THE FLAT DEFORMATION FUNCTOR

BRIAN CONRAD

Introduction

Let E_{/Q} be a semistable elliptic curve and p a prime. In the proof that E is modular, properties of the local representation

ρ_{E,p}|_{D_{p}} : D_{p} = Gal(\overline{Q}_{p}/Q_{p}) → Aut(Ta_{p}(E(\overline{Q}_{p}))) ≃ GL_{2}(Z_{p})

play an essential role. If E has either ordinary or multiplicative reduction at p, then one can describe the essential properties of ρ_{E,p}|_{D_{p}} quite explicitly in representation-theoretic terms. However, if E has supersingular reduction at p, then the situation is more subtle. More precisely, the residual representation \overline{ρ}_{E,p}|_{D_{p}} = ρ_{E,p}|_{D_{p}} mod p is absolutely irreducible (which already distinguishes this case from the other two cases) and for all n ≥ 1, the finite discrete D_{p}-module ρ_{E,p}|_{D_{p}} mod p^{n} has the property that the corresponding Q_{p}-group scheme arises as the generic fiber of a finite flat Z_{p}-group scheme (necessarily commutative, with order p^{2n}).

In order to study the supersingular case, Wiles considers a ‘flat’ deformation problem which makes critical use of the theory of finite flat group schemes. The representability of the deformation functor in this case, as well as the (abstract) ‘computation’ of the corresponding representing ring (when p ≠ 2), were worked out by Ramakrishna in his thesis [26]. The central tool he uses is Fontaine’s work on finite flat group schemes [12] (made more ‘explicit’ by the work of Fontaine-Laffaille [13, §9]). For p ≠ 2, Fontaine constructed an equivalence of categories between finite flat commutative Z_{p}-group schemes with p-power order and a category consisting of finite-length Z_{p}-modules with certain extra structures. This allows one to reformulate the study of certain group schemes over Z_{p} into the study of certain modules. Since it is far easier to manipulate and construct modules than it is to manipulate and construct group schemes, this reformulation is very useful.

We begin with an explanation of why ‘flat’ representations are a natural thing to consider and outline the role they play in Wiles’ proof. After relating flat representations to local Galois cohomology, we formulate and partially prove the main results (due to Ramakrishna). We then review certain fundamental results due to Fontaine [12]. With these in our toolbox, we complete the proof of Ramakrishna’s theorem concerning the ‘explicit’ structure of the deformation ring which represents the ‘flat deformation functor’. This yields a theorem about Galois cohomology; we also prove a similar cohomological result in the residually split ordinary case (where the deformation functor is not known to be representable). Throughout, we assume a familiarity with the basic notions in the deformation theory of Galois representations, as given in [22], and the theory of finite flat commutative group schemes, as given in [35] [Note to editors: I make use of connected-étale sequence and p-divisible groups in some of the proofs; should I say something about this point in an appendix, as I presume [35] won’t discuss these points?]. Our discussion provides a ‘concrete’ application of these ideas.

Much of what we say over Z_{p}, Q_{p}, and F_{p} is valid somewhat more generally, but we will stick with the more concrete setting when it is convenient to do so. Also, though we will frequently use Néron models of elliptic curves, we do not use anything from the theory of Néron models; all we need is that an integral Weierstrass model in P_{Z_{p}}^{2} with a smooth closed fiber (and therefore a smooth generic fiber) admits the structure of a group scheme extending the canonical group scheme structure on the generic fiber. This can be proven directly [33, Ch IV, Remark 5.4.1].

§0. Notation

We fix algebraic closures \overline{Q} and \overline{Q}_{ℓ} (for all places ℓ of Q), as well as embeddings ι_{ℓ} : \overline{Q} ↪ \overline{Q}_{ℓ}. For any field K with a fixed separable closure K_{s}, let G_{K} = Gal(K_{s}/K). Let D_{ℓ} ⊆ G_{Q} denote the decomposition group at ℓ arising from ι_{ℓ}. When ℓ ≠ ∞, let I_{ℓ} ⊆ D_{ℓ} denote the inertia subgroup. All finite extensions of Q_{ℓ} are understood to lie inside of \overline{Q}_{ℓ} and for ℓ ≠ ∞, all finite fields of characteristic ℓ are understood to be subfields of the residue field \overline{F}_{ℓ} naturally attached to \overline{Q}_{ℓ}. These choices are not important, but making them at the start simplifies the exposition below.

For Σ a finite set of places of Q, let Q_{Σ} ⊆ \overline{Q} denote the maximal subextension unramified outside of Σ, with G_{Σ} = Gal(Q_{Σ}/Q) the corresponding quotient of G_{Q}. Finally, when a prime p is fixed under discussion, let ϵ : D_{p} → Z_{p}^{×} denote the p-adic cyclotomic character and let

ω = ϵ mod p : D_{p} → F_{p}^{×}

denote its reduction modulo p. We allow for the possibility p = 2 unless we explicitly say otherwise. Finally, we adopt a common abuse of terminology: when we say ‘finite flat S-group scheme,’ it is always understood that we mean ‘finite flat commutative S-group scheme’ (of finite presentation over S — but our bases are always noetherian). We sometimes omit reference to S when the context makes it clear. When S = Spec(L) with L a field, we will abbreviate this to ‘finite’ L-group scheme. The notation X_{/S} denotes an S-scheme X. If S = Spec(A), then this is usually written X_{/A} instead. Also, for certain standard group schemes G over Z or F_{p}, such as G_{m}, α_{p}, and μ_{p}, we write G_{/T} to denote the base extension to T, and we sometimes omit reference to T when the context makes it clear.

§1. Motivation and Flat Representations

Fix a semistable elliptic curve E_{/Q} and a prime p. Suppose p is odd, the restriction of \overline{ρ}_{E,p} to Gal(\overline{Q}/Q(\sqrt{(\frac{−1}{p})p})) is absolutely irreducible, and \overline{ρ}_{E,p} is modular (e.g., p = 3 and \overline{ρ}_{E,3}|_{Gal(\overline{Q}/Q(\sqrt{−3}))} is absolutely irreducible [39, pp. 541-542]). Wiles’ proof provides an inductive procedure to show that for successively larger and larger n, the representations ρ_{E,p} mod p^{n} are ‘modular’ (in an appropriate sense). Moreover, sufficiently tight control is kept on the level (as well as other properties) of the modular forms which are constructed so that one can ‘pass to the limit’ and conclude that ρ_{E,p} is ‘modular’. In order to carry out this procedure, there is an extremely delicate balancing act to handle, with (abstract) deformation rings on one side and (concrete) Hecke rings on the other side. The latter provide a link to modular forms and representations ‘coming from modular forms,’ whereas the former provide a link to the particular representation of interest, ρ_{E,p}, which we want to prove ‘comes from a modular form.’ The relation between the two different types of rings — leading to the proof that they’re isomorphic — is supplied by a numerical criteria from commutative algebra [29]. The hard part is to check that this numerical criteria actually can be applied! In order to do this, one has to prove highly non-obvious theorems about the commutative algebra properties of the rings in question. This requires a very detailed understanding of both the deformation rings and the Hecke rings.

Refined knowledge about Hecke rings can be obtained via methods of algebraic geometry [37], essentially because Hecke rings act on algebro-geometric objects such as Jacobians of modular curves. However, refined knowledge about deformation rings is supplied by completely different methods, namely techniques from Galois cohomology. The problem of estimating the sizes of certain ‘global’ Galois cohomology groups is the central issue, and is the means by which the ‘flat deformation functor’ enters into Wiles’ proof. The cohomology groups whose order must be bounded are of the form H_{D}^{1}(G_{Σ}, ad(ρ)), where ρ : G_{Σ} → GL_{2}(A) is a continuous representation, with A a suitable local artin ring having finite residue field k of characteristic p. Here, D is a well-chosen G_{Σ}-deformation problem for \overline{ρ} = ρ mod 𝔪_{A}. In practice, \overline{ρ} is essentially \overline{ρ}_{E,p}, up to extension of scalars on the residue field k. The deformation problem D imposes ‘local’ conditions on G_{Σ}-deformations of \overline{ρ} at the finite set of places Σ, including p and ∞. In the application to elliptic curves, the other places in Σ consist of the places where E_{/Q} has bad reduction (though in his arguments, Wiles also needs to consider other sets Σ′ which include auxiliary primes used in the study of the corresponding ‘minimal’ deformation problem). The study of ‘type D’ deformations of \overline{ρ} translates cohomologically into studying elements of H^{1}(G_{Σ}, ad(ρ)) whose local restrictions at places in Σ satisfy certain conditions; these special elements constitute the A-submodule H_{D}^{1}(G_{Σ}, ad(ρ)).

As is explained in [38] and [39, Prop 1.6], for any finite discrete G_{Σ}-module X with p-power order (e.g., X = E[p^{n}](\overline{Q})), the problem of estimating the size of H_{D}^{1}(G_{Σ}, X) reduces essentially to the problem of estimating the size of certain D_{ℓ}-cohomology groups for the places ℓ ∈ Σ. We say ‘essentially’ because there is another ‘dual’ group H_{D^{∗}}^{1}(G_{Σ}, X^{∗}) that must be considered, too. But at a critical stage in what Wiles calls the ‘minimal deformation problem’, this extra factor is not hard to handle (this essentially follows from the injectivity of ε_{Q} in [39, (3.1)]).

Among the sizes of the various D_{ℓ}-cohomology groups to be estimated, the only one which turns out to present serious difficulties is the case ℓ = p. Wiles considers two types of D_{p}-cohomology groups, both modelled on semistability of E at p and based on an attempt to capture the essential properties of ρ_{E,p}|_{D_{p}} in terms of deformation theory and Galois cohomology. Particularly in view of recent work of Diamond [10], these local deformation-theoretic/Galois-cohomological conditions at p are the central reason that semistability conditions enter into Wiles’ proof. Any attempt to prove the full Taniyama-Shimura Conjecture will almost certainly have to first involve the formulation of local deformation conditions at p which accurately describe cases with additive reduction.

Before precisely defining the D_{p}-cohomology group and the deformation functor that will concern us, we will first prove some preliminary results. We now allow p = 2. In order to state the first result, we recall the notion of a fundamental character (this will occur repeatedly below, albeit in a limited context). The tame inertia group I_{p}^{t} of Q_{p} admits a canonical isomorphism

I_{p}^{t} ≃ \underleftarrow{lim} F_{p^{n}}^{×},

where the (surjective) transition maps are given by taking norms (recall our convention about finite fields in the Notation section above). This follows readily from the structure theorem for tamely ramified finite extensions of local fields. See [31, §1] for more details and generality. The projection

ψ_{n} : I_{p} ↠ I_{p}^{t} ↠ F_{p^{n}}^{×}

is called the fundamental character of level n (for Q_{p}). Explicitly, if π ∈ Z_{p} is a uniformizer and α_{n} ∈ \overline{Q}_{p}^{×} satisfies α_{n}^{p^{n}−1} = π, then for g ∈ I_{p},

ψ_{n}(g) = \frac{g(α_{n})}{α_{n}}.

For example, ψ_{1} = ω|_{I_{p}} (check!). We remind the reader that the notion of a fundamental character is not functorial. That is, if U is an open subgroup of D_{p} with fixed field K, then we can define fundamental characters for K in a similar manner, but ψ_{n}|_{U} is usually not a fundamental character for K (even if K has residue field F_{p}).

Upon choosing a basis for F_{p^{2}} over F_{p}, we get an injection of groups

j_{2} : F_{p^{2}}^{×} ↪ GL_{2}(F_{p}).

Observe that as a 2-dimensional F_{p}-representation for I_{p}, j_{2} ∘ ψ_{2} is semisimple and therefore irreducible (as the character ψ_{2} does not take all of its values in F_{p}^{×}). This representation remains irreducible under any extension of scalars of odd degree over F_{p}. However, once we extend scalars so that F_{p^{2}} lies in the field of scalars, the representation decomposes into a direct sum of the two distinct characters, ψ_{2} and ψ_{2}^{p} (viewed as taking values in the field of scalars).

Theorem 1.1 Let E_{/Q_{p}} be a semistable elliptic curve. If the reduction type is either ordinary or multiplicative, then

ρ_{E,p} ≃ (\begin{matrix}ϵχ & ∗ \\ 0 & χ^{−1} \end{matrix}) ,

with χ : D_{p} → Z_{p}^{×} a continuous unramified character. In particular,

\overline{ρ}_{E,p} ≃ (\begin{matrix}ω\overline{χ} & ∗ \\ 0 & \overline{χ}^{−1} \end{matrix}) ,

with \overline{χ} : D_{p} → F_{p}^{×} a continuous unramified character.

If the reduction type is good supersingular, then the 2-dimensional F_{p}-representation

\overline{ρ}_{E,p}|_{I_{p}} : I_{p} → Aut(E[p](\overline{Q}_{p})) ≃ GL_{2}(F_{p})

is isomorphic to the fundamental character of level 2. Moreover, \overline{ρ}_{E,p} is absolutely irreducible and Q_{p}⊗_{Z_{p}}ρ_{E,p} is irreducible.

Remarks. The irreducibility features of the supersingular case are quite a contrast with the relatively simple (and ‘reducible’) representation-theoretic description of the other cases. The supersingular case therefore requires a completely different treatment in Wiles’ proof, and that is the purpose of the ‘flat’ deformation functor, as we shall see. The convenience of the general description of the representations in Theorem 1.1 (when combined with Theorem 1.2) is why semistable cases are much more accessible (at present) than cases with additive reduction.

The results in Theorem 1.1 for \overline{ρ}_{E,p} in cases of good reduction were first proven by Serre [31, §1.11, Prop 11, 12] using formal groups. Since group schemes will be fundamental in our work below, we give a different proof that uses the theory of group schemes, particularly general theorems of Raynaud [27, §3] which were motivated by (but do not logically depend upon) the results of Serre.

We will use Raynaud’s results to prove (in Theorem 1.8) a generalization of the good reduction cases under the hypothesis p ≠ 2; however, the case of elliptic curves merits special treatment since in the ‘reducible’ cases we can interpret the unramified quotient χ^{−1} quite concretely, as is shown in the proof below.

Proof. First we consider good ordinary and bad multiplicative reduction. In both cases, we wish to produce an unramified rank 1 quotient of ρ_{E,p}. If we call the character on such a quotient χ^{−1} and recall that det ρ_{E,p} = ϵ, the rest is then clear. Suppose the reduction is bad multiplicative. In this case, the theory of Tate models supplies us with a D_{p}-module isomorphism E(\overline{Q}_{p}) ≃ (\overline{Q}_{p}^{×}/q^{Z})(χ) for some q ∈ pZ_{p} and some continuous unramified character χ : D_{p} → ⟨−1⟩ [33, Ch V, Thm 5.3]. Choosing consistent p^{n}th roots of 1 in \overline{Q}_{p}^{×} then gives rise to a rank 1 subrepresentation inside of ρ_{E,p} and the quotient by this is visibly free of rank 1 and unramified.

Now suppose the reduction is good ordinary. Observe that the Néron model E_{/Z_{p}} of E_{/Q_{p}} is an elliptic curve over Z_{p} whose p-divisible group has a closed fiber with non-trivial connected and étale factors (this is essentially the definition of what it means for the closed fiber \overline{E}_{/F_{p}} of the Néron model to be an ordinary elliptic curve). Thus, the p-divisible group of E_{/Z_{p}} has non-trivial connected and étale factors (since formation of the connected-étale sequence of a finite flat group scheme over a henselian local base is compatible with base change by a local map of base rings, such as Z_{p} → F_{p}). Passing to the generic fiber over Q_{p} and then to \overline{Q}_{p}-points, the non-trivial connected-étale sequence over Z_{p} gives rise to the desired decomposition of ρ_{E,p}, since a finite étale cover of Spec(Z_{p}) has a generic fiber Spec(L), with L a finite product of finite unramified extensions of Q_{p} (and base change preserves the exactness property of a short exact sequence of finite flat group schemes and thus of p-divisible groups). In particular, we can interpret the unramified quotient of ρ_{E,p} = Ta_{p}(E(\overline{Q}_{p})) as precisely Ta_{p}(\overline{E}(\overline{F}_{p})), via the reduction map on points; the theory of p-divisible groups, particularly the existence of a functorial connected-étale sequence, is what ensures that the natural map between these two Tate modules is surjective (and in particular is not zero).

Lastly, consider the case in which E has good supersingular reduction at p. Assume that we have established the desired description of \overline{ρ}_{E,p}|_{I_{p}} in terms of the fundamental character of level 2. Let us see how to prove the other assertions. First of all, \overline{ρ}_{E,p} must be absolutely irreducible. This is because if it has a stable line over \overline{F}_{p}, then the eigencharacter along this line gives a character on D_{p} whose restriction to I_{p} is either ψ_{2} or ψ_{2}^{p} (the two characters occuring in the decomposition of the semisimple \overline{F}_{p} ⊗_{F_{p}} \overline{ρ}_{E,p}|_{I_{p}}). But conjugation by a Frobenius element on I_{p} interchanges ψ_{2} and ψ_{2}^{p} (check!), so neither ψ_{2} nor ψ_{2}^{p} extends to an \overline{F}_{p}^{×}-valued character on D_{p}. This proves that \overline{ρ}_{E,p} is (absolutely) irreducible. If Q_{p} ⊗_{Z_{p}} ρ_{E,p} is reducible, then we can scale a generator of a D_{p}-stable line so that it is part of a basis for the lattice ρ_{E,p}. Reducing this mod p then would contradict the irreducibility of \overline{ρ}_{E,p}.

It remains to prove that \overline{ρ}_{E,p}|_{I_{p}} has the asserted form. We postpone the proof of this until after Theorem 1.7, as it will require some results from the theory of finite flat group schemes. The careful reader can check that this does not lead to circular reasoning.

■

Theorem 1.1 shows that from the point of view of Galois representations, it is natural to combine the cases of ordinary and multiplicative reduction into one case, called ‘ord’ by Wiles, and to treat the case of supersingular reduction separately; this latter case is called ‘fl’ by Wiles, as its study involves the theory of finite flat group schemes in order to circumvent the lack of upper triangular representations in the supersingular case. We want to formulate ‘abstract’ properties satisfied by the deformation ρ_{E,p} of \overline{ρ}_{E,p} (and require these for all deformations of \overline{ρ}_{E,p} that we study). The ‘ord’ condition has an obvious analogue for a representation into GL_{2}(R), where R is any topological Z_{p}-algebra. However, it is not clear how to formulate a representation-theoretic condition which both captures the supersingular Tate module representations and also makes sense in deformation theory. For example, ‘irreducibility’ of a GL_{2}(R)-representation is a bad notion when R is not a field. The correct condition is supplied by the theory of finite flat group schemes, and it is a very subtle condition from a representation-theoretic viewpoint.

It is worth pointing out at this stage that the finite flat group scheme techniques we will use to study the supersingular case are also needed at a critical technical step in the ‘minimal’ deformation problem for some ‘ord’ cases (essentially because of [39, (3.1)]; also see Example 1.3(i) below). This point is explained in [9], and it shows that what we are about to do is not only needed in the supersingular cases [Note to the editors: please let me know if this issue is not explicitly addressed in [9], as I might forget to check; it is quite important, albeit in a technical way].

The result which makes finite flat group schemes a relevant notion for us is the following fundamental fact, whose most ‘natural’ proof requires a serious use of algebraic geometry to handle the case of torsion levels divisible by p, which is the case of interest to us here. In fact, Theorem 1.2 below was implicitly invoked in the proof of Theorem 1.1 when we used the theory of p-divisible groups to treat the ordinary reduction case. If in Theorem 1.2 one only cared about torsion levels which are prime to p, an argument using just the criteria of Néron-Ogg-Shafarevich (for elliptic curves) and Galois descent of rings [3, §6.2, Ex B] would suffice.

Theorem 1.2 Let E_{/Q_{p}} have good reduction. Then for all m ≠ 0, the Q_{p}-group scheme E[m] which is canonically attached to the D_{p}-representation E[m](\overline{Q}_{p}) is the generic fiber of a finite flat group scheme over Z_{p} (in fact, the m-torsion on the Néron model over Z_{p} is such a finite flat Z_{p}-group scheme).

Proof. We will first give a ’natural’ proof that uses general principles in algebraic geometry. At the end we will give an ad hoc proof which is not at all ‘natural’, but uses only the theory of elliptic curves over fields and basic facts from commutative algebra and the theory of schemes.

Let E_{/Z_{p}} denote the Néron model of E_{/Q_{p}}. This is an abelian scheme over Z_{p} with relative dimension 1. It suffices to prove that if A → Spec(R) is any abelian scheme of relative dimension d over an affine noetherian base (i.e., a proper smooth group scheme with d-dimensional connected fibers, so necessarily geometrically connected [15, IV, 4.5.13]), then the multiplication map m_{A} : A → A is finite and flat. Base extension by the identity section of A_{/R} then yields the desired conclusion that A[m] → Spec(R) is finite and flat. The general principle is to reduce to the case in which R is an algebraically closed field and then to appeal to the ‘classical’ theory of abelian varieties (or just elliptic curves). We give the argument in the case of arbitrary relative dimension for conceptual clarity. It should be remarked that the finiteness and flatness of m_{A} holds for abelian schemes A over an arbitrary base scheme, but the method of proof is to use a lot of technicalities with limit arguments and constructible sets to reduce to the case over an affine noetherian base. More precisely, one needs to use [15, IV, 4.6.5(i), 6.7.8(i), 8.6.3.1, 8.9.1(iii), 8.10.5(xii), 11.2.6(ii), 12.1.7(iv), 12.2.1(x)].

Back to the proof that m_{A} is finite and flat. Certainly m_{A} is proper. It is also quasi-finite, as can be checked on geometric fibers, in which case it is a standard result in the theory of abelian varieties over algebraically closed fields [23, p. 62] (or for elliptic curves in our setting of interest [32, Ch III, Prop 4.2(a)]). It follows that m_{A} : A → A is a quasi-finite and proper map between noetherian schemes, so by [15, IV, 8.11.1] it is necessarily finite. Let us sketch the idea behind the proof in [15] that for a morphism f : X → Y which is proper and quasi-finite, with Y locally noetherian, f is necessarily finite (one can avoid noetherian and even finite presentation hypotheses [15, IV, 18.12.4], but more work is then needed). The essential content of this statement is that f is affine. Even in the ‘concrete’ setting of an elliptic curve E over Z_{p} given by an explicit integral Weierstrass model in P_{Z_{p}}^{2}, it is not at all a priori obvious that the ‘multiplication by m’ map on E is actually affine (but we will see below that one can prove by elementary ad hoc means that E[m] is affine).

To prove that f : X → Y as above is finite, one uses limit arguments to reduce to the case where the base Y is a local noetherian ring B, and then by faithfully flat descent one reduces to the case where B is complete. There is a structure theorem for quasi-finite separated morphisms to such a base [15, IV, 6.2.6]. This structure theorem says that X breaks up into a disjoint union of a finite B-scheme X_{0} and another open and closed subscheme X_{1} which has empty closed fiber. But in our setting, X_{1} → Spec(B) is proper and thus closed, so the image of X_{1} is a closed subset of the local scheme Spec(B), so the image must contain the closed point if X_{1} is non-empty. Since X_{1} has empty closed fiber, we conclude that X_{1} is empty (!), so X = X_{0} is indeed finite (and in particular affine) over Spec(B).

For flatness, we observe that since A → Spec(R) is flat and of finite presentation, by the fiber-by-fiber criteria for flatness [15, IV, 11.3.10(2)] it suffices to check the flatness of m_{A} : A → A along fibers over Spec(R), so we may suppose that R is a field, even algebraically closed (by faithful flatness of field extensions). By the classical theory of abelian varieties [23, p. 62], m_{A} is a map between two smooth, irreducible varieties of the same dimension d over an algebraically closed field, with fibers over all closed points of dimension d − d = 0. It is then a consequence of the fundamental ‘local criteria for flatness’ that such a map is necessarily flat [20, Cor, Thm 23.1].

Now we sketch an ‘elementary’ proof that E[m] is the generic fiber of a finite flat Z_{p}-group scheme (this proof will work for any elliptic curve over the fraction field of a discrete valuation ring). Let E ↪ P_{Z_{p}}^{2} denote an integral ‘good’ Weierstrass model for E_{/Q_{p}}. One can check without appealing to the general theory of Néron models that E has the structure of a group scheme over Z_{p} which extends that on the generic fiber E_{/Q_{p}} [33, Ch IV, Remark 5.4.1]. It remains to check that the Z_{p}-group scheme E[m] is finite and flat over Z_{p}. By the classical theory of elliptic curves over fields, we know that the generic and closed fibers are both finite schemes (over Q_{p} and F_{p} respecively) with the same rank, namely m^{2}. Note that the closed fiber might not be reduced!

Assume for the moment that we know E[m] is an affine scheme, say of the form Spec(R) for a Z_{p}-algebra R. Since Spec(R) → Spec(Z_{p}) is universally closed, by [1, Exer. 35] (this is the essential commutative algebra input) we see that R is integral over Z_{p}. As it is of finite type by construction, R is finite over Z_{p}. The equality of the generic and closed fiber dimensions then yields freeness over Z_{p} (by the structure theorem for finite Z_{p}-modules), and hence flatness.

It remains to show that E[m] is an affine scheme. Since it is at least a closed subscheme of P_{Z_{p}}^{2}, if we can find an open affine in P_{Z_{p}}^{2} which contains E[m], then we will have exhibited E[m] as a closed subscheme of an affine scheme, and so E[m] must be affine. Since the closed fiber of E[m] is finite over F_{p}, we can find some homogenous polynomial \overline{f} ∈ F_{p}[X, Y, Z] such that V (\overline{f}) ⊆ P_{F_{p}}^{2} does not meet the closed fiber of E[m]. Choose a lifting of \overline{f} to a homogenous polynomial f ∈ Z_{p}[X, Y, Z]. Since V (f) and E[m] are closed subschemes of P_{Z_{p}}^{2}, if their intersection is non-empty, it contains a closed point x. Such a point maps to the closed point of Spec(Z_{p}) by the properness of P_{Z_{p}}^{2} → Spec(Z_{p}). Our assumption on V (\overline{f}) therefore rules out the existence of such a point x, so E[m] is a closed subscheme of the open affine scheme Spec(Z_{p}[X, Y, Z]_{(f)}) = P_{Z_{p}}^{2} \ V (f) inside of P_{Z_{p}}^{2}.

■

We note in passing that it is not enough in the proof of Theorem 1.2 to have a proper, smooth, integral model for E_{/Q_{p}} (e.g., a ‘good’ projective integral Weierstrass model); it is critical to know that such a model can be chosen that is a group scheme (compatible with the group scheme structure on the generic fiber).

The converse to Theorem 1.2 is also true in a strong sense, though we will not need it in any of our proofs. More precisely, if E[p^{r}] is the generic fiber of a finite flat group scheme over Z_{p} for all r ≥ 1, then E_{/Q_{p}} has good reduction. This is a special case of a more general theorem of Grothendieck’s on p-divisible groups [17, IX, Cor 5.10], which says in particular that an abelian variety over Q_{p} has good reduction if and only if its p-divisible group has ‘good reduction’ in the sense that it is the generic fiber of a p-divisible group over Z_{p}. This is the correct analogue of the Néron-Ogg-Shafarevich criteria ‘at p’. In order to apply this, we need to invoke Raynaud’s theorem [27, 2.3.1], which asserts that Grothendieck’s criterion is equivalent to the a priori weaker condition that each torsion level of the p-divisible group is the generic fiber of a finite flat group scheme over Z_{p} (i.e., we do not need to assume that there is any compatibility between the Z_{p}-group schemes for each p-power torsion level). Keep in mind that it is essential for the converse of Theorem 1.2 that we have a condition on large torsion levels. This is illustrated by the first part of the following example.

Example 1.3

(i) It is possible that E_{/Q_{p}} can have bad reduction (and so E[p^{n}] is not the generic fiber of a finite flat Z_{p}-group scheme for some large n) while E[p] has good reduction (i.e., it is the generic fiber of a finite flat Z_{p}-group scheme). For example, if E_{/Q_{p}} has bad split multiplicative reduction with non-integral j(E) a pth power in Q_{p}^{×}, then the theory of Tate models shows

\overline{ρ}_{E,p} ≃ ω ⊕ 1,

which is certainly the D_{p}-representation attached to the generic fiber of a finite flat group scheme over Z_{p}, namely μ_{p} ×_{Z_{p}} \underline{Z/p}.

For E_{/Q} such that E_{/Q_{p}} has bad reduction and E[p]_{/Q_{p}} has good reduction, Wiles’ proof studies ρ_{E,p} as an ‘ord’ deformation of \overline{ρ}_{E,p}. But in the critical associated ‘minimal’ deformation problem, the ‘flat’ methods of this article are needed; see [39, (3.1)].

(ii) If E_{/Q_{p}} has good ordinary reduction of the form

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

then the p-torsion on the Néron model E_{/Z_{p}} of E_{/Q_{p}} is a finite flat group scheme over Z_{p} with order p^{2}, though it is not easy to describe this Z_{p}-scheme explicitly. It can be shown that E[p] fits into a short exact sequence of finite flat commutative Z_{p}-group schemes. Such a sequence must have the form

0 → μ_{p} → E[p] → \underline{Z/p} → 0,

except when E[p] ≃ μ_{p} ×_{Z_{p}} \underline{Z/p}, in which case there is also a sequence corresponding to the other way of splitting E[p]. Theorems 1.6 and 1.7 supply the results needed to justify this claim. We leave this as an exercise (it will not be needed).

Recall that for a field F of characteristic 0, all finite F -group schemes are étale over F [35], and so with a choice of an algebraic closure \overline{F}, we get an order-preserving equivalence of categories between finite flat group schemes over F and finite discrete modules over G_{F} = Gal(\overline{F} /F). This allows us to single out certain Galois modules as special:

Definition 1.5 Let K be a field of characteristic 0 that is the fraction field of a henselian (e.g., complete) discrete valuation ring R with residual characteristic p. Fix an algebraic closure \overline{K}. We say that a continuous representation

ρ : G_{K} → Aut(M)

on a finite abelian p-group M is R-flat (or just flat when R is understood from the context) if there exists a finite flat group scheme H_{/R} such that the G_{K}-representation H(\overline{K}) is isomorphic to ρ (or, equivalently, such that the finite K-group scheme canonically attached to ρ is the generic fiber of a finite flat group scheme over R).

The ‘flatness’ refers to the essential property of H → Spec(R). The representation ρ is not being required to be flat over anything. Also, it should be emphasized that flatness as defined above is really a property of finite Galois modules (though in Definition 2.1 we will extend it in a formal way to more general Galois modules, such as Tate modules of elliptic curves). Note that in the setting of the above definition, the H which arises necessarily has order equal to the size of the abelian group M (since by R-flatness, this can be checked after passage to a geometric fiber over the generic point).

If there were many different choices for H, the notion of ‘flatness’ as defined above would not be a natural one to use. In addition, if the representation ρ had extra structure such as that of a vector space over a large finite field k (with the k-action commuting with the G_{K}-action of ρ), then we would like such extra endomorphism structure to extend to H, or else the concept of a ‘flat’ representation would likely be too weak to use. Fortunately, we have the following two fundamental results due to Raynaud.

Theorem 1.6 (Raynaud) Let R, K, \overline{K}, and G_{K} be as above. Consider the covariant functor

F_{R} : H ⇝ H(\overline{K})

from finite flat p-power order R-group schemes to flat G_{K}-modules. When e(R) < p − 1 (so p ≠ 2), F_{R} is an equivalence of categories. Moreover, the category of finite flat p-power order R-group schemes is an abelian category via the usual scheme-theoretic kernel and quotient constructions.

Suppose e(R) = p − 1 and V is a continuous k-linear representation of G_{K} on a one-dimensional vector space over a finite field k of characteristic p. Assume that the F_{p}[G_{K}]-module underlying V is simple (i.e., the natural map F_{p}[G_{K}] → k is surjective). Then either there is a unique (up to canonical isomorphism) finite flat R-group scheme H with H(\overline{K}) ≃ V, or else there are two such R-group schemes, one étale and the other multiplicative (i.e., with an étale Cartier dual).

The essential image of the functor F_{R} is stable under passage to subrepresentations, quotients, and (finite) direct sums without restriction on e(R).

Proof. The proof of full faithfulness consists of reducing to the case in which H(\overline{K}) is a simple representation, and then one writes down the most general form of H that could possibly give rise to the given representation (see Theorem 1.7 and the discussion preceding it). We discuss the quasi-inverse functor below. The proof of the stability of the essential image under various constructions is much simpler and is proved by the method of ‘scheme-theoretic closure’. See [27, 2.1, 3.3.2(3), 3.3.6] for further details.

Though [27, 2.1] refers to the fppf topology for formation of quotients, in our setting it is possible to get away with a more naive construction of quotients that exploits Cartier duality for finite flat group schemes. One essentially defines the quotient by a closed subgroup scheme to be the Cartier dual of a suitable kernel. We omit the details of this alternate construction (though there is some work needed to verify flatness), as this would be too much of a digression here, except we note that such an alternate construction supplies an elementary proof that base change preserves short exactness and that applying Cartier duality to a short exact sequence of finite flat group schemes preserves the property of being short exact. See [5, §2.2] for further details.

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One can describe the ‘quasi-inverse’ to the functor in Theorem 1.6 when e(R) < p − 1. In down-to-earth terms, consider a flat representation ρ with a representation space V_{ρ} whose underlying group is a finite abelian p-group. With our choice of \overline{K}, there is (via Galois descent) canonically attached to V_{ρ} a finite K-group scheme Spec(K_{ρ}) whose representation on \overline{K}-points is V_{ρ}. The essential content of Theorem 1.6 is that inside of K_{ρ} there is a unique finite R-subalgebra O_{ρ} which has generic fiber K_{ρ} and which is ‘stable’ under the group law (or, rather, K-Hopf algebra) morphisms on K_{ρ}. This O_{ρ} is the affine R-algebra for the unique finite flat R-group scheme whose generic fiber representation is V_{ρ}. The functor ρ ⇝ Spec(O_{ρ}) is the sought-after ‘quasi-inverse’ functor.

The idea behind Raynaud’s proof is to build everything up from an analysis of finite flat R-group schemes with generic fiber representations that are simple of p-power order. This generalizes earlier results of Oort-Tate [25, Thm 2] in the case of finite flat group schemes of order p. Consider a (possibly non-simple) finite discrete G_{K}-representation V with p-power order. How many (if any) finite flat R-group schemes H admit V as a generic fiber representation (i.e., V ≃ H(\overline{K}) as G_{K}-modules)? By the theory of Galois descent of schemes [3, §6.2, Ex B], or else by an ad hoc modification of the proof of classical Galois descent for fields [32, Ch II, Lemma 5.8.1], knowledge of such an H is equivalent to knowledge of V and of H ×_{R} R^{sh}, with R^{sh} the strict henselization of R (e.g., when R = Z_{p}, R^{sh} = Z_{p}^{un}; see [3, §2.3, Prop 10, 11]). Thus, the essential case is when R is strictly henselian, so we can replace R by R^{sh} (and G_{K} by the inertia subgroup I_{K} = G_{K^{sh}}).

For R strictly henselian and V simple, k = End_{G_{K}} (V) is a finite division ring and so by Wedderburn’s Theorem is a finite field with characteristic p (here, we temporarily abandon our usual convention that such a field is equipped with an embedding into \overline{F}_{p}). Hence, V is canonically an irreducible k[G_{K}]-module with finite k-dimension. Because R is a strictly henselian discrete valuation ring with residue characteristic p, G_{K} has a pro-p normal subgroup (wild inertia) with a pro-prime-to-p abelian quotient (tame inertia). To justify this structure for G_{K}, note that the integral closure of R in a finite (necessarily separable) extension of K is finite as an R-module. Since R is a strictly henselian discrete valuation ring, such an integral closure is also a (strictly henselian) discrete valuation ring; thus, we can easily pass to the completion of R in place of R and then use [31, §1] to analyze G_{K}. We conclude from the structure of G_{K} that V is a tame (and therefore abelian) representation [30, Ch IX, Lemma 2], so from the structure theory of semisimple rings we see that dim_{k}(V) = 1. That is, V is given by a continuous character

ψ : G_{K} → k^{×}.

Raynaud suceeded in describing precisely the ψ which can arise in this way. The description is in terms of fundamental characters. Since we have only discussed fundamental characters for Q_{p}, we will state the result in that limited context with R = Z_{p}^{un} (which suffices for our purposes).

Theorem 1.7 (Raynaud) Consider a continuous character ψ : G_{Q_{p}^{un}} = I_{p} → k^{×}, with k a finite field of characteristic p. Choose an embedding k ↪ \overline{F}_{p}. Let V be a one-dimensional k-vector space with I_{p}-action given by ψ. Then V is Z_{p}^{un}-flat if and only if

ψ = ψ_{n}^{e},

where |k| = p^{n}, ψ_{n} is the fundamental character of level n, and e = e_{0} + e_{1}p + ⋯ + e_{n−1}p^{n−1}, with 0 ≤ e_{i} ≤ e(Z_{p}^{un}) = 1.

Proof. See [27, 3.4.3] for details. The proof of ‘only if’ proceeds as follows. Let H_{/Q_{p}^{un}} be the finite ppower order commutative Q_{p}^{un}-group scheme associated to V, so the k-action on the (one-dimensional) flat I_{p}-representation space V can be viewed as a k-action on H. The key step is to prove that this action necessarily extends to a k-action on certain finite flat Z_{p}^{un}-group schemes which have generic fiber H [27, 3.3.1]. We remark in passing that one can exploit this to show that there is a unique finite flat Z_{p}^{un}-group scheme with generic fiber representation V when p > 2 and there are at most two such Z_{p}^{un}-group schemes when p = 2 and V is a simple F_{p}[I_{p}]-module; this is used in the proof of Theorem 1.6.

Note that the choice of embedding k ↪ \overline{F}_{p} is harmless, since changing this merely has the effect of cyclically permuting the ‘digits’ e_{i}.

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The example of μ_{2} and \underline{Z/2} over Z_{2} shows that the full faithfulness in Theorem 1.6 is false when p = 2, since the generic fibers of μ_{2} and \underline{Z/2} over Q_{2} are isomorphic, but the closed fibers over F_{2} are not (one is reduced, the other is not), so μ_{2} and \underline{Z/2} are not Z_{2}-isomorphic (as schemes, let alone as group schemes). Note that this also explains why Theorem 1.6 treats the case e(R) = p − 1 separately.

Theorem 1.7 and the discussion preceding it show that for χ : D_{p} → F_{p}^{×} an unramified continuous character, ω^{i}χ : D_{p} → F_{p}^{×} is flat if and only if i ≡ 0, 1 mod p − 1. The flatness condition is a therefore a very severe constraint on a representation. Though Theorem 1.2 shows that flat representations arise naturally from elliptic curves, the general problem of describing all flat representations (not just the ones as in Theorem 1.7) is very subtle. An indirect answer to this question was given by Fontaine in a special case, and we will discuss his theory in §4.

Gathering together our results so far, we can complete the proof of Theorem 1.1 and in the supersingular reduction case can give a description of properties of ρ_{E,p} which make sense for representations into GL_{2}(R), where R is any complete local noetherian ring with a finite residue field of characteristic p. This will be the starting point for the deformation-theoretic study of the supersingular case. First, we complete the proof of Theorem 1.1.

Proof. (of Theorem 1.1, continued)

By Theorem 1.7 and the discussion preceding it, it suffices to check that in the supersingular case, \overline{ρ}_{E,p}|_{I_{p}} is irreducible (and then we can apply Theorem 1.7 with k = F_{p^{2}}). If \overline{ρ}_{E,p}|_{I_{p}} is reducible, then by Theorem 1.6, the diagonal characters are Z_{p}^{un}-flat. Since det \overline{ρ}_{E,p} = ω, we see from Theorem 1.7 and the fact that the level 1 fundamental character is ψ_{1} = ω|_{I_{p}} that there exists a short exact sequence of the form

0 → χ_{1} → \overline{ρ}_{E,p}|_{I_{p}} → χ_{2} → 0,

where {χ_{1}, χ_{2}} = {ω|_{I_{p}}, 1}. By Theorem 1.6, it follows that there exists a short exact sequence of finite flat Z_{p}^{un}-group schemes

0 → G_{1} → E^{un}[p] → G_{2} → 0

which on \overline{Q}_{p}-points realizes the given decomposition of \overline{ρ}_{E,p}|_{I_{p}}. Here, E_{/Z_{p}^{un}}^{un} is the Néron model of the good reduction curve E_{/Q_{p}^{un}}. It happens to be the case that E_{/Z_{p}^{un}}^{un} is the base extension of the Néron model of E_{/Q_{p}} [3, §7.2, Thm 1(ii)], but we will not need this. What we will need is that E_{/Z_{p}^{un}}^{un} has a supersingular

closed fiber. This is clear naively, since j(E_{/Q_{p}^{un}}) = j(E_{/Q_{p}}) is integral, and the image of this in \overline{F}_{p} is the j-invariant of the closed fiber. Since the reduction type is determined by the j-invariant, we see that E_{/Z_{p}^{un}}^{un} has supersingular reduction.

Now assume p ≠ 2 (so ω|_{I_{p}} is non-trivial). Let G denote whichever of G_{1} or G_{2} has a trivial generic fiber representation. We claim that G_{/Z_{p}^{un}} is étale. If this were true, then passing to the closed fiber would prove that the étale factor of the finite flat group scheme E^{un}[p] ×_{Z_{p}^{un}} \overline{F}_{p} is non-trivial, and so there exist non-trivial p-torsion geometric points on the closed fiber, contradicting the supersingularity condition.

Hence, for the case of p ≠ 2 we are reduced to checking the claim that if H_{/Z_{p}^{un}} is a finite flat group scheme with p-power order and H(\overline{Q}_{p}) has trivial inertial action, then H → Spec(Z_{p}^{un}) is étale. The generic fiber representation of G_{Q_{p}^{un}} = I_{p} is trivial, so the full faithfulness in Theorem 1.6 implies that that H_{/Z_{p}^{un}} is a constant group scheme and so it is clearly étale over Z_{p}^{un}. This completes the proof of Theorem 1.1 for odd p.

Now consider the case p = 2. This will require (at the end) more advanced results from algebraic geometry; since the case p = 2 is only being included for completeness and is not used in Wiles’ proof, the reader can skip this case.

Note that each G_{i} has trivial inertial generic fiber representation. Clearly μ_{2} and \underline{Z/2} are two nonisomorphic Z_{2}^{un}-group schemes with trivial generic fiber representations, so by Theorem 1.6 we see that each G_{i} is Z_{2}^{un}-isomorphic to μ_{2} or \underline{Z/2}. Since E^{un}[p] is connected, G_{1} and G_{2} are connected. Thus, each G_{i} is isomorphic to μ_{2}.

Applying Cartier duality to the short exact sequence

0 → μ_{2} → E^{un}[p] → μ_{2} → 0,

we get another short exact sequence. If E^{un}[p] were self-dual, then the middle term in the Cartier-dualized short exact sequence would be E^{un}[p] . However, the outer terms now are isomorphic to \underline{Z/2} over Z_{2}^{un}, so we would have a contradiction.

In order to check the self-duality of E^{un}[p], it is not enough to know that the generic fiber is self-dual (via the Weil pairing), since we cannot apply Theorem 1.6 (with p = 2) to the non-simple generic fiber representation. The essential problem is that we do not know naively that the Cartier dual of E^{un}[p] is still connected (otherwise results of Raynaud and Fontaine extending Theorem 1.6 for p = 2 could be used).

This connectedness is only known after we prove the stronger assertion of self-duality. Self-duality follows from the Cartier-Nishi duality theorem [24, Cor 1.3(ii)], applied to the morphism p : E^{un} → E^{un}, and the canonical autoduality of elliptic curves [19, 2.1.2].

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When E_{/Q_{p}} is an elliptic curve with good supersingular reduction, we can give a list of special properties of the representation \overline{ρ} = \overline{ρ}_{E,p} and its deformation class to GL_{2}(Z_{p}) represented by ρ = ρ_{E,p}. Namely, det \overline{ρ} = ω, \overline{ρ} is flat (Theorem 1.2) and absolutely irreducible (Theorem 1.1), while det ρ = ϵ and for every n ≥ 1, the finite p-power order discrete D_{p}-module ρ mod p^{n} is flat (Theorem 1.2). The absolute irreducibility of \overline{ρ} indicates that we are not in the ‘ord’ case. Keeping in mind the converse to Theorem 1.2, these conditions will motivate our definition (in the next section) of the ‘flat deformation functor’ as a deformation problem for D_{p}-representations. The remarkable fact that we will be able to explicitly describe the deformation ring associated to this deformation problem (Theorem 3.5) will allow us to reverse the usual process ‘H^{1} tells us about the deformation ring’ to get information about a local H^{1} term from knowledge about a deformation ring. This ring-theoretic knowledge will be obtained by methods based on the work of Fontaine and Ramakrishna. As we said earlier, this local H^{1} data is then pieced together with other Galois cohomological data (at other places in Σ) to tell us about the size of a global H^{1} term, and thereby gives us a grip on a global deformation ring (e.g., allows us to show it is isomorphic to a Hecke ring after a lot more work).

Using Raynaud’s results (Theorems 1.6, 1.7), we can somewhat generalize the ‘good reduction’ parts of Theorem 1.1 when p ≠ 2. This is useful insofar as it will allow us to state the Main Theorem 3.3 with fewer hypotheses (and it is an aesthetically pleasing result too).

Theorem 1.8 Let R be a complete local noetherian ring with finite residue field k of characteristic p ≠ 2. Choose a flat representation \overline{ρ} : D_{p} → GL_{2}(k) with det \overline{ρ}|_{I_{p}} = ω|_{I_{p}} and a continuous lift ρ : D_{p} → GL_{2}(R) which gives rise to an element of Dfl ; \overline{ρ}(R) (see Definition 2.1, with O = W (k) there).

Let

φ_{2} : I_{p} →^{ψ_{2}} F_{p^{2}}^{×} ↪ GL_{2}(F_{p}) ↪ GL_{2}(k)

be the map arising from a choice of an F_{p}-basis of F_{p^{2}}.

(i) If \overline{ρ} is reducible, then there exist continuous unramified characters \overline{χ}_{i} : D_{p} → k^{×} such that

\overline{ρ} ≃ (\begin{matrix}ω\overline{χ}_{1} & ∗ \\ 0 & \overline{χ}_{2} \end{matrix}) .

Otherwise \overline{ρ} is absolutely irreducible and \overline{ρ}|_{I_{p}} ≃ φ_{2} (and so \overline{ρ}|_{I_{p}} ⊗_{k} \overline{F}_{p} ≃ ψ_{2} ⊕ ψ_{2}^{p}).

(ii) If \overline{ρ} is reducible, then there exist unique continuous unramified characters χ_{i} : D_{p} → R^{×} such that

ρ ≃ (\begin{matrix}ϵχ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}) .

In particular, det ρ|_{I_{p}} = ϵ|_{I_{p}} and χ_{i} mod 𝔪_{R} = \overline{χ}_{i}.

(iii) If \overline{ρ} is irreducible, then det ρ|_{I_{p}} = ϵ|_{I_{p}}.

Remarks. It is essential in Theorem 1.8(i) that we assume a condition on det \overline{ρ}|_{I_{p}}. Otherwise one could construct counterexamples in the reducible case using Raynaud’s Theorem 1.7 and in the irreducible case using an unramified character D_{p} ↠ k_{2}^{×}, with k_{2} the quadratic extension of k.

In Theorem 1.8(ii),(iii) the main examples of such R to keep in mind are O/λ^{n} and (O/λ^{n})[ε] = (O/λ^{n})[T]/(T^{2}), with O and λ as defined in the beginning of §2.

The proof we give for (iii) involves Raynaud’s work on determinants of p-divisible groups and so ultimately relies on the Zariski-Nagata theorem on purity of the branch locus [16, p. 118]. With extra work, the proof of (iii) can be extended to the case p = 2 also. The reader should appreciate that there is a substantial amount of algebraic geometry lying behind the assertion in (iii). An alternative proof (valid at least for p ≥ 5 and perhaps also for p = 3) can probably be obtained by a brute-force calculation using Fontaine’s ideas from §4, together with [13, §6, §9]. But this alternate procedure is not very insightful.

Proof. (i) Suppose \overline{ρ} is reducible, say with diagonal characters η_{1}, η_{2} : D_{p} → k^{×}. By local class field theory for Q_{p}, η_{j}|_{I_{p}} = ω^{n_{j}}|_{I_{p}} as k^{×}-valued characters, for suitable n_{j} ∈ Z/(p − 1). Using the observations after Theorem 1.7, together with the fact that

η_{1}η_{2}|_{I_{p}} = det \overline{ρ}|_{I_{p}} = ω|_{I_{p}},

we see that {n_{1}, n_{2}} = {0, 1}. It remains to show that if

\overline{ρ} ≃ (\begin{matrix}η_{1} & ∗ \\ 0 & η_{2} \end{matrix})

with η_{1} unramified and η_{2}|_{I_{p}} = ω|_{I_{p}}, then \overline{ρ} splits. Equivalently, we must show that \overline{ρ} has a non-zero unramified quotient.

Consider the connected-étale sequence of the finite flat Z_{p}-group scheme G with generic fiber representation \overline{ρ},

0 → G^{0} → G → G^{et} → 0.

Recall the following:

Unramifiedness criteria (U): for p ≠ 2, a flat D_{p}-representation is unramified if and only if the corresponding Z_{p}-group scheme is étale over Z_{p}.

To prove this, we can make a base change to Z_{p}^{un}, and then use the argument which proved the irreducibility of \overline{ρ}_{E,p} in the supersingular case in Theorem 1.1 (this amounts to the fact that for odd p, constant finite group schemes over Q_{p}^{un} admit only constant extensions to finite flat group schemes over Z_{p}^{un}). If p = 2 this critical fact is not true (consider μ_{2} over Z_{2}^{un}).

Combining (U), Theorem 1.6, and the universal properties characterizing G^{0} and G^{et} in the category of finite flat Z_{p}-group schemes, it follows that the generic fiber representation of G^{et} is precisely the maximal unramified quotient of \overline{ρ} as an F_{p}[D_{p}]-module. However, by functoriality of the connected-étale sequence, we see that the canonical k-action on G gives rise to compatible canonical k-actions on G^{0} and G^{et}. In other words, the generic fiber representation of G^{et} is also the maximal unramified quotient of \overline{ρ} as a k[D_{p}]-module. Thus, if \overline{ρ} is not split as a k[D_{p}]-module, then G^{et} is trivial and so G is a connected group scheme. But all subrepresentations of \overline{ρ} arise as generic fiber representations of finite flat closed subgroup schemes of G, and these are all necessarily connnected, too. Hence, such representations must be ramified, by (U). This contradicts the existence of the unramified subrepresentation η_{1}.

Now consider the case in which \overline{ρ} is irreducible. Since det \overline{ρ}|_{I_{p}} = ω|_{I_{p}} and p is odd, it follows that \overline{ρ} is absolutely irreducible (the argument is a local analogue of the global theorem that for p ≠ 2, an odd continuous irreducible two-dimensional representation of G_{Q} over k is absolutely irreducible). The normality of I_{p} in D_{p} then implies that \overline{ρ}|_{I_{p}} is semisimple and therefore tame [30, Ch IX, Lemma 2]. Thus, \overline{ρ}|_{I_{p}} is abelian, so

\overline{ρ}|_{I_{p}} ⊗_{k} \overline{F}_{p} ≃ χ_{1} ⊕ χ_{2}

with continuous characters χ_{i} : I_{p} → \overline{F}_{p}^{×}. These eigencharacters must take their values in the quadratic extension k_{2} of k and χ_{1} ≠ χ_{2} since \overline{ρ} is absolutely irreducible (and therefore non-abelian). The absolute irreducibility of \overline{ρ} then forces a Frobenius element of D_{p} to interchange the two inertial lines, so from the well-known explicit description of the conjugation action of Frobenius on tame inertia, we obtain

χ_{2} = χ_{1}^{p}, χ_{1} = χ_{2}^{p}.

Thus, χ_{1} : I_{p} → F_{p^{2}}^{×} and χ_{1}(I_{p}) does not lie in F_{p}^{×}.

The flatness of \overline{ρ} implies (by Theorem 1.6) that Theorem 1.7 can be applied to χ_{1}, so χ_{1} is equal to either ψ_{2} or ψ_{2}^{p}. This proves that the semisimple k[I_{p}]-modules \overline{ρ}|_{I_{p}} and φ_{2} are isomorphic over \overline{k} = \overline{F}_{p}. By the Brauer-Nesbitt Theorem, \overline{ρ}|_{I_{p}} ≃ φ_{2}.

(ii) Our argument will follow the method suggested by the brief sketch given in the proof of [8, Lemma 2.19(b)]. Note that we can immediately reduce to the case in which R is artinian (and therefore finite), so ρ arises as the generic fiber of the finite flat Z_{p}-group scheme G. By Theorem 1.6 (p ≠ 2), R acts on G. Let

0 → G^{0} → G → G^{et} → 0

denote the connected-étale sequence of G. By the universal properties of the connected-étale sequence, it follows that R acts on G^{0} and G^{et}, so the generic fibers give rise to a short exact sequence of R[D_{p}]-modules, with ρ in the middle. We will show that both G^{0} and G^{et} have generic fiber R-modules which are free of rank 1 and that the characters on these ‘lines’ are of the desired type.

Now comes the critical step where we exploit the theory of finite flat group schemes. We claim that all Z_{p}[D_{p}]-module Jordan-Hölder factors of the generic fiber representation of G^{0} are ramified. The reason is quite simple. All such representations are flat and the corresponding finite flat Z_{p}-group scheme fits into a ‘decomposition series’ for the connected group scheme G^{0}. Since all closed subgroup schemes and quotients of a connected object in the category of finite flat Z_{p}-group schemes are again connected (as connectedness can be determined on the closed fiber and base change commutes with formation of short exact sequences of finite flat group schemes). A non-trivial connected finite flat Z_{p}-group scheme is not étale and its generic fiber representation must therefore be ramified (by the criteria (U) above). This proves the claim concerning Jordan-Hölder factors of the generic fiber representation of G^{0}.

Let V_{ρ} be the representation space underlying ρ and let V_{ρ}^{et} denote the (non-zero) maximal unramified abelian group quotient. This is the generic fiber representation of G^{et} and so it has a canonical R-module structure. Due to the form of \overline{ρ}, V\overline{ρ} = V_{ρ}/𝔪_{R} → V_{ρ}^{et}/𝔪_{R} is surjective with V_{ρ}^{et}/𝔪_{R} non-zero, so V_{ρ}^{et}/𝔪_{R} is exactly 1-dimensional over k. Hence, V_{ρ}^{et} has a single generator as an R-module. In order to show that it is a free R-module of rank 1, it suffices to check that it has the same R-length as R. On the other hand, the kernel V_{ρ}^{0} of V_{ρ} ↠ V_{ρ}^{et} is the generic fiber representation of G^{0} and consequently all of its Jordan-Hölder factors (either as an R[D_{p}]-module or even as a Z_{p}[D_{p}]-module) are ramified. Since formation of the connected-étale sequence is compatible with the local base change Z_{p} → Z_{p}^{un}, similar reasoning proves the same ‘ramified’ Jordan-Hölder factor property for V_{ρ}^{0}, with I_{p} replacing D_{p}.

Since the R-action and D_{p}-action on V_{ρ} commute, the form of \overline{ρ} shows that all R[I_{p}]-module Jordan-Hölder factors of the rank 2 free R-module V_{ρ} are 1-dimensional k-vector spaces of the form k(1) or k. Note that since p ≠ 2, k and k(1) are not isomorphic as D_{p}-modules. Also, clearly all R[I_{p}]-module Jordan-Hölder factors of V_{ρ}^{et} have the form k, and all such factors of V_{ρ}^{0} have the form k(1). Because V_{ρ} is free of rank 2 as an R-module, and the quotient V_{ρ}^{et} has a single R-module generator, we see that the number of unramified Jordan-Hölder factors is less than or equal to the number of ramified ones.

We claim that it is enough to show that V_{ρ}^{0}/𝔪_{R} is 1-dimensional over k. Indeed, this implies that V_{ρ}^{0} and V_{ρ}^{et} are both quotients of R as R-modules, so since V_{ρ} is free of rank 2 over R, comparing lengths proves that V_{ρ}^{et} and V_{ρ}^{0} are free of rank 1 over R. Set ρ′ = Hom_{R}(ρ, R(1)). Injecting R(1) into a finite product of copies of (Q_{p}/Z_{p})(1) gives an injection of ρ′ into a finite product of copies of the flat Cartier dual Hom_{Z_{p}}(ρ, (Q_{p}/Z_{p})(1)), so ρ′ is a flat representation (by Theorem 1.6). A consideration of Jordan-Hölder factors then shows that ρ′ mod 𝔪_{R} is reducible with cyclotomic inertial determinant (so (i) may be applied) and we readily conclude that V_{ρ}^{0} = Hom_{R}(V_{ρ′}^{et}, R(1)). Thus, the inertial actions on V_{ρ}^{et} and V_{ρ}^{0} are respectively trivial and cyclotomic, as desired. The uniqueness of the χ_{i} and the equality \overline{χ}_{i} = χ_{i} mod 𝔪_{R} are clear.

We now verify that the right exact sequence

0 → V_{ρ}^{0}/𝔪_{R} → V_{ρ}/𝔪_{R} → V_{ρ}^{et}/𝔪_{R} → 0

is exact, from which the desired 1-dimensionality of V_{ρ}^{0}/𝔪_{R} follows. Choose a finite set of generators X = {x_{i}} for 𝔪_{R} and consider the obvious R[D_{p}]-linear map

⨁_{x∈X} ρ → ρ

with cokernel ρ mod 𝔪_{R}. If we extend this to a diagram arising from the connected-étale sequences associated to each side, we can apply the snake lemma in the abelian category of flat D_{p}-representations. The resulting coboundary map must be 0 because any map between a connected and an étale finite flat group scheme over a local base ring must be trivial. This implies that V_{ρ}^{0}/𝔪_{R} → V_{ρ}/𝔪_{R} is injective.

(iii) The proof we give is taken from [7, Thm 13.1(i)], which considers a slightly more general setting. The main point to note is that the proof uses Ramakrishna’s Theorem 3.5 below, so when reading the proof of Theorem 3.5, note that the present result is never used, and thus there is no circular reasoning. The idea of the proof is to use an observation of Faltings’ [39, pp.457-8] and an auxiliary trick to reduce to the special case where O = Z_{p}. In that case we can use Raynaud’s fundamental work on determinants of p-divisible groups [27, Thm 4.2.1].

Using O = W (k) in Theorem 3.5 below, we know that a universal ‘flat’ W (k)-deformation ring Rfl ; \overline{ρ} exists and that it is isomorphic to W (k)[[T_{1}, T_{2}]]. If we could show that the associated universal ‘flat’ representation has inertial determinant ϵ|_{I_{p}}, then we’d be done. The observation of Faltings mentioned above ensures that we can always extend the field k without loss of generality. Thus, assume k is large enough so that there exists a continuous unramified character χ : D_{p} → k^{×} such that ω^{−1} det \overline{ρ} = χ^{2} (this is possible since D_{p}/I_{p} ≃ Ẑ has no non-trivial 2-torsion). Recall that flatness of a D_{p}-representation is unaffected by unramified twisting (due to Galois descent, as mentioned after Theorem 1.6). Thus, twisting by the unramified Teichmüller lift of χ^{−1} gives (via Yoneda’s Lemma) an isomorphism between the universal flat deformation rings of \overline{ρ} and \overline{ρ}χ^{−1}. This reduces us to the case in which det \overline{ρ} = ω.

It is relatively straightfoward to check that up to isomorphism, there is only one continuous representation D_{p} → GL_{2}(k) which has determinant ω and which is isomorphic to φ_{2} on I_{p} (and in particular, this D_{p}- representation is self-dual). In addition, this unique representation is defined over F_{p}. Now applying (i) and Faltings’ observation ‘in reverse’, we reduce to the case k = F_{p}, so Rfl ; \overline{ρ} = Z_{p}[[T_{1}, T_{2}]]. Because this universal ring has such a special form, we see that to prove that the universal flat deformation has inertial determinant ϵ|_{I_{p}}, it is enough to check this on all Z_{p}-valued points. But a Z_{p}-valued point is the same thing as a p-divisible group over Z_{p} with p-torsion representation \overline{ρ}.

We are therefore reduced to checking that if ρ : D_{p} → GL_{2}(Z_{p}) is the generic fiber representation of a p-divisible group Γ over Z_{p} with ρ mod p ≃ \overline{ρ}, then det ρ|_{I_{p}} = ϵ|_{I_{p}}. By [27, Thm 4.2.1], it is enough to check that Γ has dimension 1. Since \overline{ρ} is ramified, clearly Γ is not étale and so dim Γ ≥ 1. The dual Γ^{∗} has generic fiber p-torsion representation \overline{ρ}^{∗} ≃ \overline{ρ}, so dim Γ^{∗} ≥ 1. By [34, §2.3, Prop 3],

dim Γ + dim Γ^{∗} = height(Γ) = 2,

so dim Γ = 1, as desired.

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Since Ramakrishna’s Theorem 3.5 is true in the residually ramified and irreducible case when p = 2 (though we omit the proof of this), Theorem 1.8(iii) is true when p = 2. This is not needed in Wiles’ proof.

§2. Defining the Functor

For the remainder of this article, we will fix the following notation. Let O denote the valuation ring of a finite extension of Q_{p}, with λ a uniformizer. Let k = O/λ denote the finite residue field of characteristic p. Choose a continuous representation

\overline{ρ} : D_{p} → GL_{N}(k)

such that \overline{ρ} is flat. Later on we will specialize to the case N = 2, p ≠ 2, and det \overline{ρ}|_{I_{p}} = ω|_{I_{p}}, but for now this is not necessary. Also, we remind the reader that even though we are ultimately interested in the case \overline{ρ} = \overline{ρ}_{E,p}|_{D_{p}}, the D_{p}-representation arising from the p-torsion of an elliptic curve E over Q, for technical reasons in Wiles’ method it is necessary to extend the field of scalars from F_{p} to a finite extension k which contains all of the eigenvalues of the finite subgroup \overline{ρ}_{E,p}(G_{Q}) ⊆ GL_{2}(F_{p}). This is one reason why it is critical that we work in the level of generality fixed above.

Let Duniv ; \overline{ρ} denote the universal deformation functor attached to \overline{ρ}, on the category Ĉ_{O} of complete local noetherian O-algebras with residue field k. If \overline{ρ} is absolutely irreducible (e.g., if \overline{ρ} = \overline{ρ}_{E,p} for E_{/Q_{p}} an elliptic curve with supersingular reduction), then Duniv ; \overline{ρ} is representable [22]. We let Runiv ; \overline{ρ} denote the representing ring. Recall, as was mentioned earlier, that our methods below will be needed to handle certain types of ‘ord’ cases (e.g., those arising from Example 1.3), so it would be too restrictive for us to assume at the outset that \overline{ρ} is (absolutely) irreducible. Nevertheless, the absolutely irreducible case is good to keep in mind, since in this case the universal deformation functor is known to be representable.

The essential problem which we need to solve for input into Wiles’ proof is to compute the orders of D_{p}-cohomology groups attached to certain deformations of \overline{ρ}. In terms of deformation functors, this will amount to computing the size of certain ‘distinguished’ subsets of Duniv ; \overline{ρ}(A) for suitable artinian objects A in Ĉ_{O} (e.g., A = O/λ^{n}). If a representing ring Runiv ; \overline{ρ} exists (e.g., if \overline{ρ} is absolutely irreducible), then

Duniv ; \overline{ρ}(A) = HomĈ_{O}(Runiv ; \overline{ρ}, A),

so if we could determine the structure of Runiv ; \overline{ρ}, then we would be on the right track. This is the basic reason for interest in deformation rings. In principle, it does not matter what the deformation ring looks like, so long as we can somehow compute the size of certain ‘distinguished’ subsets of Duniv ; \overline{ρ}(A) (and these subsets will make sense even when the universal deformation ring is not known to exist). However, in many cases (e.g., supersingular cases) the computations we will need to make are consequences of a much more precise structure theorem for a ‘restricted’ universal deformation ring, and so we will aim to prove this structure theorem and then will derive its consequences for the ‘restricted’ deformation problem of interest. This also seems to be a more natural way to proceed, since the existence of a representing ring begs the question of figuring out what it is. So let us start by first defining the ‘flat deformation functor.’

Definition 2.1 For A in Ĉ_{O}, define Dfl ; \overline{ρ}(A) to be the subset in Duniv ; \overline{ρ}(A) consisting of those deformations of \overline{ρ} to GL_{N}(A) whose representative liftings

ρ : D_{p} → GL_{N}(A)

have the property that for all n ≥ 1, the finite D_{p}-module ρ mod 𝔪_{A}^{n} is flat.

A few comments are in order concerning this definition. First of all, note that the condition on the lifting ρ can be checked on any single representative in the same deformation class of \overline{ρ}, as flatness is a property of the isomorphism class of a finite discrete D_{p}-module. Also, by Theorem 1.6, we see that it suffices to check the ‘flatness’ constraint on ρ mod 𝔪_{A}^{n} just for sufficiently large n. If A is artinian, this is the same as saying that the finite D_{p}-module ρ is flat, in the sense of Definition 1.5. Of course, for any continuous representation ρ : D_{p} → GL_{N}(A), ρ mod 𝔪_{A}^{n} is a finite D_{p}-module with p-power order because A is a noetherian local ring with a finite residue field of characteristic p. Lastly, in [39, p. 457], the definition analogous to Definition 2.1 is given in terms of a condition on ρ mod 𝔞 for all open ideals 𝔞 in A, not just the ideals 𝔪_{A}^{n} for all n ≥ 1. However, since the ideals 𝔪_{A}^{n} give a base of opens around 0 in A, it follows that every ρ mod 𝔞 is a quotient of some ρ mod 𝔪_{A}^{n}, so by Theorem 1.6 it follows that the above definition is equivalent to that given in [39].

Example 2.2 Let E_{/Q_{p}} have good reduction and \overline{ρ} = \overline{ρ}_{E,p}. Let O = Z_{p} and A = O. Then ρ = ρ_{E,p} represents an element in Dfl ; \overline{ρ}(A). This uses Theorem 1.2 and the fact that ρ mod 𝔪_{A}^{n} is nothing other than E[p^{n}](\overline{Q}_{p}). This is the primordial example to keep in mind and is the main reason why we care about Dfl ; \overline{ρ} here.

Though Example 2.2 shows that the sets Dfl ; \overline{ρ}(O/λ^{n}) are the main sets in which we are interested, we also see that at this point we have done nothing of mathematical substance with these sets other than define them. It might even appear hopeless to say anything about such abstractly defined sets. We need a technique for understanding and constructing flat deformations. There is a theory due to Fontaine which will enable us to actually construct (albeit in a somewhat indirect manner) many flat deformations of \overline{ρ}. When combined with the fact that Dfl ; \overline{ρ} is a representable functor in certain cases (something we will prove shortly), we will be able to get enough of a hold on the representing ring that we can say exactly what it is! This will then make it essentially an easy exercise to answer the questions we will have about flat deformations (when Dfl ; \overline{ρ} is a representable functor).

To begin, we need to justify a couple of facts about Dfl ; \overline{ρ}. For example, at this point it is not even clear if this is a functor! To make this clearer, consider the simpler (and essential) case of a map φ : A → B in Ĉ_{O} with A and B both artinian. We then have a natural map of sets

Duniv ; \overline{ρ}(φ) : Duniv ; \overline{ρ}(A) → Duniv ; \overline{ρ}(B),

which, in concrete terms, just takes a representation into GL_{N}(A) and applies φ to the matrix entries to give a representation into GL_{N}(B). What we would most like is Duniv ; \overline{ρ}(φ) to map the subset Dfl ; \overline{ρ}(A) into Dfl ; \overline{ρ}(B). In other words, given a flat representation into GL_{N}(A), if we map the matrix entries into B, then we want the resulting representation into GL_{N}(B) to be flat. This is not obvious (until one sees the proof)! That this is the case is part of the next result.

Theorem 2.3 (Ramakrishna) The association A ⇝ Dfl ; \overline{ρ}(A) from Ĉ_{O} to Set is a subfunctor of Duniv ; \overline{ρ}. That is, given any morphism φ : A → B in Ĉ_{O},

Duniv ; \overline{ρ}(φ)(Dfl ; \overline{ρ}(A)) ⊆ Dfl ; \overline{ρ}(B).

If \overline{ρ} is has trivial centralizer (e.g., if it is absolutely irreducible), then Dfl ; \overline{ρ} is representable, by an object Rfl ; \overline{ρ} in Ĉ_{O}. The resulting natural map

Runiv ; \overline{ρ} → Rfl ; \overline{ρ}

is surjective.

Proof. Since \overline{ρ} is assumed to be flat, we see that Dfl ; \overline{ρ}(k) is a one element set; that is,

Dfl ; \overline{ρ}(k) ⊆ Duniv ; \overline{ρ}(k) = \{\overline{ρ}\}

is not empty. By Theorem 1.6, the property of a finite D_{p}-module being flat is preserved under passage to subrepresentations, quotients, and (finite) direct sums. It therefore follows from [22] that Dfl ; \overline{ρ} is a subfunctor of Duniv ; \overline{ρ} as claimed and that when \overline{ρ} is absolutely irreducible, Dfl ; \overline{ρ} is represented by a quotient ring of Runiv ; \overline{ρ}.

If \overline{ρ} has trivial centralizer (but is not assumed to be absolutely irreducible), and if we can show that all lifts of \overline{ρ} have a trivial centralizer, then the method of proof in [22] can still be applied (i.e., the only property of residual absolute irreducibility which is needed is the fact that lifts have trivial centralizer).

So we are left with the problem of proving that if

\overline{ρ} : G → GL_{N}(k)

is a representation of any group G (with k any field) and \overline{ρ} has trivial centralizer, then for any complete local noetherian ring A with residue field k and any lift

ρ : G → GL_{N}(A)

of \overline{ρ}, ρ has trivial centralizer. Passing to the limit, we are reduced to the case in which A is a local artin ring and we induct on the length of A as an A-module, the case of length 1 being our original hypothesis on \overline{ρ}.

Choose non-zero x ∈ 𝔪_{A}, so by induction ρ mod x has trivial centralizer. Let c ∈ M_{N}(A) commute with the action of ρ, so c ≡ a mod xM_{N}(A) for some a ∈ A. Replacing c by c − a, we can assume c = xc′ for some c′ ∈ M_{N}(A). Since c centralizes ρ, we see that c′ centralizes ρ mod ann(x). By our inductive length assumption and the fact that ann(x) ⊊ A, c′ is congruent to a scalar matrix modulo the annihilator of x, so we conclude as desired that c = xc′ is a scalar matrix.

■

Note that the only input from the theory of finite flat group schemes in the above proof occurs when we invoke [22], which uses only the fact that the ‘flatness’ condition on a finite D_{p}-module has certain formal properties; namely, it is preserved under passage to (finite) direct sums, subrepresentations, and quotients.

This does not make any deep use of the theory of finite flat group schemes (and in particular, this part of Theorem 1.6 is not the hard part; far from it, in fact).

§3. Local Galois Cohomology and Deformation Theory

We now review some results given in [22] and formulate them in a way that will be convenient for our purposes. We then will state the main result concerning the orders of local H^{1}’s. When \overline{ρ} has trivial centralizer (so Rfl ; \overline{ρ} is known to exist!), we will give an interpretation of this main result as a statement about the structure of Rfl ; \overline{ρ}. The proofs of these results can be reduced to constructing ‘enough’ flat representations. The actual construction of such representations will be carried out in §5, as it will first require a review of Fontaine’s approach to the theory of finite flat group schemes (to be discussed in §4).

Pick an artinian object A in Ĉ_{O} and a flat lifting ρ of \overline{ρ} to GL_{N}(A), so the deformation given by ρ is an element of Dfl ; \overline{ρ}(A). There are two examples to keep in mind, with A = O/λ^{n}, N = 2, and p ≠ 2. These are

ρ = (O ⊗_{Z_{p}} ρ_{E,p}) mod λ^{n} and ρ = ρ_{f,λ} mod λ^{n}.

We have \overline{ρ} = k ⊗_{F_{p}} \overline{ρ}_{E,p} and E_{/Q_{p}} an elliptic curve with good reduction in the first example and \overline{ρ} = ρ_{f,λ} mod λ in the second, with f a weight 2 newform having level prime to p and O the completion of the integer ring O_{f} ⊆ C at a prime above p.

Consider H^{1}(D_{p}, ad(ρ)) in our general setting. It is shown in [22] that as an A-module, this is naturally identified with Ext_{A[D_{p}]}^{1}(ρ, ρ). This is an A-module that classifies isomorphism classes of exact sequences of A[D_{p}]-modules

0 → ρ → X → ρ → 0

in which X, necessarily an A[D_{p}]-module with finite A-length, has a D_{p}-action which is trivial on an open subgroup of D_{p}. That is, the D_{p}-action must be continuous with respect to the discrete topology on the finite set underlying X. Inside of Ext_{A[D_{p}]}^{1}(ρ, ρ), there is a natural subset Ext_{A[D_{p}]}^{1,fl}(ρ, ρ) consisting of those elements with a representative short exact sequence in which X is a flat representation (note that X automatically has p-power order). This condition, of course, then holds for any choice of representative short exact sequence.

¿From the formal properties of the ‘flat’ condition on representations (Theorem 1.6), it follows that

Ext_{A[D_{p}]}^{1,fl}(ρ, ρ) ⊆ Ext_{A[D_{p}]}^{1}(ρ, ρ)

is not just a subset, but is an A-submodule [22]. Via the A-module isomorphism

Ext_{A[D_{p}]}^{1}(ρ, ρ) ≃ H^{1}(D_{p}, ad(ρ)),

we can define a corresponding A-submodule

H_{fl}^{1}(D_{p}, ad(ρ)) ⊆ H^{1}(D_{p}, ad(ρ)).

See [22] for a more detailed explanation of all of this. Beware that, despite what the notation may suggest, H_{fl}^{1}(D_{p}, ad(ρ)) is best thought of as a functor of ρ, not of ad(ρ). We will give some ad hoc definitions of H_{fl}^{1}(D_{p}, ∗) for other ∗’s, but there is not a general definition of an H_{fl}^{1} functor in our situation. Note that H_{fl}^{1}(D_{p}, ad(ρ)) is exactly the H^{1} term whose size Wiles needs to tightly control in his ‘local-to-global’ Galois cohomology estimate (for suitable A and ρ).

Let’s give an easy (but important!) example of some elements in H_{fl}^{1}(D_{p}, ad(ρ)).

Example 3.1 For an unramified continuous additive homomorphism χ : D_{p} → A, we can consider X_{χ} which has the ‘block matrix form’

X_{χ} ≃ (\begin{matrix}ρ & χρ \\ 0 & ρ \end{matrix}) .

When restricted to I_{p}, this is just ρ|_{I_{p}} ⊕ ρ|_{I_{p}}, so X_{χ}|_{I_{p}} is certainly the generic fiber representation of a finite flat group scheme G over Z_{p}^{un}; namely, if ρ is the generic fiber representation of a finite flat Z_{p}-group scheme H, then we can take

G = (H ×_{Z_{p}} Z_{p}^{un}) ×_{Z_{p}^{un}} (H ×_{Z_{p}} Z_{p}^{un}).

It follows from standard facts about Galois descent of rings that G canonically descends to a finite flat Z_{p}-group scheme whose generic fiber representation is X_{χ}. This is the same thing that we needed in the discussion of descent following Theorem 1.6.

Though Example 3.1 gives an A-line of ‘flat’ elements in H^{1}(D_{p}, ad(ρ)), we can construct even more. However, we should first point out a subtlety in the definition of H_{fl}^{1}(D_{p}, ad(ρ)) which is easy to overlook but which will slightly complicate our life. Assume for now that p does not divide N. In the case of interest, N = 2 and p is odd, so this is not a problematic assumption. Since N is then invertible in A, if we let ad^{0}(ρ) ⊆ ad(ρ) denote the A-submodule consisting of elements with trace 0, then there is a canonical isomorphism of A[D_{p}]-modules

ad^{0}(ρ) ⊕ A ≃ ad(ρ).

The D_{p}-action on the scalar line A is of course trivial. We have a canonical isomorphism of A-modules

H^{1}(D_{p}, ad^{0}(ρ)) ⊕ H^{1}(D_{p}, A) ≃ H^{1}(D_{p}, ad(ρ)).

Before connecting this up with H_{fl}^{1}, we should mention that the reason for interest in ad^{0}(ρ) is that under the identification of H^{1}(D_{p}, ad(ρ)) with the set of ‘infinitesimal’ deformations of ρ to GL_{N}(A[ε]), H^{1}(D_{p}, ad^{0}(ρ)) corresponds to those deformations whose determinant is the ‘same’ as det ρ (via the canonical inclusion A^{×} ↪ A[ε]^{×}). This is explained in [22] and is of interest because of the fact that we are primarily interested in studying deformations of \overline{ρ}_{E,p} with a ‘fixed’ determinant, namely the cyclotomic character. A cyclotomic determinat condition is but one of several deformation-theoretic conditions satisfied by the Tate module deformation ρ_{E,p}. The reason it is useful to impose such conditions on the deformations we consider is that they ‘cut down’ on the size of the corresponding deformation ring — if it exists — and so we can expect the deformation ring to encode more and more refined information about the Tate module representation ρ_{E,p}.

Now we return to a general setting. We define

H_{fl}^{1}(D_{p}, ad^{0}(ρ)) = H^{1}(D_{p}, ad^{0}(ρ)) ∩ H_{fl}^{1}(D_{p}, ad(ρ))

and

H_{fl}^{1}(D_{p}, A) = H^{1}(D_{p}, A) ∩ H_{fl}^{1}(D_{p}, ad(ρ)),

so we have the A-linear inclusion

ι_{ρ} : H_{fl}^{1}(D_{p}, ad^{0}(ρ)) ⊕ H_{fl}^{1}(D_{p}, A) ↪ H_{fl}^{1}(D_{p}, ad(ρ)).

We might expect this map to be an isomorphism. But at the present time it isn’t known how to prove this in general. Nevertheless, for the type of ρ which arise in Wiles’ proof, we can prove the result. It should be emphasized that this is actually not needed; without this, the Main Theorem 3.3 below would merely be an inequality of the form ≤, which is adequate for Wiles’ needs. But it seems like a good policy to prove results in as strong a form as possible (who knows what will be useful in the future?), so we give here a proof that ι_{ρ} is sometimes an isomorphism.

Lemma 3.2 Let A and ρ be as above, with N = 2, p ≠ 2, and det \overline{ρ}|_{I_{p}} = ω|_{I_{p}}. Then ι_{ρ} is an isomorphism and

Hom_{cont}(D_{p}/I_{p}, A) ↪ H_{fl}^{1}(D_{p}, A)

is an isomorphism (i.e., |H_{fl}^{1}(D_{p}, A)| = |A|).

Remark. The above result applies, in particular, if A = O/λ^{n}. This is the only case which arises in Wiles’ method. Also, the proof in the residually irreducible case ultimately relies on Theorem 1.8(iii), whose proof requires a vast amount of algebraic geometry. Thus, the reader can skip Lemma 3.2 in the residually irreducible case and insert inequalities where appropriate in our later arguments (i.e., Theorem 3.3) and Wiles’ method will still go through (and will yield equalities after the method succeeds).

Proof. Consider the composite map

H_{fl}^{1}(D_{p}, ad(ρ)) ↪ H^{1}(D_{p}, ad(ρ)) → H^{1}(D_{p}, A).

The lemma is readily seen to be equivalent to the assertion that the image is exactly the A-submodule Hom_{cont}(D_{p}/I_{p}, A) inside of H_{fl}^{1}(D_{p}, A). Note that it is not a priori clear that the image even lies in H_{fl}^{1}(D_{p}, A). Since Hom_{cont}(D_{p}/I_{p}, A) is the kernel of the natural restriction map

H^{1}(D_{p}, A) → H^{1}(I_{p}, A),

the lemma is equivalent to the statement that the natural map

H_{fl}^{1}(D_{p}, ad(ρ)) → H^{1}(I_{p}, ad(ρ))

has image inside of H^{1}(I_{p}, ad^{0}(ρ)).

Under the identification [22]

H_{fl}^{1}(D_{p}, ad(ρ)) ≃ D_{ρ}^{fl}(A[ε]),

our assertion is equivalent to the statement that if

ρ_{1} : D_{p} → GL_{2}(A[ε])

is a flat deformation of ρ, then det ρ_{1}|_{I_{p}} = det ρ|_{I_{p}}. Since both ρ and ρ_{1} are flat deformations of \overline{ρ}, both have the same inertial determinant, namely ϵ|_{I_{p}} (treat the cases in which \overline{ρ} is reducible and irreducible separately, using Theorem 1.8).

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There does not (at the present time) exist a satisfactory theory of any sort of H_{fl}^{1} functor in our setting. For us, H_{fl}^{1}(D_{p}, ad(ρ)) is primarily an artifice for singling out a special submodule of ‘distinguished’ cohomology classes, and we need to find out how many of them there are in certain special cases.

We now are in a position to state an important result that Wiles needs. This is the main result of the present article.

Main Theorem 3.3 Let \overline{ρ} : D_{p} → GL_{2}(k) be flat with det \overline{ρ}|_{I_{p}} = ω|_{I_{p}} and p ≠ 2. Let

ρ : D_{p} → GL_{2}(O/λ^{n})

be a flat lifting of \overline{ρ}. Then

|H_{fl}^{1}(D_{p}, ad^{0}(ρ))| = |H^{0}(D_{p}, ad^{0}(ρ))| · |O/λ^{n}|.

Remarks. Using Theorem 1.8(ii),(iii), the hypotheses imply that det ρ|_{I_{p}} = ϵ|_{I_{p}}. However, in the residually irreducible case the proof requires sophisticated methods. The condition is automatic in the application to elliptic curves, so the reader can insert this as an extra hypothesis in Theorem 3.3 and thereby bypass Theorem 1.8(iii) without affecting the applicability of Theorem 3.3 to the study of semistable elliptic curves over Q.

Since Lemma 3.2 in the residually irreducible case requires Theorem 1.8(iii), the reader who would prefer to avoid Theorem 1.8(iii) should insert ≤ in place of = in Theorem 3.3 above. Such an inequality suffices for the successful application of Wiles’ methods to elliptic curves.

Before addressing the proof of Theorem 3.3, we make a few observations about what it says. The H^{0} term is nothing other than the number of trace 0 matrices in M_{2}(O/λ^{n}) which commute with the action of ρ. For example, if \overline{ρ} is not split, then by Theorem 1.8(i) we see that \overline{ρ} has trivial centralizer. We saw in the proof of Theorem 2.3 that this implies that ρ must also have trivial centralizer. In particular, the only trace 0 matrix commuting with ρ is the zero matrix, so |H^{0}(D_{p}, ad^{0}(ρ))| = 1 in these cases.

Now we prove Theorem 3.3. By Lemma 3.2, the injection ι_{ρ} is an isomorphism and

|H_{fl}^{1}(D_{p}, O/λ^{n})| = |O/λ^{n}|,

so we have

|H_{fl}^{1}(D_{p}, ad^{0}(ρ))| = |O/λ^{n}|^{−1}|H_{fl}^{1}(D_{p}, ad(ρ))|.

Therefore, it suffices to prove Lemma 3.4 Under the hypotheses in Theorem 3.3, we have

|H_{fl}^{1}(D_{p}, ad(ρ))| = |H^{0}(D_{p}, ad^{0}(ρ))| · |O/λ^{n}|^{2}.

Since Theorem 1.8(iii) is true for p = 2 and Theorem 4.5 below can be extended to the case p = 2 with some connectedness hypotheses, Lemma 3.4 can be extended to the case p = 2 when \overline{ρ} is irreducible. We omit the arguments needed for this case, as it is not needed in Wiles’ proof.

The idea of the proof of Lemma 3.4 is quite simple: we will write down every flat representation corresponding to an element in H_{fl}^{1}(D_{p}, ad(ρ)) ≃ D_{ρ}^{fl}((O/λ^{n})[ε]) [22] and just count how many we have! This is not the most conceptually satisfying way to proceed, and in the case where \overline{ρ} has trivial centralizer, Fontaine has said that he can give a more ‘pure thought’ proof (with a small amount of computation required). We will say more about this below. We use a computational ‘brute force’ proof because it involves a really clever idea, and also because it is needed (at present) to handle the cases with non-trivial centralizer at the residual level. Regardless of how one proves Lemma 3.4, Fontaine’s work on finite flat group schemes is indispensible.

Before proving Lemma 3.4, we make some remarks in the case when p ≠ 2, \overline{ρ} as above has trivial centralizer, and ρ = ρ_{0} mod λ^{n} for some

ρ_{0} : D_{p} → GL_{2}(O)

lifting \overline{ρ}, with ρ_{0} corresponding to an element in Dfl ; \overline{ρ}(O). In practice, ρ_{0} is the local restriction of a modular lifting of \overline{ρ} = \overline{ρ}_{E,p} ⊗_{F_{p}} k, but we will see in Theorem 3.5 below that some ρ_{0} as above always exists. The following analysis serves as a good indication of what Lemma 3.4 really means.

Since ρ has a trivial centralizer, Lemma 3.4 amounts to the assertion

|H_{fl}^{1}(D_{p}, ad(ρ))| = |O/λ^{n}|^{2}.

By Theorem 2.3, the functor Dfl ; \overline{ρ} is represented by some Rfl ; \overline{ρ} in Ĉ_{O}. Let us interpret Lemma 3.4 as an assertion about Rfl ; \overline{ρ}. In the case n = 1, Lemma 3.3 says that the k-vector space H_{fl}^{1}(D_{p}, ad(\overline{ρ})) is 2-dimensional. Recalling the fundamental isomorphism relating Galois cohomology and deformation theory [22]

Hom_{k}(𝔪/(𝔪^{2}, λ), k) ≃ H_{fl}^{1}(D_{p}, ad(\overline{ρ})),

with 𝔪 the maximal ideal of Rfl ; \overline{ρ}, we see that choosing a k-basis of 𝔪/(𝔪^{2}, λ) gives rise to a surjection of rings

π : O[[T_{1}, T_{2}]] ↠ Rfl ; \overline{ρ}.

We now consider whether π is an isomorphism.

Let 𝔭 denote the kernel of the natural map Rfl ; \overline{ρ} ↠ O induced by ρ_{0}. By suitable change of coordinates, we can assume 𝔭 is the image of (T_{1}, T_{2}) under π, so 𝔭 has two generators. The method of proof of [39, Prop 1.2] produces an O-linear isomorphism

H_{fl}^{1}(D_{p}, ad(ρ)) ≃ Hom_{O}(𝔭/𝔭^{2}, O/λ^{n})

(this construction uses the choice of the representation ρ_{0} corresponding to 𝔭). Hence, Lemma 3.4 says that the O/λ^{n}-module 𝔭/(𝔭^{2}, λ^{n}𝔭), which has two generators, is in fact free of rank 2. Passing to the limit (using ρ_{0} mod λ^{m} with m ≥ n), this says that 𝔭/𝔭^{2} is a free O-module of rank 2. This fact is not at all obvious. Moreover, passing to the direct limit on cohomology gives

H_{fl}^{1}(D_{p}, ad(ρ_{0}) ⊗_{O} (K/O)) =^{def} \underrightarrow{lim} H_{fl}^{1}(D_{p}, ad(ρ_{0} mod λ^{m})) ≃ (K/O) ⊕ (K/O),

so this cohomology module is actually p-divisible as a group. This is all very important in [39, Prop 1.9(v)]. What sort of structure must Rfl ; \overline{ρ} have if 𝔭/𝔭^{2} is free of rank 2? One possibility is that the surjection π above is an isomorphism. This is much stronger than the statement that 𝔭/𝔭^{2} ≃ O ⊕ O (e.g., a priori, we could have Rfl ; \overline{ρ} ≃ O[[T_{1}, T_{2}]]/(T_{1}, T_{2})^{2}, which has a unique O-valued point 𝔭 and 𝔭/𝔭^{2} ≃ O ⊕ O). Nevertheless, if π is an isomorphism, this is a ‘good’ explanation for the O-module freeness. This may seem like a lot to ask for, but Ramakrishna came up with a very clever way to prove that indeed, π is an isomorphism.

Theorem 3.5 (Ramakrishna) Assume p ≠ 2. For a flat representation \overline{ρ} : D_{p} → GL_{2}(k) with trivial centralizer and with det \overline{ρ}|_{I_{p}} = ω|_{I_{p}},

Rfl ; \overline{ρ} ≃ O[[T_{1}, T_{2}]].

In particular, the natural map Dfl ; \overline{ρ}(O) → Dfl ; \overline{ρ}(O/λ^{n}) is surjective for all n ≥ 1.

Proof. The centralizer hypothesis ensures that Rfl ; \overline{ρ} exists. Let 𝔪 be the maximal ideal of Rfl ; \overline{ρ}. Using Fontaine’s theory from §4 (this requires p ≠ 2), we will write down in §5 all short exact sequences of flat k[D_{p}]-modules

0 → \overline{ρ} → X → \overline{ρ} → 0

and count that there are |k|^{2} of them (up to equivalence). Since we have k-linear isomorphisms

Hom_{k}(𝔪/(𝔪^{2}, λ), k) ≃ H_{fl}^{1}(D_{p}, ad(\overline{ρ})) ≃ Ext_{k[D_{p}]}^{1,fl}(\overline{ρ}, \overline{ρ}),

we obtain a surjection O[[T_{1}, T_{2}]] ↠ Rfl ; \overline{ρ}. Let I denote the kernel.

A natural approach to proving I = 0 is to try to use methods from commutative algebra. But our knowledge of the commutative algebra properties Rfl ; \overline{ρ} is (right now) quite minimal. Instead, we will exploit the defining property of Rfl ; \overline{ρ} in the following remarkable manner. Choose f ∈ I. If we can show that the induced injective map on O/λ^{n}-valued points

HomĈ_{O}(Rfl ; \overline{ρ}, O/λ^{n}) ↪ HomĈ_{O}(O[[T_{1}, T_{2}]], O/λ^{n})

is a bijection, then it follows that f(t_{1}, t_{2}) ∈ λ^{n}O for all t_{1}, t_{2} ∈ λO and all n ≥ 1. Thus, f vanishes on the open λ-adic unit disc and so by a basic result from non-archimedean analysis we may conclude that f = 0!

To show that the injection on O/λ^{n}-valued points is a bijection, all we have to show is that both sides have the same size. Since the right side trivially has size |λO/λ^{n}O|^{2} = |k|^{2(n−1)}, what we need to show is

| HomĈ_{O}(Rfl ; \overline{ρ}, O/λ^{n})| = |k|^{2(n−1)}.

But | HomĈ_{O}(Rfl ; \overline{ρ}, O/λ^{n})| actually means something: it is the number of flat deformations of \overline{ρ} to GL_{2}(O/λ^{n}). We again use Fontaine’s theory (p ≠ 2) to simply write down all such possible deformations (this will be done in §5) and count the number of possibilities; it turns out to be exactly what we want.

■

Note that the above proof shows that any surjection O[[T_{1}, T_{2}]] ↠ Rfl ; \overline{ρ} is an isomorphism (which is to be expected, since any surjection from a noetherian ring to itself must be an isomorphism). One may ask if the isomorphism in Theorem 3.5 can be chosen ‘naturally’. That is, can one ‘interpret’ what the parameters T_{1} and T_{2} actually mean? In the proof of Theorem 3.5, we chose arbitrarily a basis of the reduced Zariski cotangent space 𝔪/(𝔪^{2}, λ) and then arbitrarily lifted these to elements of 𝔪. Is there a natural way to make these choices? Recent work of Fontaine and Mazur suggests that natural choices might exist. In the special case of Theorem 3.5, when O = Z_{p}, p ≥ 5, and \overline{ρ} is absolutely irreducible, they give an explicit parameterization of HomĈ_{O}(Rfl ; \overline{ρ}, O) which assigns explicit meaning to T_{1} and T_{2} [14, Thm B2(ii)]. One of the ingredients for this is an analogue of the ideas to be described in the next section.

The fact that Rfl ; \overline{ρ} is a power series ring should have a cohomological interpretation as a vanishing condition on an appropriately defined H^{2} (compare with [21, §1.6, Prop 2] in the ‘unrestricted’ case). Fontaine has said that he can prove such a vanishing condition directly, using [2, Lemma 4.4] and a ‘cohomological dimension’ argument, thereby giving a conceptual proof that Rfl ; \overline{ρ} is a power series ring. One then needs to do the calculation that dim_{k} Ext_{k[D_{p}]}^{1,fl}(\overline{ρ}, \overline{ρ}) = 2 as above in order to prove that the number of variables is 2. This procedure has the advantage of working for GL_{N} and thereby highlights the role of GL_{2} as an artifact needed only for the residual Ext^{1} calculation (which requires Theorem 1.8(i)). Also, one can use the theory of Fontaine-Laffaille to bypass the residual Ext^{1} calculation and to compute directly the k-dimension of H_{fl}^{1}(D_{p}, ad(\overline{ρ})). This produces the number 2 via a completely different (but much more complicated) calculation that explains more conceptually where the ‘2’ comes from. See [4, pp. 6-11, esp. Thm 5] for further details.

It is fairly easy to use Theorem 3.5 to deduce Lemma 3.4 in the case of a residually trivial centralizer. Indeed, Theorem 3.5 proves that for any flat lift ρ : D_{p} → GL_{2}(O/λ^{n}) of \overline{ρ}, there exists ρ_{0} : D_{p} → GL_{2}(O) lifting ρ with ρ_{0} giving an element in Dfl ; \overline{ρ}(O). Hence, we can apply the discussion preceding Theorem 3.5, and combining this with Rfl ; \overline{ρ} ≃ O[[T_{1}, T_{2}]], Lemma 3.4 follows if \overline{ρ} has a trivial centralizer. In §5, we will give a direct proof of Lemma 3.4 in all residually reducible cases, as well as complete the unfinished steps in the proof of Theorem 3.5.

We conclude this section by showing how imposing extra deformation conditions cuts down quite a lot on the ‘size’ of the deformation ring. For example, suppose

\overline{ρ} : D_{p} → GL_{2}(k)

is flat and irreducible, with cyclotomic determinant on inertia. Also assume p ≠ 2. Ramakrishna showed using Tate local duality that

Runiv ; \overline{ρ} ≃ O[[X_{1}, X_{2}, X_{3}, X_{4}, X_{5}]],

where 5 = dim_{k} ad(\overline{ρ}) + h^{0}(D_{p}, ad(\overline{ρ})) [26, Thm 4.1], [39, pp. 457-8] (remark: the cohomological calculations used to prove [26, Thm 4.1] can be simplified by the use of a k-linear version of Tate local duality rather than just an F_{p}-linear version). By Theorem 3.5, we see that imposing a flatness condition on the deformations of \overline{ρ} yields the quotient Rfl ; \overline{ρ} ≃ O[[T_{1}, T_{2}]] of Runiv ; \overline{ρ}. Moreover, by [7, Thm 13.1(ii)], it follows that if we impose the added deformation condition that the determinant be some fixed O^{×}-valued character χ lifting det \overline{ρ}, with χ|_{I_{p}} = ϵ|_{I_{p}}, then the resulting deformation ring is isomorphic to a quotient of the form O[[T]]. ¿From an intuitive viewpoint, if we required the D_{p}-representations to actually come from global representations of a suitably restricted type, then this would cut down on the deformation ring even more. Hence, one way to think about why the global deformation rings considered by Wiles are so ‘small’ is that the functors they represent involve a lot of (local) constraints!

There is one interesting property of all of the Galois deformation rings which have been computed, whether local or global: they are flat over O. Does this fact have any deep meaning?

§4. Fontaine’s Approach to Finite Flat Group Schemes.

The classical theory of complex (or real) Lie groups can be ‘linearized’ insofar as the theory of Lie algebras often allows one to translate theorems and constructions concerning (connected) Lie groups into issues concerning Lie algebras. Since the Lie algebra only perceives tangential information at the origin, an attempt to construct a theory of Lie algebras in the context of algebraic groups is reasonable as long as the group schemes are reduced (since tangent spaces can’t distinguish between a scheme and the underlying reduced subscheme). In particular, everything is fine in characteristic 0. However, in characteristic p there are many group schemes which are not reduced. This is the most important fact about the theory of group schemes in characteristic p. Any attempt at constructing a ‘Lie algebra’ theory for non-reduced group schemes over a field of positive characteristic must use more subtle infinitesimal information than that detected at the level of tangent spaces.

Let k be any field of characteristic p > 0. For the study of finite k-group schemes (recall the commutativity hypotheses) there is a theory of Dieudonne modules which serves as a good analogue to the theory of Lie algebras, up to the fact that the Dieudonne theory is contravariant (like a cotangent space rather than a tangent space). In fact, the theory of Dieudonne modules covers a much wider class of commutative k-group schemes than the finite ones, but we restrict ourselves to this case, as it is all that we will need. Before discussing the basic ingredients of this theory, we mention that Fontaine’s idea (following Grothendieck) is that a finite flat group scheme G over Z_{p} should be classified by specifying its closed fiber (a finite F_{p}- group scheme, or equivalently, a ‘Dieudonne module’), together with some additional ‘lifting data’. In other words, he proposed a refinement of the theory of Dieudonne modules which would create an analogue to the theory of Lie algebras for finite flat Z_{p}-group schemes. Fontaine’s theory actually applies to finite flat group schemes over any base W (k) with k a perfect field of characteristic p. We will only discuss the case k = F_{p}, as this case is all we shall need and it avoids some technical Frobenius-semilinearity issues (thereby making computations simpler). Omitting this Frobenius-semilinearity may create some mistaken impressions about the shape of the more general theory, so what follows should be taken as a slightly skewed perspective on Fontaine’s theory.

The reason that it is preferable to classify a finite flat Z_{p}-group scheme by its closed fiber, together with ‘extra data,’ rather than by its generic fiber, together with ‘extra data,’ is that closed fibers can be classified by the very explicit linear-algebraic notion of a Dieudonne module. The generic fibers, on the other hand, constitute the entire theory of finite discrete modules over D_{p}. Since the structure of D_{p} is a still quite a mystery, using Theorem 1.6 to ‘classify’ finite flat Z_{p}-group schemes via flat representations is not a very useful ‘classification.’

Let us now give the fundamental classification of finite F_{p}-group schemes via Dieudonne modules. An essentially self-contained development of the general theory of Dieudonne modules is given in [11, Ch II]. Define the ‘Dieudonne ring’ D = Z_{p}[F, V]/(F V − p) to be a ring with the variables F (‘Frobenius’) and V (‘Verscheibung’). By a finite D-module, we will mean a D-module with finite Z_{p}-length (and so the underlying abelian group is finite). The category of finite D-modules is an abelian category in an evident way. The category of finite F_{p}-group schemes with p-power order is also an abelian category, using scheme-theoretic kernels and quotients. This is one of the essential ingredients in the proof of

Theorem 4.1 (Dieudonne-Cartier) There exists a contravariant additive anti-equivalence of abelian categories M : G ⇝ M(G) from the category of finite F_{p}-group schemes of p-power order to the category of finite D-modules. Moreover, the order of G is equal to the order of M(G) (i.e., p^{ℓ_{Zp} (M(G))}).

Proof. See [11, Ch III] for Fontaine’s proof, where it is obtained from more general considerations in the setting of certain formal commutative group schemes over an arbitrary perfect field of characteristic p.

■

For a finite F_{p}-group scheme G, the Dieudonne module of G is the finite D-module M(G) from Theorem 4.1. Just to give a hint as to where the construction of M(G) comes from, let us look at finite abelian p-groups, which are the same thing as finite C-group schemes of p-power order. Cartier duality G ⇝ G^{∗}, with

G^{∗}(T) = Hom_{T} (G ×_{C} T, G_{m}_{/T})

for C-schemes T, becomes on C-points just classical duality of finite abelian p-groups:

G ⇝ G^{∗} = Hom(G, C^{×}) = Hom(G, Q/Z) = Hom(G, Q_{p}/Z_{p}).

The Q_{p}/Z_{p} term is a direct limit of the groups Z/p^{n} = W_{n}(F_{p}), where W_{n} is the ‘Witt ring scheme of length n.’ Thus, if we could somehow exploit the group/ring-scheme theoretic properties of the W_{n}’s, we could try to construct a ‘schemified version’ of Q_{p}/Z_{p} as a formal scheme. Considering group scheme maps into such an object would give a reasonable candidate for a ‘linear algebra’ object attached to G. The actual construction of M(G) requires some care (e.g., W_{n} is a finite type scheme, not finite or even formal), but the above gives the flavor of the basic idea.

The essential content of Theorem 4.1 is that one can pass from ‘linear algebra data’ such as a finite D-module and produce something as subtle as a (possibly non-reduced) F_{p}-group scheme. By considering how the functor M is constructed as a sort of dual and trying to mimic ‘double duality’ to get a quasiinverse to M, the general proof of Theorem 4.1 uses a finite D-module M to define a functor (analogous to a ‘double dual’) from finite F_{p}-algebras to abelian groups. Then a general (and essentially formal) ‘prorepresentability’ theorem of Grothendieck’s is invoked. See [5, §1.4] for a precise formulation and proof of this ‘pro-representability’ theorem in the form needed. In this way, one gets a commutative formal F_{p}-group scheme whose affine ring R_{M} is an inverse limit of finite F_{p}-algebras over an enormous index set. One then has to show that R_{M} is actually finite over F_{p} (and that the finite F_{p}-group scheme G_{M} = Spec(R_{M}) has M(G_{M}) ≃ M naturally in M). But at least R_{M} is a ring to work with, albeit an abstract one (so Theorem 4.1 is not a complete black hole).

There are more refined versions of Theorem 4.1 which translate various notions from the theory of finite F_{p}-group schemes over into the language of finite D-modules. We give a limited sampling that is all we shall need.

Lemma 4.2 Let G be a finite F_{p}-group scheme with p-power order, M = M(G). Then G is étale if and only if F (M) = M (or, equivalently, F : M → M is a Z_{p}-linear isomorphism) and G is connected if and only if the action of F is nilpotent. Define M^{∗} = Hom_{Z_{p}}(M, Q_{p}/Z_{p}) and let F and V act on M^{∗} as duals to the actions of V and F on M respectively. In this way, M^{∗} has the structure of a finite D-module and there is a natural isomorphism of finite D-modules

φ_{G} : M(G^{∗}) → M^{∗},

with G^{∗} the Cartier dual of G.

The étale criteria in Lemma 4.2 corresponds to the fact that a finite F_{p}-algebra R is étale if and only if r ↦ r^{p} is an automorphism of R. The connectedness criteria corresponds to the fact [11, Ch I, Rem 9.5.2] that a finite connected F_{p}-group scheme always has an affine ring R = F_{p}[X_{1}, …, X_{m}]/(X_{j}^{p^{n_{j}}}), so it necessarily has p-power order, and for n ≥ max(n_{j}) the nth iterate of r ↦ r^{p} kills the augmentation ideal of R. The isomorphism φ_{G} is constructed in [11, Ch II, §5] in a very indirect manner. In particular, it is not at all clear from the definition that φ_{G} is a ‘symmetric’ duality pairing in the sense that the natural diagram

M(G)^{∗∗}  ⟶^{φ_{G}^{∗}} M(G^{∗})^{∗}
≃ ↑  ↑ φ_{G^{∗}}
M(G) ⟵^{M(α_{G})} M(G^{∗∗})

commutes, where α_{G} : G ≃ G^{∗∗} is the canonical isomorphism. The construction of φ_{G} involves the Artin-Hasse exponential function (for the prime p) and checking the commutativity is not easy. Here is how to do it. A similar duality pairing is defined in Oda’s thesis [24, Thm 3.19], where the commutativity is checked (it requires a long and careful analysis of Witt schemes and the Artin-Hasse exponential). By [11, Cor 3, Prop 5.3] and [24, Cor 3.15], the definitions of the Dieudonne module functor in [11] and [24] are isomorphic. There is a strong similarity between Oda’s duality construction and Fontaine’s definition of φ_{G}, though Fontaine’s definition is slightly more natural than Oda’s. One then has to check that the work in [24] can be applied to prove that the above diagram commutes (i.e., that the two duality constructions coincide). [Note to the editors: I have not actually checked this final point yet; I will try to check it soon]

In what we do, we will never need to know that the above diagram commutes. What we will need is the existence of the (natural) isomorphism φ_{G}.

Example 4.3 By Theorem 4.1, M(\underline{Z/p}), M(μ_{p}), and M(α_{p}) are all 1-dimensional over F_{p}. The actions of F and V on these are given as follows: G = \underline{Z/p}: F = 1, V = 0, G = μ_{p}: F = 0, V = 1, G = α_{p}: F = V = 0. By Lemma 4.2, the only thing which remains to be checked is that V = 1 when G = μ_{p}. This follows from [11, Ch I, §8.7; Ch III, Prop 4.3].

In general, it is possible that non-isomorphic finite flat Z_{p}-group schemes can have isomorphic closed fibers. In the notation of [25, Rem 5, pp.15-16], if π^{3} = p and we work in the category of finite flat group schemes over Z_{p}[π], then G_{π,Z_{p}[π]}^{π^{2}} and G_{π^{2},Z_{p}[π]}^{π} are non-isomorphic, yet they have isomorphic closed fiber α_{p}. With more work, one can construct examples over Z_{p} as well. In other words, the analogue to Theorem 1.6 for passage to the closed fiber is false: the functor G ⇝ G ×_{Z_{p}} F_{p} is not fully faithful. The emphasis here is on the ‘fully’ part; it is shown in the course of the proof of Theorem 4.5 below that this functor is faithful for p ≠ 2.

Thus, if one wishes to describe finite flat Z_{p}-group schemes in terms of ‘linear algebra data’, then one needs to find some ‘extra structure’ within M(G ×_{Z_{p}} F_{p}) that encodes the lifting G to Z_{p}. Using the affine ring of G_{/Z_{p}}, Fontaine constructs a Z_{p}-submodule of ‘logarithms’

L(G) ⊆ M(G ×_{Z_{p}} F_{p})

which is not necessarily stable under F and V but which satisfies the following two properties:

(1) V |_{L(G)} : L(G) → M(G ×_{Z_{p}} F_{p}) is injective

(2) The natural Z_{p}-linear composite map

L(G)/p → M(G ×_{Z_{p}} F_{p})/p ↠ M(G ×_{Z_{p}} F_{p})/F

is an isomorphism. This is valud even if p = 2.

For a finite F_{p}-group scheme H, M(H)/F can be canonically identified with the cotangent space of H at its origin [11, Ch III, Prop 4.4(ii)], so the second condition above shows that L(G) provides some sort of ‘minimal basis’ lifting of the cotangent space, partially explaining the name ‘logarithm’. [11, Ch IV] develops the ideas which motivate the construction of L(G), as well as the techniques which are needed to actually construct it. The brief article [12] gives an outline of the actual construction, whose justification relies on the theory of p-divisible groups; nearly the entire contents of the book [11] are needed for this! For a more detailed explanation of [12], see [7, §1].

It is now reasonable to make the following definition, following Fontaine.

Definition 4.4 The category SH^{f} of finite Honda systems (over Z_{p}) consists of pairs (L, M) with M a finite D-module and L a Z_{p}-submodule satisfying the properties (1) and (2) above. The notion of a morphism is defined in the obvious manner.

One can show directly that SH^{f} is an abelian category, using the obvious candidates for kernel and cokernel as the kernel and cokernel objects. See [13, §1, §9], noting that [13, Prop 9.10] provides a translation of SH^{f} into the language of finite filtered modules \underline{MF} _{tor}^{f} as is used in [13, Prop 1.8]. In addition, the construction of L(G) is sufficiently natural for a finite flat Z_{p}-group scheme G so that

LM : G ⇝ (L(G), M(G ×_{Z_{p}} F_{p}))

is an additive contravariant functor from the category of finite flat Z_{p}-group schemes to the category SH^{f}. We have the following fundamental fact, whose proof makes essential use of the theory of p-divisible groups:

Theorem 4.5 (Fontaine [12]) For p ≠ 2, the additive contravariant functor LM is an anti-equivalence of abelian categories.

The idea of the proof is to simultaneously show that all finite flat Z_{p}-group schemes embed into p-divisible groups over Z_{p} and to invoke Fontaine’s classification theory for such p-divisible groups (when p ≠ 2) in terms of a ‘finite free’ analogue of the notion of a finite Honda system. The special fact about p-divisible groups used in all of this is that over a field k of characteristic p (where the Dieudonne theory classifies various commutative group schemes when the field is perfect), a connected p-divisible group is the same thing as a finite-dimensional formal Lie group Γ such that for all finite k-algebras R, every element of the abelian group Γ(R) has p-power order.

Theorem 4.5, in conjunction with a good understanding of the construction of LM (e.g., the theory of Dieudonne modules, Lemma 4.2, etc.), enables us to translate questions concerning the construction and study of finite flat Z_{p}-group schemes (for p ≠ 2) into analogous questions in the setting of SH^{f}, where (in principle) everything is just a matter of ‘linear algebra’! There are more refined questions one could ask, such as whether one can give an explicit description of the functor that passes from a finite Honda system to the generic fiber D_{p}-representation of the associated finite flat Z_{p}-group scheme. This can be done and is the essential point of [13, §9]. We will not need this. However, we note in passing that this point has led to the misunderstanding that [13] is critical to the proof of Ramakrishna’s theorem and Wiles’ proof of the modularity of semistable elliptic curves over Q. This is not true. Everything which Ramakrishna and Wiles need is contained in Theorem 4.5, which was proven by Fontaine long before [13] was written. Nevertheless, we remind the reader that [13, Prop 9.12] can still be useful in the present setting; the main issue that [13, Prop 9.12] enables us to handle is the analysis of generic fiber representation ‘tensor’ constructions such as ‘determinant’; see also [13, §6.13(b)] and its corrected form in [6, §7.11]. Generic fiber representation ‘tensor’ constructions are difficult to study solely from the point of view of group schemes because the representation-theoretic notion of a tensor product has no analogue in the context of group schemes.

Before using Theorem 4.5 to construct Z_{p}-flat representations of D_{p} in the next section, we give a modified formulation which is used to handle the cases in which some finite extension O of Z_{p} acts on the representation space. Keeping in mind Raynaud’s full faithfulness result (Theorem 1.6), it is not hard to deduce a variant of Theorem 4.5 in the following manner.

Let A be a finite local Z_{p}-algebra and let D_{A} = A[F, V]/(F V − p). We can define the notion of a finite D_{A}-module in the obvious way; the category of such objects forms an A-linear subcategory of the category of finite D-modules. In a similar way, we can define a finite A-Honda system (L, M) by replacing Z_{p} by A in the definition of a finite Honda system (still using L/p and not L/𝔪_{A} in condition (2)). We then get a category SH_{A}^{f} which one readily checks is an abelian subcategory of SH^{f}, with the ‘forgetful functor’ SH_{A}^{f} → SH^{f} exact. We then have

Corollary 4.6 For A as above and p ≠ 2, the functor LM induces an A-linear anti-equivalence of abelian categories between the category of finite flat Z_{p}-group schemes with an A-action on the generic fiber representation and the category SH_{A}^{f}. If A = O or O/λ^{n} and G is a finite flat Z_{p}-group scheme with an A-action on its generic fiber representation ρ, then ρ and M(G ×_{Z_{p}} F_{p}) are non-canonically isomorphic as A-modules.

Proof. The only point we need to check is the final one. For a finite-length O-module N, it is clear that the abstract O-module structure of N is determined by the O-module structure of λN and the value of dim_{k}(N[λ]). Arguing par devissage and using the fact that the dimension of a k-vector space is determined by the underlying F_{p}-vector space, we are reduced to the case O = Z_{p} and objects killed by p. It now suffices to invoke the fact that p^{ℓ_{Zp} (M(G×_{Zp} F_{p}))} is equal to the order of G ×_{Z_{p}} F_{p}, which is equal to the order of G.

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§5. Applications to Flat Deformations

We now apply Fontaine’s theory to complete the proof of Lemma 3.4 and Ramakrishna’s Theorem 3.5. For the proof of Lemma 3.4 in the residually reducible (i.e., ‘ord’) cases, we give a variant on the arguments in [26] and [8, §2.5]. Our argument is different insofar as we make direct use of the results in §4, rather than work in the language of Fontaine-Laffaille modules. This makes the role of the Dieudonne theory more explicit and clarifies the role of the theory of finite flat group schemes in the calculations.

After finishing the proof of Lemma 3.4 in the residually reducible case, we will carry out the unfinished calculations from our earlier sketch of the proof of Theorem 3.5 (which, as we have already seen, implies Lemma 3.4 in the residually irreducible case).

Let \overline{ρ} and ρ be as in Lemma 3.4, with p ≠ 2 and \overline{ρ} possibly irreducible. By Theorem 1.6, there is a canonical finite flat Z_{p}-group scheme G(ρ) with generic fiber representation ρ. By Corollary 4.6, we obtain an object

LM(ρ) = (L(ρ), M(ρ)) =^{def} (L(G(ρ)), M(G(ρ) ×_{Z_{p}} F_{p}))

in SH_{O}^{f}. As an O-module, M(ρ) is free of rank 2 over O/λ^{n}, by Corollary 4.6. Observe that we have natural O-linear isomorphisms

H_{fl}^{1}(D_{p}, ad(ρ)) ≃ D_{ρ}^{fl}((O/λ^{n})[ε]) ≃ Ext_{(O/λ^{n})[D_{p}]}^{1,fl}(ρ, ρ) ≃ Ext_{SH_{O/λ^{n}}^{f}}^{1} (LM(ρ), LM(ρ)).

Using the above chain of isomorphisms, the following gives a more precise version of Lemma 3.4.

Theorem 5.1 As O/λ^{n}-modules, there is a non-canonical isomorphism

Ext_{SH_{O/λ^{n}}^{f}}^{1} (LM(ρ), LM(ρ)) ≃ (O/λ^{n}) ⊕ (O/λ^{n}) ⊕ H^{0}(D_{p}, ad^{0}(ρ)).

Proof. For any finite O-Honda system (L, M), the composite map

L/p → M/p → M/F M

is an isomorphism of O-modules. If we apply ⊗_{O}k, we get a composite isomorphism

L/λ → M/λ → (M/F M) ⊗_{O} k,

so the map L/λ → M/λ is injective. Thus, L is necessarily an O-module direct summand of M.

Consider the pair (L(ρ), M(ρ)). We claim that L(ρ) ≠ 0 and L(ρ) ≠ M(ρ), so therefore L(ρ) is a rank 1 free O/λ^{n}-module direct summand of M(ρ). If L(ρ) = 0, then M(ρ)/F (M(ρ)) = 0, so by Lemma 4.2, G(ρ) ×_{Z_{p}} F_{p} is étale. But then G(ρ) is étale over Z_{p}, in which case ρ is unramified. This contradicts the fact that \overline{ρ} = ρ mod λ has a ramified determinant.

If L(ρ) = M(ρ), then the injectivity of V on L(ρ) implies that V is an automorphism on M(ρ). Hence, by Lemma 4.2, the Cartier dual of G(ρ) is a finite étale Z_{p}-group scheme, so the Cartier dual ρ^{∗} of ρ is unramified. Thus, the Cartier dual \overline{ρ}^{∗} of \overline{ρ} is unramified, since \overline{ρ}^{∗} ≃ ρ^{∗}[λ]. This is inconsistent with the classification of possibilities for \overline{ρ} in Theorem 1.8(i).

Choose a basis e_{1}, e_{2} of M(ρ) over O/λ^{n} such that e_{2} is a basis for the direct summand L(ρ). Consider an extension (L, M) of LM(ρ) by itself in the abelian category SH_{O/λ^{n}}^{f}. By the very nature of the abelian

category structure of SH_{O/λ^{n}}^{f} (i.e., the construction of kernels and cokernels), it follows that M must be free of rank 4 as an O/λ^{n}-module and L must be free of rank 2 as an O/λ^{n}-module. Now choose an abstract rank 4 free O/λ^{n}-module M with a chosen basis m_{1}, m_{2}, m_{3}, m_{4} and we set L to be the submodule spanned by m_{2} and m_{4}. We fix a short exact sequence of O/λ^{n}-modules

0 → M(ρ) →^{j} M →^{h} M(ρ) → 0

determined by j(e_{1}) = m_{1}, j(e_{2}) = m_{2} and h(m_{3}) = e_{1}, h(m_{4}) = e_{2}. Our problem is to count the number of ways (up to equivalence respecting L) we can impose a D-module structure on M compatible with the D-module structure on M(ρ) via j and h. The main point is that since LM(ρ) is an object in SH_{O/λ^{n}}^{f}, any

such D-module structure on (L, M) structure would have to make (L, M) an object in SH_{O/λ^{n}}^{f} (and so the

resulting sequence would be a short exact sequence in SH_{O/λ^{n}}^{f}). We will check this below.

We will only consider the case in which \overline{ρ} is reducible, since otherwise we have seen in §3 that Theorem 5.1 follows from Theorem 3.5, whose proof we will finish later. Since \overline{ρ} is reducible, by Theorem 1.8(ii) there exist continuous unramified characters χ_{i} : D_{p} → (O/λ^{n})^{×} and a short exact sequence of (O/λ^{n})[D_{p}]-modules

0 → ϵχ_{1} → ρ → χ_{2} → 0.

Lemma 4.2 and the contravariance of LM enable us to modify our choice of e_{1} so that (0, (O/λ^{n})e_{1}) is a subobject of LM(ρ), corresponding to the unramified quotient χ_{2}. In addition, we have F (e_{1}) = ve_{1} and V (e_{1}) = v^{−1}pe_{1} for some v ∈ (O/λ^{n})^{×}.

Set V (e_{2}) = ae_{1} + ue_{2} and F (e_{2}) = be_{1} + ce_{2}. Since the subrepresentation ϵχ_{1} of ρ has an unramified Cartier dual, it follows that u ∈ (O/λ^{n})^{×}. Finally, the conditions F V = V F = p force c = u^{−1}p and a = −v^{−1}ub, so with respect to the ordered basis {e_{1}, e_{2}} of M(ρ), we have matrices

F_{M(ρ)} = (\begin{matrix}v & b \\ 0 & u^{−1}p \end{matrix}) , V_{M(ρ)} = (\begin{matrix}v^{−1}p & −v^{−1}ub \\ 0 & u \end{matrix}) .

Since we can write the actions of F and V on M in the ‘block matrix’ form

F_{M} = (\begin{matrix}F_{M(ρ)} & X \\ 0 & F_{M(ρ)} \end{matrix}) , V_{M} = (\begin{matrix}V_{M(ρ)} & −Y \\ 0 & V_{M(ρ)} \end{matrix}) ,

the conditions F V = V F = p yield the matrix equations

XV_{M(ρ)} = F_{M(ρ)}Y, V_{M(ρ)}X = Y F_{M(ρ)}.

We claim that with such data, (L, M) is necessarily an object in SH_{O/λ^{n}}^{f}. Since V |_{L(ρ)} is injective, V |_{L} is injective. In order to deduce that L/p → M/F M is an isomorphism from the corresponding fact for L(ρ) and M(ρ), the crux of the argument is to check that the sequence

0 → M(ρ)/F (M(ρ)) → M/F (M) → M(ρ)/F (M(ρ)) → 0

is actually exact on the left. In terms of the explicit basis for M(ρ), this reduces to the statement that if m ∈ M(ρ) is of the form X(m′) with F_{M(ρ)}(m′) = 0, then m ∈ F_{M(ρ)}(M(ρ)). Using length considerations, it is a straightfoward consequence of the axioms for a finite Honda system that the sequence

0 → M(ρ)/V (M(ρ)) →^{F} M(ρ)/p → M(ρ)/F (M(ρ)) → 0

is exact (and not just right exact). Thus, F_{M(ρ)}(m′) = 0 implies that m′ = V_{M(ρ)}(m_{0}), so

m = X(m′) = XV_{M(ρ)}(m_{0}) = F_{M(ρ)}Y (m_{0}) ∈ F_{M(ρ)}(M(ρ)),

as desired.

Define the O/λ^{n}-module

E(ρ) = {(X, Y) ∈ M_{2}(O/λ^{n}) × M_{2}(O/λ^{n}) | XV_{M(ρ)} = F_{M(ρ)}Y, V_{M(ρ)}X = Y F_{M(ρ)}}.

Inside of here is a submodule of ‘commutators’

C(ρ) = {([F_{M(ρ)}, A], −[V_{M(ρ)}, A]) | A = (a_{ij}) ∈ M_{2}(O/λ^{n}), a_{12} = 0}

(the a_{12} = 0 condition corresponds to preserving the line L(ρ) = (O/λ^{n})e_{2} inside of M(ρ)). It follows from [18, Ch III] that we have an O/λ^{n}-module isomorphism

Ext_{SH_{O/λ^{n}}^{f}}^{1} (LM(ρ), LM(ρ)) ≃ E(ρ)/C(ρ).

We now will determine the O/λ^{n}-module structures of E(ρ) and C(ρ). Choose X = (x_{ij}) and Y = (y_{ij}) in M_{2}(O/λ^{n}). The condition that (X, Y) ∈ E(ρ) is easily checked to be equivalent to the simultaneous constraints

x_{21} = u^{−1}vy_{21}, x_{22} = u^{−1}(by_{21} + u^{−1}py_{22}), y_{11} = v^{−1}(v^{−1}px_{11} − by_{21}),

and

y_{12} = v^{−1}(−v^{−1}ubx_{11} + ux_{12} − by_{22}),

with x_{11}, x_{12}, y_{21}, and y_{22} arbitrarily chosen in O/λ^{n}. Thus,

E(ρ) ≃ (O/λ^{n})^{⊕4}.

In order to determine the O/λ^{n}-module structure of C(ρ), we note that there is an obvious surjection

q : {A ∈ M_{2}(O/λ^{n}) | a_{12} = 0} ↠ C(ρ)

and the kernel is identified (as an O/λ^{n}-module) with all O/λ^{n}-module endomorphisms of M(ρ) which stabilize L(ρ) and commute with the actions of F and V — in other words, we have an O-linear isomorphism

ker(q) ≃ End_{SH_{O/λ^{n}}^{f}} (LM(ρ)) ≃ H^{0}(D_{p}, ad(ρ)),

where the second isomorphism is O-linear because of the linearity properties of the functor in Corollary 4.6. It is straightfoward to compute that for A = (a_{ij}) ∈ M_{2}(O/λ^{n}) with a_{12} = 0, A ∈ ker(q) if and only if a_{21} = 0 and b(a_{11} − a_{22}) = 0, so

H^{0}(D_{p}, ad(ρ)) ≃ (O/λ^{n}) ⊕ (O/λ^{n})[b].

Thus, we obtain

C(ρ) ≃ (O/λ^{n})^{⊕3}/((O/λ^{n}) ⊕ (O/λ^{n})[b]) ≃ (O/λ^{n}) ⊕ (b · (O/λ^{n}))

and

H^{0}(D_{p}, ad^{0}(ρ)) ≃ (O/λ^{n})[b] ≃ (O/λ^{n})/b.

Since

E(ρ)/C(ρ) ≃ (O/λ^{n})^{⊕2} ⊕ (O/λ^{n})/b,

we are done.

■

We now complete the proof of Theorem 3.5 (which requires \overline{ρ} to have trivial centralizer). We will handle the residually reducible and irreducible cases separately. First, we need to check that | Ext_{SH_{k}^{f}}^{1} (LM(\overline{ρ}), LM(\overline{ρ}))| =

|k|^{2} in order to obtain a surjection

π : O[[T_{1}, T_{2}]] ↠ Rfl ; \overline{ρ}.

In the residually reducible case, this is just the calculation in the proof of Theorem 5.1, with n = 1. Now consider the case in which \overline{ρ} is irreducible. By Theorem 1.8(i), \overline{ρ}|_{I_{p}} is self-dual. In particular, the closed fiber of G(\overline{ρ}) is connected with a connected dual, so by Lemma 4.2, F and V act in a nilpotent manner on M(\overline{ρ}). Thus, ker(V) ≠ 0, so we have a natural k-linear isomorphism

ker(V) ⊕ L(\overline{ρ}) ≃ M(\overline{ρ}),

giving two natural lines in M(\overline{ρ}). Let L(\overline{ρ}) = ke_{2} and ker(V) = ke_{1}, so with respect to this basis, we get the matrix

V_{M(\overline{ρ})} = (\begin{matrix}0 & \overline{b} \\ 0 & 0 \end{matrix}) ,

with \overline{b} ≠ 0. The condition F V = V F = p = 0 on M(\overline{ρ}) yields

F_{M(\overline{ρ})} = (\begin{matrix}0 & \overline{a} \\ 0 & 0 \end{matrix}) ,

for some \overline{a} ∈ k. Applying the above reasoning to the irreducible flat ‘connected’ dual \overline{ρ}^{∗}, we see that \overline{a} ≠ 0 by Lemma 4.2.

Thus, V = \overline{c}F, with \overline{c} = \overline{a}^{−1}\overline{b} ∈ k^{×}. It won’t matter for us what the value of \overline{c} ∈ k^{×} is, but we mention for completeness that \overline{c} is determined by the unramified character

ω^{−1} det \overline{ρ} : D_{p} → k^{×},

with \overline{c} = −1 when det \overline{ρ} = ω (see [7, Lemma 6.1] for more details). Defining E(\overline{ρ}) and C(\overline{ρ}) as in the proof of Theorem 5.1, we compute that E(\overline{ρ}) is 4-dimensional over k, while C(\overline{ρ}) is 2-dimensional over k, so

Ext_{SH_{k}^{f}}^{1} (LM(\overline{ρ}), LM(\overline{ρ})) ≃ E(\overline{ρ})/C(\overline{ρ})

is 2-dimensional over k, as desired.

As we saw in the sketch of the proof of Theorem 3.5 earlier, it remains to check that

| Dfl ; \overline{ρ}(O/λ^{n})| = |k|^{2(n−1)}.

In fact, the surjection π already gives us ≤, so it is enough to simply construct |k|^{2(n−1)} distinct flat deformations of \overline{ρ} to GL_{2}(O/λ^{n}). First consider the residually irreducible case. We will list |k|^{2(n−1)} objects X = (L_{X}, M_{X}) in SH_{O/λ^{n}}^{f} with M_{X} free of rank 2 over O/λ^{n} and

X[λ] ≃ LM(\overline{ρ})

in SH_{O}^{f} (recall that LM is contravariant). Fix a ∈ O/λ^{n} lifting \overline{a} ∈ k^{×} (where \overline{a} is defined via a matrix for F_{M(\overline{ρ})} as above). Choose any α, β ∈ O/λ^{n} with α ≡ 0 mod λ and β mod λ = \overline{b} (recall \overline{b} ∈ k^{×} from above). Define M_{X} to be free with basis e_{1}, e_{2} and define L_{X} = (O/λ^{n})e_{2}. Also, define

F_{M_{X}} = (\begin{matrix}α & a \\ pβ^{−1}a^{−1} & 0 \end{matrix}) , V_{M_{X}} = (\begin{matrix}0 & βa \\ pa^{−1} & −αβ \end{matrix}) .

It is easy to check that for each of the |k|^{2(n−1)} different choices of (α, β), the corresponding (L_{X}, M_{X}) is an object in SH_{O/λ^{n}}^{f} and that different pairs (α, β) give rise to non-isomorphic flat deformations of \overline{ρ} of the desired type. When O = W (k), [26] gives a non-explicit direct proof that | Dfl ; \overline{ρ}(O/λ^{n})| = |k|^{2(n−1)} for \overline{ρ} irreducible or reducible as in Theorem 3.5.

Before finishing off the residually reducible case in Theorem 3.5 (with trivial centralizer), note that what we have done so far completes the proof of Lemma 3.4 in all cases, which is what is needed in Wiles’ method.

Back to the reducible flat \overline{ρ} in Theorem 3.5. By the argument used to prove Theorem 5.1, we can choose a k-basis \{e_{1}, e_{2}\} for M(\overline{ρ}) with L(\overline{ρ}) = ke_{2} and

F_{M(\overline{ρ})} = (\begin{matrix}\overline{v} & \overline{b} \\ 0 & 0 \end{matrix}) , V_{M(\overline{ρ})} = (\begin{matrix}0 & −\overline{v}^{−1}\overline{u}\overline{b} \\ 0 & \overline{u} \end{matrix}) ,

with \overline{u}, \overline{v} ∈ k^{×} and \overline{b} ∈ k. Fix b ∈ O/λ^{n} lifting \overline{b}. For each of the |k|^{2(n−1)} pairs (u, v) with u, v ∈ O/λ^{n} lifting \overline{u} and \overline{v} respectively, define

F_{M} = (\begin{matrix}v & b \\ 0 & u^{−1}p \end{matrix}) , V_{M} = (\begin{matrix}v^{−1}p & −v^{−1}ub \\ 0 & u \end{matrix}) .

It is easy to check that in this way, (L, M) acquires the structure of an object in SH_{O/λ^{n}}^{f} with λ-torsion isomorphic to LM(\overline{ρ}), so the corresponding (O/λ^{n})[D_{p}]-module gives rise to a flat deformation of \overline{ρ} to GL_{2}(O/λ^{n}). Moreover, different pairs (u, v) are readily checked to give rise to non-isomorphic deformations. This concludes the proof of Theorem 3.5.

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