote the corresponding abelian extensions of K.

Proposition 4.3 (i) The natural surjection H′′ → H′ is an isomorphism.

(ii) If N ≡ −1 mod 4, the natural surjection H′ → H is an isomorphism and E = E′. If N ≡ 1 mod 4, the kernel of this surjection has order two and E′/E is a quadratic extension that is ramified at two.

(iii) The group H′ is cyclic.

(iv) Let D_{2}(E′/Q) denote the decomposition group of some prime of E′ lying over 2, and let I_{2}(E′/Q) denote the inertia subgroup of D_{2}(E′/Q). Then I_{2}(E′/Q) has index at most two in D_{2}(E′/Q). If furthermore E′/E is a quadratic extension, then D_{2}(E′/Q) is dihedral of order 8.

Proof. The groups H′ and H′′ sit inside the following exact diagram:

[figure: O_{K}^{×}; (O_{K}/π_{K}^{3})^{×}; ψ; H′′; H; 0; O_{K}^{×}; (O_{K}/π_{K}^{2})^{×}; H′; H; 0.]

To prove that the map H′′ → H′ is an isomorphism, it suffices to show that the kernel of ψ maps to zero in H′′; in other words, that the kernel of ψ consists of the images of global units. Since π_{K}^{2} = (2), we see that the kernel of ψ is equal to {±1}; this completes the proof of (i).

The proof of (ii) is even more straightforward: it follows immediately from a consideration of the units in O_{K} and the exact sequence

O_{K}^{×} → (O_{K}/π_{K}^{2})^{×} = (O_{K}/2)^{×} → H′ → H → 0.

We now turn to proving (iii). For this, it suffices to prove that H′/2 ≅ Z/2. Note that since the non-trivial element of Gal(K/Q) acts on H′ via h ↦ h^{−1}, we see that the extension K′ of K corresponding by class field theory to H′/2 is abelian over Q. If H′/2 were isomorphic to Z/2 ⊕ Z/2 (rather than Z/2), then since H is cyclic, this would imply that there exists a subextension of K′, quadratic over K and of conductor 2. Such an extension would again be abelian over Q. Using the Kronecker-Weber theorem, it is easy to check that there are no such quadratic extensions of K. Thus H′ must be cyclic, as claimed.

If N ≡ −1 mod 4 then E′ = E = K(\sqrt{−1}) and (iv) is obvious. If E′/E is a quadratic extension, then N ≡ 1 mod 4 and the class of π_{K} has order two in H. Thus the decomposition group D_{2}(E/K) at 2 in the Hilbert class field has order exactly 2. Since K/Q is ramified at 2, we see that the decomposition group D_{2}(E/Q) has order four, and that the inertia subgroup I_{2}(E/Q) has order two. If E′ = E then this completes the proof of (iv). If instead E′/E is quadratic, then E′/E is ramified at 2, implying that D_{2}(E′/Q) has order 8, and that I_{2}(E′/Q) has order 4. Since D_{2}(E′/K) ⊆ Gal(E′/K) ≅ H′ is cyclic, by (iii), and since Gal(E′/Q) is dihedral, it follows that D_{2}(E′/Q) is dihedral of order 8. □

Let (V, L, ρ) be an object of Def(A) for some Artinian local F_{2}-algebra, and let F denote compositum of K with the fixed field of the kernel of ρ. The following result greatly restricts the possibilities for F.

Theorem 4.4 The field F is contained in the strict 2-Hilbert class field E of K.

Proof. Since A is assumed to be of characteristic 2, the natural map Z_{2}^{×} → A^{×} has trivial image, and thus the image of ρ is contained in SL_{2}(A). Since I_{N} acts trivially on both L and V/L, we deduce that inertia at N acts through an abelian group of exponent 2, and thus through a cyclic group of order at most 2.

Lemma 4.5 The extension F/K is unramified at all finite primes outside π_{K}. Moreover, if K^{ab}/K is the maximal abelian extension of K contained in F, then the finite part of the conductor of K^{ab}/K divides π_{K}^{3}.

Proof. The Galois group Gal(F/Q) embeds into Gal(K/Q) × ρ(G_{Q}). In particular, it is of 2-power order, and so the image of an inertia group I_{N} at N in Gal(F/Q) is cyclic of two-power order. As observed above, ρ(I_{N}) is a quotient of I_{N} of order at most two. On the other hand, since K/Q is a quadratic extension that is ramified at N, we see that I_{N} surjects onto the order two group Gal(K/Q). It follows that the image of I_{N} in Gal(F/Q) has trivial intersection with Gal(F/K), and so F/K is unramified at the prime above N.

By definition, ρ is unramified outside 2 and N, and so it remains to prove the result about the conductor of K^{ab}/K. Since the compositum of extensions of conductor dividing π_{K}^{3} has conductor at most π_{K}^{3}, it suffices to prove the result for extensions K′/K with cyclic Galois group. Suppose such an extension K′/K with Galois group Z/2^{k}Z had conductor divisible by π_{K}^{4}. Then the conductor discriminant formula says that the discriminant Δ_{K′/K} is the product over all characters of Z/2^{k}Z of the corresponding conductor:

Δ_{K′/K} = ∏_{χ} 𝔣_{χ}.

Since Z/2^{k}Z has exactly 2^{k−1} faithful characters, restricting the product to this set we find that the discriminant is divisible by at least (π_{K})^{4·2^{k−1}}. Recall that the root discriminant of a number field L/Q is defined to be the positive real number δ_{L} := |Δ_{L/Q}|^{1/[L:Q]}. Let Δ_{L/Q,p} be the largest power of p diving Δ_{L/Q}. Define the p-root discriminant δ_{L,p} of L/Q to be |Δ_{L/Q,p}|^{1/[L:Q]}. The divisibility of discriminants proved above implies a lower bound for the 2-root discriminant of K′, and thus of F. Explicitly,

δ_{F,2} ≥ δ_{K′,2} = δ_{K,2}N_{K/Q}(Δ_{K′/K})^{1/[K′:Q]} ≥ 2 · 2 = 4.

Yet the Fontaine bound ([3], Theorem 1) for finite flat group schemes over Z_{2} killed by 2 implies that δ_{F,2} < 2^{1+ \frac{1}{2−1}} = 4. Thus the result follows by contradiction. □

We will strengthen this lemma step-by-step, until we eventually establish the theorem.

Lemma 4.6 The extension F/K is cyclic, and is contained in the field E′.

Proof. The preceding lemma, together with part (i) of Proposition 4.3, shows that the extension of K cut out by any abelian quotient of Gal(F/K) is contained in E′′ = E′. Part (iii) of the same proposition then implies that any such quotient is cyclic. Thus Gal(F/K) is a 2-group with no non-cyclic abelian quotients, and so is itself cyclic. The result follows. □

We now turn to a more careful study of the ramification at 2. Corollary 2.9 shows that V_{/Q_{2}} has a unique prolongation to a finite flat group scheme M_{/Z_{2}}, that the action of A on V prolongs to an action of A on M, and that the connected-étale sequence

0 → M^{0} → M → M^{ét} → 0

induces a two-step filtration of V by free A-modules of rank one.

Lemma 4.7 The action of inertia at 2 on M^{0}(\overline{Q}_{2}) and M^{ét}(\overline{Q}_{2}) is trivial.

Proof. This is clear for M^{ét}(\overline{Q}_{2}), since étale implies unramified. It follows for M^{0}(\overline{Q}_{2}) from the Cartier self-duality of M_{/\overline{Q}_{2}}. □

Lemma 4.8 If σ ∈ G_{Q_{2}} then σ^{2} acts trivially on V.

Proof. Let us choose a basis of V compatible with its filtration arising from the connectedétale sequence of M, and write the action of σ on V as a matrix over A in terms of this basis:

σ = ( \begin{matrix}1 + a & b \\ 0 & 1 + c \end{matrix} )

Part (iv) of Proposition 4.3 implies that σ^{2} lies in the inertia subgroup. Thus it must fix M^{0}(\overline{Q}_{2}) and M^{ét}(\overline{Q}_{2}). Computing σ^{2}, we find that (1 + a)^{2} = (1 + c)^{2} = 1, and so a^{2} = c^{2} = 0. Since the determinant of σ is 1, we see that (1 + c) = (1 + a)^{−1} = 1 + a. Now computing σ^{2} we find that it is trivial. □

Conclusion of proof of Theorem 4.4: If E′ = E, then by Lemma 4.6 there is nothing more to prove. Otherwise, Proposition 4.3 implies that the D_{2}(E′/Q) is dihedral of order 8. We have seen that for any σ ∈ G_{Q_{2}}, the element σ^{2} acts trivially. Thus the image ρ_{|G_{Q_{2}}} factors through an exponent 2 group, which is therefore abelian. Yet the dihedral group of order 8 is not abelian, and hence F is contained in a proper subfield of E′ that is Galois over Q. All such subfields lie inside E. □

Corollary 4.9 If 2^{m} denotes the order of H, then there exists a surjection R → F_{2}[X]/X^{n} if and only if n ≤ 2^{m−1}. Furthermore, any such surjection is unique up to applying an automorphism of F_{2}[X]/X^{n}.

Proof. Corollary 3.4 implies that there exists a surjection R → F_{2}[X]/X^{n} if and only if there exits (V, L, ρ) ∈ Def(F_{2}[X]/X^{n}) with the property that the traces of ρ generate F_{2}[X]/X^{n} as an F_{2}-algebra (or equivalently, with the property that there is an element of G_{Q} whose image under ρ has trace congruent to X mod X^{2}.)

Lemma 4.10 Let A be an F_{2}-algebra, and let U ∈ SL_{2}(A).

(i) U^{2} = I + Trace(U)U.

(ii) For any k ≥ 1, we have that Trace(U^{k}) ∈ Trace(U)A.

(iii) If U ∈ SL_{2}(A), then

U^{2^{k}} = (∑_{i=0}^{k−1} Trace(U)^{2^{k}−2^{k−i}}) I + Trace(U)^{2^{k}−1}U,

for any k ≥ 1.

Proof. Any 2 × 2 matrix U over the ring A satisfies the identity U^{2} = Det(U)I + Trace(U)U. Part (i) is a particular case of this identity, and parts (ii) and (iii) follow by induction. □

Theorem 4.4 shows that ρ factors as

G_{Q} → Gal(E/Q) → SL_{2}(F_{2}[X]/X^{n}).

Now Gal(E/Q) is a dihedral group of order 2^{m+1}; indeed, we may write

Gal(E/Q) = ⟨σ, τ|σ^{2^{m}} = τ^{2} = (στ)^{2} = 1⟩,  (12)

where σ generates Gal(E/K), and τ generates the image of I_{N} in Gal(E/Q).

Part (i) of Lemma 4.10 shows that any element of order two in the image of ρ has vanishing trace. Since any element of Gal(E/Q) that is not of order two is a power of σ, we conclude from part (ii) of the same lemma that all the traces of ρ lie in the ideal of F_{2}[X]/X^{n} generated by Trace(ρ(σ)). Since the trace of any element in the image of \overline{ρ} is zero, we see that this ideal is contained in the maximal ideal of F_{2}[X]/X^{n}. Applying part (iii) of Lemma 4.10, we deduce that Trace(σ)^{2^{m−1}} = 0 (since σ^{2^{m}} = 1, and so ρ(σ^{2^{m}}) = I).

Thus, on the one hand, the only way that X can arise as a trace of ρ is if Trace(σ) ≡ X mod X^{2}. On the other hand, if this condition holds, then X^{2^{m−1}} = 0, and hence n ≤ 2^{m−1}. This proves one direction of the “if and only if” statement of the corollary.

Let us now prove the uniqueness assertion, assuming that we are given a surjective map R → F_{2}[X]/X^{n}. Since the corresponding triple (V, L, ρ) deforms (\overline{V} , \overline{L}, \overline{ρ}), and since σ has non-trivial image in Gal(Q(\sqrt{−1})/Q), while τ generates the image of I_{N} in Gal(E/Q), we may choose a basis of V such that σ and τ act through matrices in SL_{2}(F_{2}[X]/X^{n}) of the form

ρ(σ) = (\begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix}) ≡ (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) mod X , ρ(τ) = (\begin{matrix}a(τ) & 0 \\ c(τ) & d(τ) \end{matrix}) ≡ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) mod X.

Now conjugating by matrices in ker(GL_{2}(F_{2}[X]/X^{n}) → GL_{2}(F_{2})) of the form

(\begin{matrix}α & 0 \\ γ & δ \end{matrix}) ,

it is easy to show that we may change our basis so that

ρ(σ) = (\begin{matrix}1 + uX & 1 \\ uX & 1 \end{matrix}) , ρ(τ) = (\begin{matrix}1 & 0 \\ uX & 1 \end{matrix}) ,

for some u ∈ (F_{2}[X]/X^{n})^{×}. Thus, after applying the inverse of the automorphism of F_{2}[X]/X^{n} induced by the map X ↦ uX, we see that we may put ρ in the form

ρ(σ) = ( \begin{matrix}1 + X & 1 \\ X & 1 \end{matrix} ) ,  ρ(τ) = ( \begin{matrix}1 & 0 \\ X & 1 \end{matrix} ) .  (13)

This proves the uniqueness statement.

Finally, one checks that the preceding formula gives a well-defined homomorphism ρ : Gal(E/Q) → SL_{2}(F_{2}[X]/X^{n}), so long as n ≤ 2^{m−1}, and that it deforms \overline{ρ}. It is certainly finite flat at 2, since the inertia group at two acts through its image in Gal(Q(\sqrt{−1})/Q). Thus, if we let L denote the line spanned by the vector (0, 1), then we obtain an object of Def(F_{2}[X]/X^{n}) of the required sort (since Trace(ρ(σ)) = X). This completes the proof of the corollary. □

Let us consider the particular case n = 2 of the preceding corollary.

Proposition 4.11 If N ≢ 1 mod 8, then Def(F_{2}[X]/X^{2}) = 0. If N ≡ 1 mod 8, then Def(F_{2}[X]/X^{2}) is one dimensional over F_{2}. Furthermore, if (V, L, ρ) corresponds to the non-trivial element, then we have the following formulas for the traces of ρ:

(i) If ℓ is an odd prime distinct from N, then

Trace(ρ(Frob_{ℓ})) = \begin{cases}0 if ℓ ≡ 1 mod 4 or ℓ is a square mod N \\ X otherwise \end{cases} .

(ii) If α_{2} denotes the eigenvalue of Frob_{2} on the rank one F_{2}[X]/X^{2}-module of I_{2}- coinvariants of V, then

α_{2} = \begin{cases}1 if − 4 is an 8th power mod N \\ 1 + X if not \end{cases} .

Proof. If N ≢ 1 mod 8 then Proposition 4.1 shows that H has order two, and Corollary 4.9 shows that any map R → F_{2}[X]/X^{2} factors through the map R → F_{2}. Thus in this case Def(F_{2}[X]/X^{2}) = 0, as claimed.

If N ≡ 1 mod 8, then conversely we conclude from Proposition 4.1 that H has order divisible by 4. Corollary 4.9 then shows that there is a unique surjection R → F_{2}[X]/X^{2}, and thus that Def(F_{2}[X]/X^{2}) is one dimensional over F_{2}. If F denotes the subextension of E over K cut out by this non-trivial deformation, then F is a dihedral extension of Q of degree 8, unramified over K, containing K(\sqrt{−1}). (Concretely, as we saw in the proof of Proposition 4.1, the field F has the form Q(\sqrt{−1}, \sqrt{ν}, \sqrt{ν̄}), for appropriate ν, ν̄.)

Recall the presentation (12) of Gal(E/Q). If we let σ̄ and τ̄ denote the image of σ and τ under the surjection Gal(E/Q) → Gal(F/Q), then Gal(F/Q) has the following presentation:

Gal(F/Q) = ⟨σ̄, τ̄|σ̄^{4} = τ̄^{2} = (σ̄τ̄)^{2} = 1⟩.

Recall from the proof of Corollary 4.9 that the only elements of Gal(F/Q) whose images under \overline{ρ} have non-zero trace (which is then equal to X) are σ̄^{±1}; that is, the elements of Gal(F/Q) that are of order 4.

If ℓ is an odd prime distinct from N, then ℓ is unramified in F. The final remark of the preceding paragraph shows that

Trace(ρ(Frob_{ℓ})) = \begin{cases}0 if Frob_{ℓ} has order 1 or 2 \\ X if Frob_{ℓ} has order 4 \end{cases} .

Now K is the maximal subfield of F fixed by σ̄, while one checks that any element of Gal(F/Q) of order two fixes at least one of the subfields Q(\sqrt{−1}) or Q(\sqrt{N}) of F. Thus we see that ρ(Frob_{ℓ}) has trace zero (as opposed to trace X) if and only if ℓ splits in at least one of the fields Q(\sqrt{−1}) or Q(\sqrt{N}). This establishes (i).

Again referring to the presentation (12) of Gal(E/Q), one easily checks that D_{2}(E/Q) is generated by στ and σ^{2^{m−1}}, with I_{2}(E/Q) being generated by στ. (Recall that 2^{m} denotes the order of H.) Thus D_{2}(F/Q) is generated by σ̄τ̄ and σ̄^{2^{m−1}}. Thus if m ≥ 3, then we see that D_{2}(F/Q) = I_{2}(F/Q), while if m = 2, then D_{2}(F/Q)/I_{2}(F/Q) is generated by the image of σ̄^{2}.

In terms of the explicit model (13) for ρ, we see that the coinvariants of I_{2}(F/Q) = ⟨σ̄τ̄⟩ on V are spanned by the image of the basis vector (0, 1), and that σ̄^{2} (which is central in Gal(F/Q), and so does act on the space of coinvariants) acts on the image of this vector as multiplication by 1 + X. Combining this computation with the discussion of the previous paragraph proves part (ii), once we recall from Proposition 4.1 that m ≥ 3 if and only if −4 is an 8th power modulo N. □

The quotient R/2 is the universal deformation ring classifying deformations of (\overline{V} , \overline{L}, \overline{ρ}) in characteristic 2. The preceding two results together imply that R/2 ≅ F_{2}[X]/X^{2^{m−1}}, where 2^{m} is the order of H; formula (13) then gives an explicit model for the universal deformation over R/2.

We close this section by observing that Theorem 1.1 follows from Corollaries 1.6 and 4.9 taken together. More generally, we see that g_{2} = 2^{m−1} − 1.

5 Explicit deformation theory: p odd

In this section we suppose that p ≥ 3, and that N is prime to p. We begin by considering the problem of analyzing deformations (V, L, ρ) ∈ Def(A), where A is an Artinian local F_{p}-algebra with residue field F_{p}. Our results will be less definitive than those obtained in the case of p = 2.

Let Δ denote the following subgroup of GL_{2}(F_{p}) ⊂ GL_{2}(A):

Δ = \{(\begin{matrix}α & 0 \\ 0 & 1 \end{matrix}) | α ∈ F_{p}^{×}\} ,

and let G′ denote the kernel of the map SL_{2}(A) → SL_{2}(F_{p}) induced by reduction modulo 𝔭 (the maximal ideal of A); note that G′ is a normal subgroup of GL_{2}(A). If we let G denote the subgroup of GL_{2}(A) generated by G′ and Δ, then G is isomorphic to the semi-direct product G′ × Δ, where Δ acts on G′ via conjugation. Explicitly, one computes that

(\begin{matrix}α & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}α^{−1} & 0 \\ 0 & 1 \end{matrix}) = (\begin{matrix}a & αb \\ α^{−1}c & d \end{matrix}) .  (14)

Lemma 5.1 Let (V, L, ρ) be an object of Def(A), and let M denotes the finite flat group scheme over Z_{p} whose generic fibre equals V. If 0 → M^{0} → M → M^{ét} → 0 denotes the connected-étale exact sequence of M, then there is a basis for V over A such that

(i) The representation ρ : G_{Q} → GL_{2}(A) has image lying in G.

(ii) The submodule M^{0}(\overline{Q}_{p}) of V (which is free of rank one, by Corollary 2.9) is spanned by the vector (1, 0).

(iii) The line L is spanned by the vector (1, 1).

Proof. By definition of the deformation problem Def, the determinant of ρ is equal to \overline{χ}_{p}. Thus im(ρ) sits in the exact sequence of groups

0 → G′ → im(ρ) → F_{p}^{×} → 0.

The order of F_{p}^{×} is coprime to the order of G′, and so this exact sequence splits. If we fix a splitting s, then one easily sees that we may choose an eigenbasis for the action of s(F_{p}^{×}) so that this group acts via the matrices in Δ. Thus condition (i) is satisfied for this basis. Condition (ii) follow directly from condition (i). The stipulations of the deformation problem Def then imply that L is spanned by a vector of the form (1, u), for some unit u ∈ A^{×}. Rescaling the second basis vector by u, we may assume that L is in fact spanned by (1, 1). □

From now on, we fix an object (V, L, ρ) ∈ Def(A), and choose a basis of V as in the preceding lemma. Thus we may regard ρ as a homomorphism G_{Q} → G ⊂ GL_{2}(A).

Lemma 5.2 If (V, L, ρ) is a non-trivial deformation, then the image of I_{N} under ρ is a

cyclic subgroup of G′ of order p. Furthermore, if (\begin{matrix}a & b \\ c & d \end{matrix}) is a generator of this cyclic group,

then neither b nor c vanishes, and neither a nor d equals 1.

Proof. Since the image under ρ of inertia at N acts trivially on each of the lines L and V/L (the determinant of ρ equals \overline{χ}_{p}, which is trivial on I_{N}), we see that I_{N} acts via an abelian group of exponent p. Since tame inertia is pro-cyclic, inertia at N must act through a group of order dividing p. If I_{N} has trivial image, then Proposition 3.2 shows that V = A ⊗_{F_{p}} \overline{V} , and thus that (V, L, ρ) is the trivial deformation, contradicting our assumption. Thus I_{N} has image of order p.

The line L is spanned by the vector (1, 1). Thus if γ = (\begin{matrix}a & b \\ c & d \end{matrix}) is a generator of the

image of I_{N}, it fixes such a vector. Since the determinant of ρ equals \overline{χ}_{p}, we see that det(γ) = 1. If a (respectively d) equals 1 then we conclude that b (respectively c) equals 0. If either b or c vanishes, one easily checks that γ must be the identity, contradicting the fact that I_{N} has non-trivial image. □

If F denotes the extension of Q cut out by the kernel of ρ, then F contains Q(ζ_{p}) (where ζ_{p} denotes a primitive pth root of unity), since det(ρ) = \overline{χ}_{p}. We let F^{ab} denote the maximal subextension of F abelian over Q(ζ_{p}).

Lemma 5.3 The p-part of the conductor of F^{ab}/Q(ζ_{p}) divides π^{2}, where π = (1−ζ_{p})Z[ζ_{p}], and the extension F/Q(ζ_{p}) has inertial degree dividing p at N and is unramified outside N and π.

Proof. Lemma 5.2 shows that the image under ρ of inertia at N is a cyclic group of order dividing p. Therefore it suffices to prove the conductor bound at π.

The image under ρ of G_{Q(ζ_{p})} lies in G′, a p-group, and so we see that Gal(F^{ab}/Q(ζ_{p})) is an abelian p-group. Thus it is a compositum of cyclic extensions of p-power degree. The conductor of a compositum of cyclic extensions is equal to the g.c.d. of the conductors of the individual cyclic extensions, and thus it suffices to bound the conductor of a cyclic subextension of F^{ab} of degree p^{k}, for some k ≥ 1.

Let F′ be such a subextension, and suppose that the conductor of F′ is divisible by π^{3}. There are (p − 1)p^{k−1} faithful characters of Z/p^{k}, and so by the conductor discriminant formula, the discriminant Δ_{F′/Q(ζ_{p})} is divisible by π^{3(p−1)p^{k−1}}. Thus the p-root discriminant of F′ (as defined in the proof of Lemma 4.5) satisfies

δ_{F′,p} ≥ δ_{Q(ζ_{p})}N_{Q(ζ_{p})/Q}(π^{3(p−1)p^{k−1}})^{1/[F′:Q]} = p^{(p−2)/(p−1)} · p^{3(p−1)/(p(p−1))}

and thus

v_{p}(δ_{F′,p}) ≥ 1 + \frac{1}{p − 1} + \frac{p − 3}{p(p − 1)}.

This violates Fontaine’s bound [3] when p ≥ 3. The result follows for F^{ab}. □

In order to apply this result, we will need to classify the relevant class fields of Q(ζ_{p}) that can arise in the situation of the preceding lemma.

Proposition 5.4 Let p be an odd prime, and let N be a prime distinct from p. For any value of i, let K_{(i)} denote the maximal abelian extension of Q(ζ_{p}) satisfying the following conditions: K_{(i)} has conductor dividing π^{2}N; the Galois group Gal(K_{(i)}/Q_{p}(ζ_{p})) has exponent p; the Galois group Gal(Q_{p}(ζ_{p})/Q) acts on Gal(K_{(i)}/Q_{p}(ζ_{p})) through the ith power of the mod p cyclotomic character \overline{χ}_{p}. Then:

(i) K_{(1)} = Q(ζ_{p}, N^{1/p});

(ii) K_{(0)} = \begin{cases}the degree p subextension of Q_{p}(ζ_{p}, ζ_{N})/Q_{p}(ζ_{p}) if N ≡ 1 mod p \\ Q(ζ_{p}) otherwise \end{cases} ;

(iii) K_{(−1)} = \begin{cases}a degree p extension of Q_{p}(ζ_{p}) if N^{2} ≡ 1 mod p \\ Q(ζ_{p}) otherwise \end{cases} .

Proof. Let E_{(i)} denote the unramified extension of Q(ζ_{p}) of exponent p corresponding to the maximal elementary p-abelian quotient of the class group of Q(ζ_{p}) on which Gal(Q(ζ_{p})/Q) acts through \overline{χ}_{p}^{i}. Then we have the short exact sequence of abelian Galois groups

0 → Gal(K_{(i)}/E_{(i)}) → Gal(K_{(i)}/Q(ζ_{p})) → Gal(E_{(i)}/Q(ζ_{p})) → 0.

Global class field theory allows us to compute the group Gal(K_{i}/E_{(i)}). Indeed, it sits in the exact sequence

(Z[ζ_{p}])^{×} ⟶ ((Z[ζ_{p}]/π^{2} × Z[ζ_{p}]/N)^{×} /{p})_{(i)} ⟶ Gal(K_{(i)}/E_{(i)}) ⟶ 0;

here the subscript (i) denotes the maximal quotient on which Gal(Q(ζ_{p})/Q) acts via \overline{χ}_{p}^{i}, and G/{m} denotes G/G^{m}.

Since the reduction mod π^{2} of the global unit ζ_{p} = 1 + (ζ_{p} − 1) generates the p-power part of (Z[ζ_{p}]/π^{2})^{×}, we may eliminate this factor from the second term of the preceding exact sequence. If we fix a prime 𝔫 over N in Z[ζ_{p}], then as in the proof of Lemma 3.9, we obtain a surjection

(Z[ζ_{p}]/𝔫)^{×}/{(p, N^{1−i} − 1)} → Gal(K_{(i)}/E_{(i)}).

Consequently, we find that Gal(K_{(i)}/E_{(i)}) is either trivial (when N^{(1−i)} ≢ 1 mod p) or of order p (when N^{(1−i)} ≡ 1 mod p).

Let us now consider the particular cases i = 1, 0, −1. The 1, 0 and −1 eigenspaces inside the class group of Q(ζ_{p}) are trivial by Kummer theory, abelian class field theory and Herbrand’s theorem respectively. Thus for these values of i, we have E_{i} = Q(ζ_{p}), and so the preceding paragraph yields a computation of Gal(K_{(i)}/Q(ζ_{p})). The explicit descriptions of K_{(i)} in the case when i = 1 or 0 are easily verified, and so we leave this verification to the reader. □

We are now in a position to determine the reduced Zariski tangent space to the deformation functor Def. We will also record some useful information regarding non-trivial elements of this tangent space (assuming that they exist).

Proposition 5.5 If p does not divide the numerator of (N −1)/12, then Def(F_{p}[X]/X^{2}) = 0; otherwise, Def(F_{p}[X]/X^{2}) is one dimensional over F_{p}. Suppose for the remainder of the statement of the proposition that we are in the second case, and let (V, L, ρ) correspond to a non-trivial element of Def(F_{p}[X]/X^{2}).

(i) If as above F denotes the extension of Q cut out by the kernel of ρ, then F is equal to the compositum K_{(1)}K_{(0)}K_{(−1)} (where the class fields K_{(i)} of Q(ζ_{p}) are defined as in the statement of the previous proposition).

(ii) If p = 3, then Gal(F/Q(ζ_{p})) ≅ Gal(K_{1}/Q(ζ_{p})) × Gal(K_{0}/Q(ζ_{p})), and the image of an appropriately chosen generator of the first (respectively second) factor under ρ has the

form (\begin{matrix}1 & −rX \\ rX & 1 \end{matrix}) (respectively (\begin{matrix}1 + rX & 0 \\ 0 & 1 − rX \end{matrix})) for some r ∈ F_{p}^{×}.

(iii) If p ≥ 5, then Gal(F/Q(ζ_{p})) ≅ Gal(K_{1}/Q(ζ_{p}))×Gal(K_{0}/Q(ζ_{p}))×Gal(K_{1}/Q(ζ_{p})), and the image of an appropriately chosen generator of the first (respectively second, re-

spectively third) factor under ρ has the form (\begin{matrix}1 & −rX \\ 0 & 1 \end{matrix}) (respectively (\begin{matrix}1 + rX & 0 \\ 0 & 1 − rX \end{matrix}) ,

respectively (\begin{matrix}1 & 0 \\ rX & 1 \end{matrix})) for some r ∈ F_{p}^{×}.

(iv) We have the following formulas for the traces of ρ:

(iv.i) If ℓ is a prime distinct from N and p, then

Trace(ρ(Frob_{ℓ})) = \begin{cases}1 + ℓ if ℓ ≡ 1 mod p or ℓ is a pth power mod N \\ 1 + ℓ + uX otherwise \end{cases} ;

here u denotes an element of F_{p}^{×}.

(iv.ii) If α_{p} denotes the eigenvalue of Frob_{p} on the rank one F_{p}[X]/X^{2}-module of I_{p}- coinvariants of V, then

α_{p} = \begin{cases}1 if p is a pth power mod N \\ 1 + uX if not \end{cases} ;

again, u denotes an element of F_{p}^{×}.

Proof. Let (V, L, ρ) be a non-trivial element of Def(F_{p}[X]/X^{2}), cutting out the extension F of Q. As above, we choose the basis of V so that the conditions of Lemma 5.1 are satisfied. Since G′ is abelian, we see that F = F^{ab}. Equation 14, together with Lemma 5.3, thus shows that F is contained in the compositum K_{(1)}K_{(0)}K_{(−1)}. Lemma 5.2 then shows that in fact F must be equal to this compositum, proving part (i) of the proposition; that furthermore, each of the extensions K_{(1)}, K_{(0)} and K_{(−1)} of Q(ζ_{p}) must be non-trivial, and thus that N ≡ 1 mod p, by proposition 5.4; and that either part (ii) or part (iii) of the proposition is satisfied, depending on whether p = 3 or p ≥ 5. (We choose the generator of each group Gal(K_{(i)}/Q(ζ_{p})) to be the image of some fixed generator of the inertia group I_{N}.)

Suppose conversely that N ≡ 1 mod p, so that each of K_{(1)}, K_{(0)} and K_{(−1)} is a nontrivial extension of Q(ζ_{p}). Write F = K_{(1)}K_{(0)}K_{(−1)}. If we fix an element r ∈ F_{p}^{×}, then we may use the formulas of parts (ii) and (iii) to define a representation ρ : Gal(F/Q) → G ⊂ GL_{2}(F_{p}[X]/X^{2}). If we let L denote the line spanned by (1, 1), then this representation will deform the representation (V, L). Thus it will provide an element of Def(F_{p}[X]/X^{2}) provided that it is finite at p. An argument as in the proof of Lemmas 3.8 and 3.9 shows that this is automatically the case when p ≥ 5, and holds provided p divides (N − 1)/12, when p = 3. This establishes the initial claim of the proposition.

It remains to prove part (iv) of the proposition. Suppose first that ℓ is a prime distinct from p and N. We may write

ρ(Frob_{ℓ}) = (\begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 + aX & bX \\ cX & 1 − aX \end{matrix}) ,

for some elements a, b, c ∈ F_{p}. Thus Trace(ρ(Frob_{ℓ})) = 1 + ℓ + (ℓ − 1)aX. This is distinct from 1+ℓ if and only if ℓ ≢ 1 mod p and a ≠ 0. The latter occurs if and only if the primes over ℓ are not split in the extension K_{(0)}/Q(ζ_{p}), which in turn is the case if and only if ℓ is not a pth power mod N. (Here we have taken into account the explicit description of K_{(0)} provided by Proposition 5.4.) Thus we proved part (iv.i).

Since the vector (1, 0) spans the subspace M^{0}(\overline{Q}_{p}) of V, the space of I_{p}-inertial coinvariants is spanned by the image of the vector (0, 1). The Frobenius element Frob_{p} acts non-trivially on the image of this vector if and only if the prime over p is not split in the extension K_{(0)} of Q(ζ_{p}), which is the case if and only if p is not a pth power mod N. This proves part (iv.ii). □

As we will see below, for p ≥ 3, the rank g_{p} + 1 of T/p over F_{p} is no longer explained by an abelian extension of number fields (and hence by a single class group), as it is in the case p = 2, but by certain more complicated solvable extensions. However, the question of whether or not g_{p} = 1 is somewhat tractable. Indeed, from Corollary 1.6 we deduce the following criterion.

Lemma 5.6 The rank g_{p} of the parabolic Hecke algebra T^{0}/p over F_{p} is greater than one (equivalently, T^{0} ≠ Z_{p}) if and only if there exists a (V, L, ρ) in Def(F_{p}[X]/X^{3}) whose traces generate F_{p}[X]/X^{3}.

In order to apply this lemma, we now assume that A = F_{p}[X]/X^{3}, so that (V, L, ρ) lies in Def(F_{p}[X]/X^{3}). As always, we assume that the basis of V is chosen so as to satisfy the conditions of Lemma 5.1. We let ρ_{n} denote the composition of ρ with the natural surjection GL_{2}(F_{p}[X]/X^{3}) → GL_{2}(F_{p}[X]/X^{n}), for n ≤ 3. Requiring the traces of ρ to generate F_{p}[X]/X^{3} is equivalent to requiring the traces of ρ_{2} to generate F_{p}[X]/X^{2}, which in turn is equivalent is to requiring that ρ_{2} be a non-trivial deformation. We assume this to be the case. Also, we let F_{n} denote the extension cut out by the kernel of ρ_{n}. Thus F_{1} = Q(ζ_{p}), and F_{3} = F.

Since we are assuming that ρ_{2} is non-trivial, Proposition 5.5 shows that p divides the numerator of (N − 1)/12, and that F_{2} is equal to the compositum of the class fields K_{(i)} (for i = 1, 0, −1).

Lemma 5.7 (i) We have F_{2} = F^{ab}, and Gal(F_{2}/F_{1}) ≅ (Z/p)^{2} (respectively (Z/p)^{3}) if p = 3 (respectively p ≥ 5).

(ii) F/F_{2} is unramified at N.

Proof. Since p ≥ 3, we see that G′ has exponent p. Lemma 5.3 and equation (14) then imply that F^{ab} ⊂ K_{(1)}K_{(0)}K_{(−1)} = F_{2}. Certainly F_{2} ⊂ F^{ab}, and so we have the equality stated in (i). The claims regarding Gal(F_{2}/F_{1}) follow from parts (ii) and (iii) of Proposition 5.5.

Part (ii) follows from the Lemma 5.2 and the fact that F_{2}/F_{1} is ramified at N. □

We now separate our analysis into two cases: p = 3, and p ≥ 5.

5.1 p = 3

Throughout this subsection we set p = 3.

Lemma 5.8 The extension F/F_{2} is unramified everywhere and has degree exactly three.

Proof. The image of ρ_{|G_{Q(\sqrt{−3})}} is a subgroup of G′ = ker(GL_{2}(F_{3}[X]/X^{3}) → GL_{2}(F_{3})) whose image in GL_{2}(F_{3}[X]/X^{2}) is isomorphic to (Z/3)^{2}, by Lemma 5.7. Thus the commutator subgroup of the image of ρ_{|G_{Q(\sqrt{−3})}} is either trivial or cyclic of order three. Thus the extension F/F_{2} has degree at most three.

Consider the representation ρ_{2}, which factors through Gal(F_{2}/Q). By assumption this yields a non-trivial element of Def(F_{3}[X]/X^{2}). Part (ii) of Lemma 5.5 thus shows that the image under ρ_{2} of the element of order three coming from the \overline{χ}_{p}^{1} extension K_{(1)} = Q(\sqrt{−3}, \sqrt[3]{N}) of Q(\sqrt{−3}) must be of the form

(\begin{matrix}1 & −rX \\ rX & 1 \end{matrix}) ,

and that the image under ρ_{2} of the element of order three coming from the \overline{χ}_{p}^{0} extension K_{(0)} of Q(\sqrt{−3}) is of the form

(\begin{matrix}1 + rX & 0 \\ 0 & 1 − rX \end{matrix}) ,

for some r ∈ F_{3}^{×}. Lifting these two elements (in any way) to GL_{2}(F_{3}[X]/X^{3}) and taking their commutator, we produce a new element in Gal(F/Q) which has a lower left-hand entry equal to r^{2}X^{2} = X^{2}. This element cannot be in the decomposition group at 3 because it doesn’t preserve M^{0}(\overline{Q}_{3}), which is generated by (1, 0). Thus F/F_{2} has order exactly three and is unramified at all primes above three. Part (ii) of Lemma 5.7 shows that the extension F/F_{2} is also unramified at all primes above N, and the lemma is proved. □

Let K = Q(\sqrt[3]{N}), and as above write K_{(1)} = K^{gal} = K(\sqrt{−3}). The extension F/K_{(1)} has degree 9, and Gal(F/K_{(1)}) = (Z/3Z)^{2}. Moreover, F/K_{(1)} is unramified everywhere. The following lemma shows that the existence of such an extension F is sufficient for the construction of a deformation ρ of the type considered here. This completes the proof of part one of each of Theorems 1.2 and 1.3.

Lemma 5.9 If N ≡ 1 mod 9, then the class group of K_{(1)} = Q(\sqrt{−3}, \sqrt[3]{N}) has 3-rank greater than or equal (equivalently, equal) to two if and only if there exists a surjection R → F_{3}[X]/X^{3}; the kernel of the corresponding deformation ρ : G_{Q} → GL_{2}(F_{p}[X]/X^{3}) then cuts out the (3, 3) unramified class field F of K_{(1)}.

Proof. The preceding discussion establishes the “if” claim, and so it suffices to prove the “only if” claim. Genus theory and a consideration of the ambiguous class predicts that the 3-rank of the class group of K_{(1)} is either one or two, and hence by assumption this rank is exactly two (see for example [4]). We let F denote the corresponding unramified (3, 3)- extension of K_{(1)}, and (as above) let F^{ab} denote the unique subextension of F abelian over Q(\sqrt{−3}). It is easily checked that F^{ab} is in fact the maximal abelian 3-power extension of Q(\sqrt{−3}) that is unramified over K_{(1)}, and that F^{ab} = K_{(1)}K_{(0)}.

Proposition 5.5 yields a Galois representation Gal(F^{ab}/Q) → GL_{2}(F_{3}[X]/X^{2}), while lemma 5.10 below shows that Gal(F/Q(\sqrt{−3})) is the unique non-abelian group of order 27 and of exponent three. It is then easy to see that one can lift the representation of Gal(F^{ab}/Q) to a representation ρ : Gal(F/Q) → GL_{2}(F_{3}[X]/X^{3}). Furthermore one checks that for any such lift, the image of I_{N} fixes an appropriate line.

To show that we have constructed an element of Def(F_{3}[X]/X^{3}), as required, it remains to show that this representation extends to a finite flat group scheme at 3, For this, it suffices to work over the maximal unramified extension of Q_{3}. Since F/Q(\sqrt{−3}) is unramified at 3 (because N ≡ 1 mod 9), the representation ρ|_{Q_{3}^{ur}} factors through a group of order two, and explicitly prolongs to a product of trivial and multiplicative group schemes. Thus ρ is indeed finite at the prime 3. □

Lemma 5.10 The Galois group Gal(F/Q(\sqrt{−3})) is the (unique up to isomorphism) nonabelian group of order 27 and of exponent three.

Proof. Let Γ = Gal(K_{(1)}/Q(\sqrt{−3})) = ⟨γ⟩. The 3-class group H of K_{(1)} is naturally a Z_{3}[Γ]-module. From class field theory we have that H/(γ −1)H is isomorphic to the Galois group over K_{(1)} of the maximal abelian 3-extension of Q(\sqrt{−3}) that is unramified over K_{(1)}; that is, to Gal(F^{ab}/K_{(1)}), a cyclic group of order 3. Thus by Nakayama’s lemma H is a cyclic Z_{3}[Γ]-module. By class field theory, the quotient H/3 is isomorphic to Gal(F/K_{(1)}).

Note that Gal(F/Q(\sqrt{−3})) sits in the exact sequence:

0 → Gal(F/K_{(1)}) ⟶ Gal(F/Q(\sqrt{−3})) ⟶ Gal(K_{(1)}/Q(\sqrt{−3})) ⟶ 0,

which is an extension of Γ ≅ Z/3Z by H/3 ≅ (Z/3Z)^{2}. The action via conjugation of Γ on H/3 is non-trivial, since otherwise H could not be cyclic as a Γ-module. Already this shows that Gal(F/Q(\sqrt{−3})) is one of the two non-abelian groups of order 27. To pin down the group precisely, we must show that it has exponent three. For this, it suffices to find a splitting of the above exact sequence (a section from Γ = Gal(K_{(1)}/Q(\sqrt{−3})) back to Gal(F/Q(\sqrt{−3}))). Since the inertia group above N in Gal(F/Q(\sqrt{−3})) has order exactly three, and maps isomorphically to Γ, the required splitting exists. □

The final result of this section provides a relation between the rank of the 3-class group of K_{(1)} and the power of 3 dividing the class number of K.

Lemma 5.11 The 3-class group of K_{(1)} = Q(\sqrt{−3}, \sqrt[3]{N}) has three rank two if the 3-class group of K = Q(\sqrt[3]{N}) (which is cyclic) is divisible by nine.

Proof. One has a class number relation between K and K_{(1)} given by h_{K_{(1)}} = h_{K}^{2}/3 · q, where q is the index of the units in K_{(1)} coming from K, K^{γ}, and Q(\sqrt{−3}) inside the full unit group. This was initially proven using analytic means by Scholz [15] (for a more recent algebraic proof, see [5]). Here, as above, γ denotes a generator of the cyclic group Γ = Gal(K_{(1)}/Q(\sqrt{−3})). If 9|h_{K}, then 27|h_{K_{(1)}}. Recall from the proof of the previous lemma that the 3-part H of the class group of K_{(1)} is a cyclic Z_{3}[Γ]-module, and satisfies the condition that H/(γ − 1)H is cyclic of order 3.

Now Z_{3}[Γ] admits no quotients H′ that are cyclic groups of order 27 with the property that H′/(γ − 1)H′ is of order 3. It follows that if H is of order divisible by 27, then it must be non-cyclic, as claimed. □

We conjecture that the converse to the preceding lemma is also true. To prove this, it would suffice to show that whenever 3∥h_{K}, the unit index q is always equal to one. We have verified this for all primes less than 50,000 for which 3∥h_{K}.

5.2 p ≥ 5

Throughout this section we assume that p ≥ 5, and that we are given a deformation to F_{p}[X]/X^{3} as in the discussion following Lemma 5.6. Proposition 5.5 and Lemma 5.7 together show that F_{2} = K_{(1)}K_{(0)}K_{(−1)}, that Gal(F_{2}/F_{1}) = (Z/p)^{3}, and that F_{2} = F^{ab}. We see that F_{2}/F_{1} is unramified at p if and only if N ≡ 1 mod p^{2}.

It follows from our determination of F_{2} that Gal(F/F_{1}) is the full kernel of the map from SL_{2}(F_{p}[x]/x^{3}) to SL_{2}(F_{p}), since all the elements of

Ker(SL_{2}(F_{p}[x]/x^{3}) → SL_{2}(F_{p}[x]/x^{2}))

are generated by commutators of lifts of elements of Ker(SL_{2}(F_{p}[x]/x^{2}) → SL_{2}(F_{p})).

Lemma 5.12 If E is a degree p Galois extension of F_{2} inside F_{3} on which the matrix

(\begin{matrix}1 + x^{2} & 0 \\ 0 & 1 − x^{2} \end{matrix})

acts non-trivially, then E/F_{2} is everywhere unramified.

Proof. Part (ii) of Lemma 5.7 shows that this extension is unramified at primes above N. To see that it is unramified at primes above p, it suffices to note that the matrix

(\begin{matrix}1 + x^{2} & 0 \\ 0 & 1 − x^{2} \end{matrix}) does not fix the vector (1, 0) (which spans M^{0}(\overline{Q}_{p})). □

Let K = Q(N^{1/p}) and L = K^{gal} = K(ζ_{p}) = K_{(1)}.

Lemma 5.13 The Hilbert class field of K has p-rank at least two.

Proof. Let us first consider the extension Gal(F/K). One sees that Gal(F/K)^{ab} ≅ (Z/pZ)^{2} × (Z/p)^{×} is explicitly generated by the images of

(\begin{matrix}1 + x^{k} & 0 \\ 0 & (1 + x^{k})^{−1} \end{matrix}) ,

for k = 1, 2, together with the image of Δ. We let H be the (p, p)-extension of K contained in F that is fixed by Δ. We will show that H is unramified over K.

We may write H as a compositum H = H_{1}H_{2}, where for each of k = 1, 2, we let

H_{k} denote a p-extension of K contained in F, on which the matrix (\begin{matrix}1 + x^{k} & 0 \\ 0 & (1 + x^{k})^{−1} \end{matrix})

acts non-trivially. If we let ζ_{N}^{+} denote an element of Q(ζ_{N}) that generates the degree p subextension over Q (so that K_{(0)} = Q(ζ_{p}, ζ_{N}^{+})), then we may take H_{1} to be K(ζ_{N}^{+}), which is clearly unramified everywhere over K (it is the genus field). We will show that H_{2} is is also unramified everywhere over K. Lemma 5.2 takes care of the primes above N, and so it remains to treat the primes above p.

We begin by proving that H_{2}(ζ_{p})/L is unramified. Lemma 5.12 shows that the extension H_{2} · F_{2}/F_{2} is unramified. Since F_{2}/L is unramified, it follows that H_{2}(ζ_{p})/L is unramified, as claimed. We now use the fact that H_{2}(ζ_{p})/L is unramified to show that H_{2}/K is unramified. We consider two cases. Suppose first that p∥N − 1. Then K is totally ramified at p, and thus if H_{2}/K is ramified we deduce that since H_{2} is Galois over K, e_{p}(H_{2}) = p^{2}, contradicting the fact that H_{2}(ζ_{p})/L is unramified. If instead N ≡ 1 mod p^{2}, then things are even easier: If H_{2}/K is ramified at at least one prime 𝔭 above p, then again using the fact that H_{2}/K is Galois we deduce that p|e_{𝔭}(H_{2}). Yet p is tamely ramified in L and therefore also in H_{2}(ζ_{p}). Thus H_{2}/K is unramified everywhere, and K has p-rank at least two. □

This completes the proof of parts two of Theorem 1.2 and Theorem 1.3. We expect (based on the numerical evidence) that the condition that the class group of K has p-rank two is equivalent to the existence of an appropriate group scheme, and thus to g_{p} > 1. Part of this could perhaps be proved by more sophisticated versions of Lemmas 5.9, 5.10, and 5.11.

6 Examples

The first example in Mazur’s table [7] where e_{2} > 1 occurs when N = 41. The class group of Q(\sqrt{−41}) is Z/8Z. Thus one has e_{2} = 3. Using gp one finds that that the class group of Q(\sqrt{−21929}) is Z/256Z. Indepedendly, using William Stein’s programmes, one finds that e_{2} = 127 for N = 21929. In Mazur’s table, e_{3} always equals 1 or 2. One has to go quite some distance before finding an example where e_{3} > 2. For N = 2143, however, one has e_{3} = 3. This is related to the fact that 2143 is the smallest prime congruent to 1 mod 9 such that the class group of the corresponding extension K_{(0)} of Q(\sqrt{−3}) (in the terminology of Proposition 5.4) has an element of order 9. The corresponding class field contributes to the maximal unramified solvable extension of K = Q(\sqrt[3]{2143}). Finally, let us note that when p = 3, Lemmas 5.9 and 5.11 show that the value of g_{p} is related to the size of the 3-power part of the class group of Q(\sqrt[3]{N}), whereas for p ≥ 5, Lemma 5.13 shows that this value is related to the p-rank of the class group of Q(\sqrt[p]{N}). As an illustration, when N = 4261, one computes that the class group of Q(\sqrt[5]{4261}) is Z/25Z. However, since the 5-rank of Z/25Z is one, it follows that e_{5} = 1.

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