ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD 3-ADIC EXERCISES.

CHRISTOPHE BREUIL, BRIAN CONRAD, FRED DIAMOND, AND RICHARD TAYLOR

Introduction

In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2).

Theorem A. If E_{/Q} is an elliptic curve, then E is modular.

Theorem B. If \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(F_{5}) is an irreducible continuous representation with cyclotomic determinant, then \overline{ρ} is modular.

We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ_{1}(N) denote the subgroup of SL_{2}(Z) consisting of matrices that

modulo N are of the form

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

The quotient of the upper half plane by Γ_{1}(N), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y_{1}(N)_{/C}. This curve has a natural model Y_{1}(N)_{/Q}, which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N. We will let X_{1}(N) denote the smooth projective curve which contains Y_{1}(N) as a dense Zariski open subset.

Recall that a cusp form of weight k ≥ 1 and level N ≥ 1 is a holomorphic function f on the upper half complex plane ℌ such that

• for all matrices

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N)

we have f((az + b)/(cz + d)) = (cz + d)^{k}f(z);

• and |f(z)|^{2}(Im z)^{k} is bounded on ℌ. The space S_{k}(N) of cusp forms of weight k and level N is a finite dimensional complex vector space. If f ∈ S_{k}(N) then it has an expansion

f(z) = ∑_{n=1}^{∞} c_{n}(f)e^{2πinz}

and we define the L-series of f to be

L(f, s) = ∑_{n=1}^{∞} c_{n}(f)/n^{s}.

The first author was supported by the CNRS. The second author was partially supported by a grant from the NSF. The third author was partially supported by a grant from the NSF and an AMS Centennial Fellowship. He was working at Rutgers University during much of the research. The fourth author was partially supported by a grant from the NSF and by the Miller Institute for Basic Science.

For each prime p |̸N there is a linear operator T_{p} on S_{k}(N) defined by

(f|T_{p})(z) = p^{−1} ∑_{i=0}^{p−1} f((z + i)/p) + p^{k−1}(cpz + d)^{−k}f((apz + b)/(cpz + d))

for any

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z)

with c ≡ 0 mod N and d ≡ p mod N. The operators T_{p} for p |̸N can be simultaneously diagonalised on the space S_{k}(N) and a simultaneous eigenvector is called an eigenform. If f is an eigenform then the corresponding eigenvalues, a_{p}(f), are algebraic integers and we have c_{p}(f) = a_{p}(f)c_{1}(f).

Let λ be a place of the algebraic closure of Q in C above a rational prime ℓ and let \overline{Q}_{λ} denote the algebraic closure of Q_{ℓ} thought of as a \overline{Q} algebra via λ. If f ∈ S_{k}(N) is an eigenform, then there is a unique continuous irreducible representation

ρ_{f,λ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(\overline{Q}_{λ})

such that for any prime p |̸Nl, ρ_{f,λ} is unramified at p and tr ρ_{f,λ}(Frob_{p}) = a_{p}(f). The existence of ρ_{f,λ} is due to Shimura if k = 2 [Sh2], to Deligne if k > 2 [De] and to Deligne and Serre if k = 1 [DS]. Its irreducibility is due to Ribet if k > 1 [Ri] and Deligne and Serre if k = 1 [DS]. Moreover ρ is odd in the sense that det ρ of complex conjugation is −1. Also, ρ_{f,λ} is potentially semi-stable at ℓ in the sense of Fontaine. We can choose a conjugate of ρ_{f,λ} which is valued in GL_{2}(𝒪_{\overline{Q}_{λ}}), and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation

\overline{ρ}_{f,λ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(\overline{F}_{ℓ}),

which, up to isomorphism, does not depend on the choice of conjugate of ρ_{f,λ}.

Now suppose that ρ : G_{Q} → GL_{2}(\overline{Q}_{ℓ}) is a continuous representation which is unramified outside finitely many primes and for which the restriction of ρ to a decomposition group at ℓ is potentially semi-stable in the sense of Fontaine. To ρ|_{Gal(\overline{Q}_{ℓ}/Q_{ℓ})} we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil group of Q_{ℓ}. We define the conductor N(ρ) of ρ to be the product over p ≠ ℓ of the conductor of ρ|_{Gal(\overline{Q}_{p}/Q_{p})} and of the conductor of the Weil-Deligne representation associated to ρ|_{Gal(\overline{Q}_{ℓ}/Q_{ℓ})}. We define the weight k(ρ) of ρ to be 1 plus the absolute difference of the two Hodge-Tate numbers of ρ|_{Gal(\overline{Q}_{ℓ}/Q_{ℓ})}. It is known by work of Carayol and others that the following two conditions are equivalent

• ρ ∼ ρ_{f,λ} for some eigenform f and some place λ|ℓ;

• ρ ∼ ρ_{f,λ} for some eigenform f of level N(ρ) and weight k(ρ) and some place λ|ℓ. When these equivalent conditions are met we call ρ modular. It is conjectured by Fontaine and Mazur that if ρ : G_{Q} → GL_{2}(\overline{Q}_{ℓ}) is a continuous irreducible representation which satisfies

• ρ is unramified outside finitely many primes,

• ρ|_{Gal(\overline{Q}_{ℓ}/Q_{ℓ})} is potentially semi-stable with its smaller Hodge-Tate number 0,

• and, in the case where both Hodge-Tate numbers are zero, ρ is odd; then ρ is modular [FM].

Next consider a continuous irreducible representation \overline{ρ} : Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{ℓ}). Serre [Se2] defines the conductor N(\overline{ρ}) and weight k(\overline{ρ}) of \overline{ρ}. We call \overline{ρ} modular if \overline{ρ} ∼ \overline{ρ}_{f,λ} for some eigenform f and some place λ|ℓ. We call \overline{ρ} strongly modular if moreover we may take f to have weight k(\overline{ρ}) and level N(\overline{ρ}). It is known from work of Mazur, Ribet, Carayol, Gross, Coleman and Voloch and others that for ℓ ≥ 3, \overline{ρ} is strongly modular if and only if it is modular (see [Di1]). Serre has conjectured that all odd, irreducible \overline{ρ} are strongly modular [Se2].

Now consider an elliptic curve E_{/Q}. Let ρ_{E,ℓ} (resp. \overline{ρ}_{E,ℓ}) denote the representation of Gal(\overline{Q}/Q) on the ℓ-adic Tate module (resp. the ℓ-torsion) of E(\overline{Q}). Let N(E) denote the conductor of E. It is known that the following conditions are equivalent.

1. The L-function L(E, s) of E equals the L-function L(f, s) for some eigenform f.

2. The L-function L(E, s) of E equals the L-function L(f, s) for some eigenform f of weight 2 and level N(E).

3. For some prime ℓ, the representation ρ_{E,ℓ} is modular.

4. For all primes ℓ, the representation ρ_{E,ℓ} is modular.

5. There is a non-constant holomorphic map X_{1}(N)(C) → E(C) for some positive integer N.

6. There is a non-constant morphism X_{1}(N(E)) → E which is defined over Q. The equivalence of these conditions depends on the work of many mathematicians, including Carayol [Ca1] and Faltings [Fa]. When these equivalent conditions are satisfied we call E modular.

It has become a standard conjecture that all elliptic curves over Q are modular, although at the time this conjecture was first suggested the equivalence of the conditions above may not have been clear. Taniyama made a suggestion along the lines (1) as one of a series of problems collected at the Tokyo-Nikko conference in September 1955. However his formulation did not make clear whether f should be a modular form or some more general automorphic form. Shimura reports that he made more precise suggestions along the lines of items (5) and (6) in private conversations with a number of mathematicians (including Weil) in the early 1960’s. He checked this for CM elliptic curves [Sh1]. (The assertion (1) for CM elliptic curves was known much earlier.) Weil published in [We] a conjecture corresponding to (2) above.

Our approach to Theorem A is a extension of the methods of Wiles [Wi] and of Wiles and one of us (R.T., [TW]). We divide the proof into three cases.

1. If \overline{ρ}_{E,5}|_{Gal(\overline{Q}/Q(\sqrt{5}))} is irreducible, we show that ρ_{E,5} is modular.

2. If \overline{ρ}_{E,5}|_{Gal(\overline{Q}/Q(\sqrt{5}))} is reducible, but \overline{ρ}_{E,3}|_{Gal(\overline{Q}/Q(\sqrt{−3}))} is absolutely irreducible, we show that ρ_{E,3} is modular.

3. If \overline{ρ}_{E,5}|_{Gal(\overline{Q}/Q(\sqrt{5}))} is reducible and \overline{ρ}_{E,3}|_{Gal(\overline{Q}/Q(\sqrt{−3}))} is absolutely reducible, then we show that E

has j-invariant 0, (11/3)^{3}, or 5(29)^{3}/2^{5} and so is modular. In each of cases 1 and 2 there are two steps. First we prove that \overline{ρ}_{E,ℓ} is modular and then that ρ_{E,ℓ} is modular. In case 1 this first step is our Theorem B and in case 2 it is a celebrated theorem of Langlands and Tunnell [L], [T]. In fact, in both cases E obtains semi-stable reduction over a tame extension of Q_{ℓ} and the deduction of the modularity of ρ_{E,ℓ} from that of \overline{ρ}_{E,ℓ} was carried out in [CDT] by an extension of the methods of [Wi] and [TW]. In the third case we have to analyse the rational points on some modular curves of small level. This we did, with Elkies’ help, in [CDT].

It thus only remained to prove Theorem B. Let \overline{ρ} be as in that theorem. Twisting by a quadratic character we may assume that \overline{ρ}|_{Gal(\overline{Q}_{3}/Q_{3})} falls into one of the following cases (see §2.2).

1. \overline{ρ} is unramified at 3.

2. \overline{ρ}(I_{3}) has order 5.

3. \overline{ρ}(I_{3}) has order 4.

4. \overline{ρ}(I_{3}) has order 12 and \overline{ρ}|_{Gal(\overline{Q}_{3}/Q_{3})} has conductor 27.

5. \overline{ρ}(I_{3}) has order 3.

6. \overline{ρ}|_{Gal(\overline{Q}_{3}/Q_{3})} is induced from a character χ : Gal(\overline{Q}_{3}/Q_{3}(\sqrt{−3})) ⟶ F_{25}^{×} such that χ(−1) = −1 and

χ(\sqrt{−3}) = χ(1 + 3\sqrt{−3}) − χ(1 − 3\sqrt{−3}),

where we use the Artin map (normalised to take uniformisers to arithmetic Frobenius) to identify χ with a character of Q_{3}(\sqrt{−3})^{×}.

We will refer to these as the f = 1,3,9,27, 81 and 243 cases respectively.

Using the technique of Minkowski and Klein (i.e. the observation that the moduli space of elliptic curves with full level 5 structure has genus 0, see for example [Kl]), Hilbert irreducibility and some local computations of Manoharmayum [M], we find an elliptic curve E_{/Q} with the following properties (see §2.2)

• \overline{ρ}_{E,5} ∼ \overline{ρ},

• \overline{ρ}_{E,3} is surjective onto GL_{2}(F_{3}),

• and

1. in the f = 1 case, either \overline{ρ}_{E,3}|_{I_{3}} ⊗ F_{9} ∼ ω_{2} ⊕ ω_{2}^{3} or

\overline{ρ}_{E,3}|_{I_{3}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

and is peu ramifié;

2. in the f = 3 case,

\overline{ρ}_{E,3}|_{I_{3}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} );

3. in the f = 9 case, \overline{ρ}_{E,3}|_{I_{3}} ⊗ F_{9} ∼ ω_{2} ⊕ ω_{2}^{3};

4. in the f = 27 case,

\overline{ρ}_{E,3}|_{I_{3}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

and is très ramifié;

5. in the f = 81 case,

\overline{ρ}_{E,3}|_{I_{3}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

and is très ramifié;

6. in the f = 243 case,

\overline{ρ}_{E,3}|_{Gal(\overline{Q}_{3}/Q_{3})} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

is non-split over \overline{Q}_{3}^{ker \overline{ρ}} and is très ramifié.

(We are using the terms très ramifié and peu ramifié in the sense of Serre [Se2]. We are also letting ω denote the mod3 cyclotomic character and ω_{2} the the second fundamental character I_{3} → F_{9}^{×}, i.e.

ω_{2}(σ) ≡ σ(\sqrt[8]{3})/\sqrt[8]{3} mod \sqrt[8]{3}.

We will often use the equality ω = ω^{−1} without further remark.) We emphasise that for a general elliptic curve over Q with \overline{ρ}_{E,5} ≅ \overline{ρ}, the representation \overline{ρ}_{E,3} does not have the above form, rather we are placing a significant restriction on E.

In each case our strategy is to prove that ρ_{E,3} is modular and so deduce that \overline{ρ} ∼ \overline{ρ}_{E,5} is modular. Again we use the Langlands-Tunnell theorem to see that \overline{ρ}_{E,3} is modular and then an analogue of the arguments of [Wi] and [TW] to conclude that ρ_{E,3} is modular. This was carried out in [Di2] in the cases f = 1 and f = 3, and in [CDT] in the case f = 9. (In these cases the particular form of \overline{ρ}_{E,3}|_{I_{3}} is not important.) This leaves the cases f = 27, 81 and 243, which are complicated by the fact that E now only obtains good reduction over a wild extension of Q_{3}. In these cases our argument relies essentially on the particular form we have obtained for \overline{ρ}_{E,3}|_{Gal(\overline{Q}_{3}/Q_{3})} (depending on \overline{ρ}_{E,5}|_{I_{3}}). We do not believe that our methods for deducing the modularity of ρ_{E,3} from that of \overline{ρ}_{E,3} would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for \overline{ρ}|_{I_{3}} we can find a choice for \overline{ρ}_{E,3}|_{Gal(\overline{Q}_{3}/Q_{3})} for which our methods work.

Following Wiles, to deduce the modularity of ρ_{E,3} from that of \overline{ρ}_{E,3}, we consider certain universal deformations of \overline{ρ}_{E,3} and identify them with certain modular deformations which we realise over certain Hecke algebras. The key problem is to find the right local condition to impose on these deformations at the prime 3. As in [CDT] we require that the deformations lie in the closure of the characteristic zero points which are potentially Barsotti-Tate (i.e. come from a 3-divisible group over the ring of integers of a finite extension of Q_{3}) and for which the associated representation of the Weil group (see for example Appendix B of [CDT]) is of some specified form. That one can find suitable conditions on the representation of the Weil group at 3 for the arguments of [TW] to work seems to be a rare phenomenon in the wild case. It is here we make essential use of the fact that we need only treat our specific pairs (\overline{ρ}_{E,5}, \overline{ρ}_{E,3}).

Our arguments follow closely the arguments of [CDT]. There are two main new features. Firstly, in the f = 243 case, we are forced to specify the restriction of our representation of the Weil group completely, rather than simply its restriction to the inertia group as we have done in the past. Secondly, in the key computation of the local deformation rings, we now make use of a new description (due to C.B.) of finite flat group schemes over the ring of integers of any p-adic field in terms certain (semi-)linear algebra data (see [Br2] and the summary [Br1]). This description enables us to make these computations. As the persistent reader will soon discover they are lengthy and delicate, particularly in the case f = 243. It seems miraculous to us that these long computations with finite flat group schemes in §7, §8 and §9 give answers completely in accord with predictions made from much shorter computations with the local Langlands correspondence and the modular representation theory of GL_{2}(Q_{3}) in §3. We see no direct connection, but can’t help thinking that some such connection should exist.

Acknowledgements. We would like to thank Barry Mazur for helpful conversations and David Pollack for help with computer calculations which we made in an earlier attempt to compute some of these local deformation rings. We are grateful to the Harvard University Clay fund for supporting C.B. during a key visit he paid to Harvard. B.C. is grateful to the Institute for Advanced Study for its stimulating environment and the University of Münster for its hospitality. F.D. is grateful to Harvard University and the Université de Paris Sud for their hospitality. R.T. is grateful to the University of California at Berkeley for its hospitality and the Miller Institute for Basic Science for their support.

Notation. In this paper ℓ will denote a rational prime. In §1.1, §4.1, §4.2 and §4.3 it will be arbitrary. In the rest of §1 and in §5 we will only assume it is odd. In the rest of the paper we will only consider ℓ = 3.

If F is a field we will let \overline{F} denote a separable closure, F^{ab} the maximal subextension of \overline{F} which is abelian over F and G_{F} the Galois group Gal(\overline{F} /F). If F_{0} is a p-adic field (i.e. a finite extension of Q_{p}) and F′/F_{0} a (possibly infinite) Galois extension we will let I_{F′/F_{0}} denote the inertia subgroup of Gal(F′/F_{0}). We will also let I_{F_{0}} denote I_{\overline{F}_{0}/F_{0}}, Frob_{F_{0}} ∈ G_{F_{0}}/I_{F_{0}} denote the arithmetic Frobenius element, and W_{F_{0}} denote the dense subgroup of G_{F_{0}} consisting of elements which map to an integer power of Frob_{F_{0}}. We will normalise the Artin map of local class field theory so that uniformisers and arithmetic Frobenius elements correspond. (We apologise for this convention, which now seems to us a bad choice. However we feel it is important to stay consistent with [CDT].) We will let 𝒪_{F_{0}} denote the ring of integers of F_{0}, ℘_{F_{0}} the maximal ideal of 𝒪_{F_{0}} and k_{F_{0}} the residue field 𝒪_{F_{0}}/℘_{F_{0}}. We will write simply G_{p} for G_{Q_{p}}, I_{p} for I_{Q_{p}} and Frob_{p} for Frob_{Q_{p}}. We will also let Q_{p^{n}} denote the unique unramified degree n extension of Q_{p} in \overline{Q}_{p}. If k is any perfect field of characteristic p we will also use Frob_{p} to denote the p^{th}-power automorphism of k and its canonical lift to the Witt vectors W (k).

We will write ϵ for the ℓ-adic cyclotomic character and sometimes ω for the reduction of ϵ modulo ℓ. We will write ω_{2} for the second fundamental character I_{ℓ} → F_{ℓ^{2}}^{×}, i.e.

ω_{2}(σ) ≡ σ(ℓ^{1/(ℓ^{2}−1)})/ℓ^{1/(ℓ^{2}−1)} mod ℓ^{1/(ℓ^{2}−1)}.

We will also use ω and ω_{2} to denote the Teichmuller lifts of ω and ω_{2}.

We will let 1 denote the trivial character of a group. We will denote by V^{∨} the dual of a vector space V. If g : A → B is a homomorphism of rings and if X_{/ Spec A} is an A-scheme we will sometimes write ^{g}X

for the pullback of X by Spec g. We adopt this notation so that ^{g}(^{h}X) = ^{gh}X. Similarly if θ : X → Y is a morphism of schemes over A we will sometimes write ^{g}θ for the pullback of θ by Spec g.

By finite flat group scheme we will always mean commutative finite flat group scheme. If F_{0} is a field of characteristic 0 with fixed algebraic closure \overline{F}_{0} we will use without comment the canonical identification of finite flat F_{0}-group schemes with finite discrete Gal(\overline{F}_{0}/F_{0})-modules, and we will say that such objects correspond. If R is an Dedekind domain with field of fractions F of characteristic 0 then by a model of a finite flat F -group scheme G we mean a finite locally free R-group scheme 𝒢 and an isomorphism i : G →^{∼} 𝒢 × F′. As in Proposition 2.2.2 of [Ra] the isomorphism classes of models for G form a lattice ((𝒢, i) ≥ (𝒢′, i′) if there exists a map of finite flat group schemes 𝒢 → 𝒢′ compatible with i and i′) and we can talk about the inf and sup of two such models. If R is also local we will call the a model (𝒢, i) local-local if its special fibre is local-local. When the ring R is understood we will sometimes simply refer to (𝒢, i), or even just 𝒢, as an integral model of G.

We will use Serre’s terminology peu ramifié and très ramifié, see [Se2].

Contents

Introduction 1 Acknowledgements. 5 Notation. 5 1. Types 7 1.1. Types of local deformations 7 1.2. Types for admissible representations. 8 1.3. Reduction of types for admissible representations. 10 1.4. The main theorems. 12 2. Examples and applications. 15 2.1. Examples. 15 2.2. Applications. 17 2.3. An extension of a result of Manoharmayum. 19 3. Admittance. 20 3.1. The case of τ_{1}. 21 3.2. The case of τ_{−1}. 21 3.3. The case of τ_{±3}. 21 3.4. The case of τ′_{i}. 21 4. New deformation problems. 22 4.1. Some generalities on group schemes. 22 4.2. Filtrations. 24 4.3. Generalities on deformation theory. 27 4.4. Reduction steps for Theorem 2.1.2. 28 4.5. Reduction steps for Theorem 2.1.4. 28 4.6. Reduction steps for Theorem 2.1.6. 29 4.7. Some Galois cohomology. 32 5. Breuil Modules. 34 5.1. Basic properties of Breuil modules. 35 5.2. Examples. 37 5.3. Relationship to syntomic sheaves. 39 5.4. Base change. 40 5.5. Reformulation. 43 5.6. Descent data. 46 5.7. Another example. 47 6. Some local fields. 48 6.1. The case of F′_{1}. 48 6.2. The case of F′_{−1}. 48 6.3. The case of F′_{3}. 49 6.4. The case of F′_{−3}. 50 6.5. The case of F′_{i}. 50 7. Proof of Theorem 4.4.1. 52 7.1. Rank one calculations. 52 7.2. Rank two calculations. 52 7.3. Rank three calculations. 55 7.4. Conclusion of proof of Theorem 4.4.1. 56 8. Proof of Theorem 4.5.1. 57 8.1. Rank one calculations. 58 8.2. Models for \overline{ρ}. 59 8.3. Completion of proof of Theorem 4.5.1. 63 9. Proof of Theorems 4.6.1, 4.6.2 and 4.6.3. 65 9.1. Rank one calculations. 65 9.2. Models for \overline{ρ}. 65 9.3. Further rank two calculations. 72 9.4. Completion of the proof of Theorem 4.6.1. 75 9.5. Completion of the proof of Theorem 4.6.3. 76 9.6. Completion of the proof of Theorem 4.6.2. 78 10. Corrigenda for [CDT]. 79 References 80

1. Types

1.1. Types of local deformations. By an ℓ-type we mean an equivalence class of two-dimensional representations

τ : I_{ℓ} → GL(D)

over \overline{Q}_{ℓ} which have open kernel and which can be extended to a representation of W_{Q_{ℓ}}. By an extended ℓ-type we shall simply mean an equivalence class of two-dimensional representations

τ′ : W_{Q_{ℓ}} → GL(D′)

over \overline{Q}_{ℓ} with open kernel.

Suppose that τ is an ℓ-type and that K is a finite extension of Q_{ℓ} in \overline{Q}_{ℓ}. Recall from [CDT] that a continuous representation ρ of G_{ℓ} on a finite dimensional K-vector space M is said to be of type τ if

1. ρ is Barsotti-Tate over F for any finite extension F of Q_{ℓ} such that τ|_{I_{F}} is trivial;

2. the restriction of W D(ρ) to I_{ℓ} is equivalent to τ;

3. the character ϵ^{−1} det ρ has finite order prime to ℓ. (For the definition of “Barsotti-Tate” and of the representation W D(ρ) associated to a potentially Barsotti-Tate representation see §1.1 and Appendix B of [CDT]. Similarly if τ′ is an extended ℓ-type then we say that ρ is of extended type τ′ if

1. ρ is Barsotti-Tate over F for any finite extension F of Q_{ℓ} such that τ′|_{I_{F}} is trivial;

2. W D(ρ) is equivalent to τ′;

3. the character ϵ^{−1} det ρ has finite order prime to ℓ. Note that no representation can have extended type τ′ unless det τ′ is of the form χ_{1}χ_{2} where χ_{1} has finite order prime to ℓ and where χ_{2} is unramified and takes an arithmetic Frobenius element to ℓ; see Appendix B of [CDT]. (Using Theorem 1.4 of [Br2], one can show that for ℓ odd one obtains equivalent definitions of “type τ” and “extended type τ′” if one weakens the first assumption to simply require that ρ is potentially Barsotti-Tate.)

Now fix a finite extension K of Q_{ℓ} in \overline{Q}_{ℓ}. Let 𝒪 denote the integers of K and let k denote the residue field of 𝒪. Let

\overline{ρ} : G_{ℓ} → GL(V)

be a continuous representation of G_{ℓ} on a finite dimensional k-vector space and suppose that End_{k[G_{ℓ}]} V = k. One then has a universal deformation ring R_{V,𝒪} for \overline{ρ} (see, for instance, Appendix A of [CDT]).

We say that a prime ideal 𝔭 of R_{V,𝒪} is of type τ (resp. of extended type τ′) if there exist a finite extension K′ of K in \overline{Q}_{ℓ} and an 𝒪-algebra homomorphism R_{V,𝒪} → K′ with kernel 𝔭 such that the pushforward of the universal deformation of ρ over R_{V,𝒪} to K′ is of type τ (resp. of extended type τ′).

Let τ be an ℓ-type and τ′ an irreducible extended ℓ-type. If there do not exist any prime ideals 𝔭 of type τ (resp. of extended type τ′), we define R_{V,𝒪}^{D} = 0 (resp. R_{V,𝒪}^{D′} = 0). Otherwise, define R_{V,𝒪}^{D} (resp. R_{V,𝒪}^{D′} ) to be the quotient of R_{V,𝒪} by the intersection of all 𝔭 of type τ (resp. of extended type τ′). We say that a deformation of \overline{ρ} is weakly of type τ (resp. weakly of extended type τ′) if the associated local 𝒪-algebra map R_{V,𝒪} → R factors through the quotient R_{V,𝒪}^{D} (resp. R_{V,𝒪}^{D′} ). We say that τ (resp. τ′) is weakly acceptable for \overline{ρ} if either R_{V,𝒪}^{D} = 0 (resp. R_{V,𝒪}^{D′} = 0) or there is a surjective local 𝒪-algebra map 𝒪[[X]] ↠ R_{V,𝒪}^{D} (resp. 𝒪[[X]] ↠ R_{V,𝒪}^{D′} ). We say that τ (resp. τ′) is acceptable for \overline{ρ} if R_{V,𝒪}^{D} ≠ 0 (resp. R_{V,𝒪}^{D′} ≠ 0) and if there is a surjective local 𝒪-algebra map 𝒪[[X]] ↠ R_{V,𝒪}^{D} (resp. 𝒪[[X]] ↠ R_{V,𝒪}^{D′} ).

If K′ is a finite extension of K in \overline{Q}_{ℓ} with valuation ring 𝒪′ and residue field k′, then 𝒪′ ⊗_{𝒪} R_{V,𝒪}^{D} (resp. 𝒪′ ⊗_{𝒪} R_{V,𝒪}^{D′} ) is naturally isomorphic to R_{V ⊗_{k}k′,𝒪′}^{D} (resp. R_{V ⊗_{k}k′,𝒪′}^{D′} ). Thus (weak) acceptability depends only on τ (resp. τ′) and \overline{ρ}, and not on the choice of K. Moreover τ (resp. τ′) is acceptable for \overline{ρ} if and only if τ (resp. τ′) is acceptable for \overline{ρ} ⊗_{k} k′.

Although it is of no importance for the sequel, we make the following conjecture, part of which we already conjectured as conjecture 1.2.1 of [CDT].

Conjecture 1.1.1. Suppose that τ is an ℓ-type and τ′ an absolutely irreducible extended ℓ-type. A deformation ρ : G_{ℓ} → GL(M) of \overline{ρ} to the ring of integers 𝒪′ of a finite extension K′/K in \overline{Q}_{ℓ} is weakly of type τ (resp. weakly of extended ℓ-type τ′) if and only if M is of type τ (resp. of extended type τ′).

If τ is a tamely ramified ℓ-type then we expect that it is frequently the case that τ is acceptable for residual representations \overline{ρ}, as in Conjectures 1.2.2 and 1.2.3 of [CDT]. On the other hand if τ (resp. τ′) is a wildly ramified ℓ-type (resp. wildly ramified extended ℓ-type) then we expect that it is rather rare that τ (resp. τ′) is acceptable for a residual representation \overline{ρ}. In this paper we will be concerned with a few wild cases for the prime ℓ = 3 which do turn out to be acceptable.

1.2. Types for admissible representations. From now on we assume that ℓ is odd. If F is a finite extension of Q_{ℓ} we will identify F^{×} with W_{F}^{ab} via the Artin map. Let U_{0}(ℓ^{r}) denote the subgroup of GL_{2}(Z_{ℓ}) consisting of elements with upper triangular mod ℓ^{r} reduction. Also let Ũ_{0}(ℓ) denote the normaliser of U_{0}(ℓ) in GL_{2}(Q_{ℓ}). Thus Ũ_{0}(ℓ) is generated by U_{0}(ℓ) and by

w_{ℓ} = ( \begin{matrix}0 & −1 \\ ℓ & 0 \end{matrix} ) . (1.2.1)

If τ is an ℓ-type, set U_{τ} = GL_{2}(Z_{ℓ}) if τ is reducible and U_{τ} = U_{0}(ℓ) if τ is irreducible. If τ′ is an extended ℓ-type with τ′|_{I_{ℓ}} irreducible, set U_{τ′} = Ũ_{0}(ℓ). In this subsection we will associate to an ℓ-type τ an irreducible representation σ_{τ} of U_{τ} over \overline{Q}_{ℓ} with open kernel, and to an extended ℓ-type τ′ with τ′|_{I_{ℓ}} irreducible an irreducible representation σ_{τ′} of U_{τ′} over \overline{Q}_{ℓ} with open kernel. We need to consider seceral cases, which we treat one at a time.

First suppose that τ = χ_{1}|_{I_{ℓ}} ⊕ χ_{2}|_{I_{ℓ}} where each χ_{i} is a character of W_{Q_{ℓ}}. Let a denote the conductor of χ_{1}/χ_{2}. If a = 0 then set

σ_{τ} = St ⊗(χ_{1} ∘ det) = St ⊗(χ_{2} ∘ det),

where St denotes the Steinberg representation of PGL_{2}(F_{ℓ}). Now suppose that a > 0. Let σ_{τ} denote the induction from U_{0}(ℓ^{a}) to GL_{2}(Z_{ℓ}) of the character of U_{0}(ℓ^{a}) which sends

( \begin{matrix}α & β \\ ℓ^{a}γ & δ \end{matrix} ) ⟼ (χ_{1}/χ_{2})(α)χ_{2}(αδ − ℓ^{a}βγ).

This is irreducible and does not depend on the ordering of χ_{1} and χ_{2}.

For the next case, let F denote the unramified quadratic extension of Q_{ℓ} and s the non-trivial automorphism of F over Q_{ℓ}. Suppose that τ is the restriction to I_{ℓ} of the induction from W_{F} to W_{Q_{ℓ}} of a character χ of W_{F} with χ ≠ χ^{s}. Let a denote the conductor of χ/χ^{s}, so that a > 0. Choose a character χ′ of W_{Q_{ℓ}} such that χ′|_{W_{F}}^{−1} χ has conductor a. If a = 1 we set

σ_{τ} = Θ(χ′|_{W_{F}}^{−1} χ) ⊗ (χ′ ∘ det),

where Θ(·) is the irreducible representation of GL_{2}(F_{ℓ}) defined on page 532 of [CDT].

To define σ_{τ} for a > 1 we will identify GL_{2}(Z_{ℓ}) with the automorphisms of the Z_{ℓ}-module 𝒪_{F}. If a is even then we let σ_{τ} denote the induction from 𝒪_{F}^{×}(1+ℓ^{a/2}𝒪_{F} s) to GL_{2}(Z_{ℓ}) of the character φ of 𝒪_{F}^{×}(1+ℓ^{a/2}𝒪_{F} s), where, for α ∈ 𝒪_{F}^{×} and β ∈ (1 + ℓ^{a/2}𝒪_{F} s),

φ(αβ) = (χ′|_{W_{F}}^{−1} χ)(α)χ′(det αβ).

If a > 1 is odd, then we let σ_{τ} denote the induction from 𝒪_{F}^{×}(1 + ℓ^{(a−1)/2}𝒪_{F} s) to GL_{2}(Z_{ℓ}) of η, where η is the ℓ dimensional irreducible representation of 𝒪_{F}^{×}(1 + ℓ^{(a−1)/2}𝒪_{F} s) such that η|_{𝒪_{F}^{×}(1+ℓ^{(a+1)/2}𝒪_{F} s)} is the direct sum of the characters

αβ ⟼ (χ′|_{W_{F}}^{−1} χχ′′)(α)χ′(det αβ)

for α ∈ 𝒪_{F}^{×} and β ∈ (1 + ℓ^{(a+1)/2}𝒪_{F} s), where χ′′ runs over the ℓ non-trivial characters of 𝒪_{F}^{×}/Z_{ℓ}^{×}(1 + ℓ𝒪_{F}).

Now suppose τ′ is an extended type such that τ′|_{I_{ℓ}} is irreducible. There is a ramified quadratic extension F/Q_{ℓ} and a character χ of W_{F} such that the induction from W_{F} to W_{Q_{ℓ}} of χ is τ′ (see §2.6 of [G]). Let s denote the non-trivial field automorphism of F over Q_{ℓ} and also let ℘_{F} denote the maximal ideal of the ring of integers 𝒪_{F} of F. Let a denote the conductor of χ/χ^{s}, so a is even and a ≥ 2. We may choose a character χ′ of W_{Q_{ℓ}} such that χ′|_{W_{F}}^{−1} χ has conductor a. We will identify GL_{2}(Q_{ℓ}) with the automorphisms of the Q_{ℓ} vector space F. We will also identify U_{0}(ℓ) with the stabiliser of the pair of lattices ℘_{F}^{−1} ⊃ 𝒪_{F}. We define σ_{τ′} to be the induction from F^{×}(1 + ℘_{F}^{a/2}s) to Ũ_{0}(ℓ) of the character φ of F^{×}(1 + ℘_{F}^{a/2}s), where

φ(αβ) = (χ′|_{W_{F}}^{−1} χχ′′)(α)χ′(det αβ),

with α ∈ F^{×} and β ∈ (1 + ℘_{F}^{a/2}s), where χ′′ is a character of F^{×}/(𝒪_{F}^{×})^{2} defined as follows. Let ψ be a character of Q_{ℓ} with kernel Z_{ℓ}. Choose θ ∈ F^{×} such that for x ∈ ℘_{F}^{a−1} we have

(χ′|_{W_{F}}^{−1} χ)(1 + x) = ψ(tr_{F/Q_{ℓ}}(θx)).

We impose the following conditions which determine χ′′:

• χ′′ is a character of F^{×}/(𝒪_{F}^{×})^{2};

• χ′′|_{𝒪_{F}^{×}} is non-trivial;

• and

χ′′(−θ(N_{F/Q_{ℓ}}ϖ)^{a/2}) = ∑_{x∈Z/ℓZ} ψ(x^{2}/N_{F/Q_{ℓ}}ϖ),

where ϖ is a uniformiser in 𝒪_{F}.

Finally if τ is an irreducible ℓ-type, choose an extended ℓ-type τ′ which restricts to τ on I_{ℓ} and set σ_{τ} = σ_{τ′}|_{U_{0}(ℓ)}.

We remark that these definitions are independent of any choices (see [G]).

Recall that by the local Langlands conjecture we can associate to an irreducible admissible representation π of GL_{2}(Q_{ℓ}) a two-dimensional representation W D(π) of W_{Q_{ℓ}}. (See §4.1 of [CDT] for the normalisation we use.)

Lemma 1.2.1. Suppose that τ is an ℓ-type and that τ′ an extended ℓ-type with τ′|_{I_{ℓ}} irreducible. Suppose also that π is an irreducible admissible representation of GL_{2}(Q_{ℓ}) over \overline{Q}_{ℓ}.

1. σ_{τ} and σ_{τ′} are irreducible.

2. If W D(π)|_{I_{ℓ}} ∼ τ (resp. W D(π) ≅ τ′) then

Hom_{U_{τ}} (σ_{τ} , π) ≅ \overline{Q}_{ℓ}

(resp.

Hom_{U_{τ′}} (σ_{τ′}, π) ≅ \overline{Q}_{ℓ}).

3. If W D(π)|_{I_{ℓ}} ≅ τ̸ (resp. W D(π) ≅ τ̸′) then

Hom_{U_{τ}} (σ_{τ} , π) = (0)

(resp.

Hom_{U_{τ′}} (σ_{τ′}, π) = (0)).

Proof. Except in the case that τ is an irreducible ℓ-type, everything follows easily from [G]. Thus, suppose that τ is an irreducible ℓ-type and that τ′ is an extension of τ to an extended ℓ-type. If δ denotes the unramified quadratic character of W_{Q_{ℓ}} then τ′ ≁ τ′ ⊗ δ and so we deduce that

σ_{τ′} ≁ σ_{τ′⊗δ} ∼ σ_{τ′} ⊗ (δ ∘ det).

Thus σ_{τ′}|_{Q_{ℓ}^{×}U_{0}(ℓ)} is irreducible. It follows that σ_{τ} is irreducible. The second and third part of the lemma for τ follow similarly.

1.3. Reduction of types for admissible representations. We begin by reviewing some irreducible representations of GL_{2}(Z_{ℓ}), U_{0}(ℓ) and Ũ_{0}(ℓ). Let σ_{1,0} denote the standard representation of GL_{2}(F_{ℓ}) over \overline{F}_{ℓ}. If n = 0, 1, ..., ℓ − 1 and if m ∈ Z/(ℓ − 1)Z then we let σ_{n,m} = Symm^{n}(σ_{1,0}) ⊗ det^{m}. We may think of σ_{n,m} as a continuous representation of GL_{2}(Z_{ℓ}) over \overline{F}_{ℓ}. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible continuous representations of GL_{2}(Z_{ℓ}) over \overline{F}_{ℓ}.

If m_{1}, m_{2} ∈ Z/(ℓ − 1)Z we will let σ′_{m_{1},m_{2}} denote the character of U_{0}(ℓ) over \overline{F}_{ℓ} determined by

( \begin{matrix}a & b \\ ℓc & d \end{matrix} ) ⟼ a^{m_{1}}d^{m_{2}}.

These representations are irreducible, mutually non-isomorphic, and exhaust the irreducible continuous representations of U_{0}(ℓ) over \overline{F}_{ℓ}.

If m_{1}, m_{2} ∈ Z/(ℓ − 1)Z, a ∈ \overline{F}_{ℓ}^{×} and m_{1} ≠ m_{2} then we let σ′_{\{m_{1},m_{2}\},a} denote the representation of

Ũ_{0}(ℓ) over \overline{F}_{ℓ} obtained by inducing the character of Q_{ℓ}^{×}U_{0}(ℓ) which restricts to σ′_{m_{1},m_{2}} on U_{0}(ℓ) and which sends −ℓ to a. If m ∈ Z/(ℓ − 1)Z and a ∈ \overline{F}_{ℓ}^{×}, then we let σ′_{\{m\},a} denote the character of Ũ_{0}(ℓ) over \overline{F}_{ℓ} which restricts to σ′_{m,m} on U_{0}(ℓ) and which sends w_{ℓ} to a. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible, finite dimensional, continuous representations of Ũ_{0}(ℓ) over \overline{F}_{ℓ}.

We will say that a reducible ℓ-type τ (resp. irreducible ℓ-type, resp. extended ℓ-type τ with irreducible restriction to I_{ℓ}) admits an irreducible representation σ of GL_{2}(Z_{ℓ}) (resp. U_{0}(ℓ), resp. Ũ_{0}(ℓ)) over \overline{F}_{ℓ}, if σ_{τ} (resp. σ_{τ}, resp. σ_{τ′}) contains an invariant 𝒪_{\overline{Q}_{ℓ}}-lattice Λ and if σ is a Jordan-Hölder constituent of Λ ⊗ \overline{F}_{ℓ}. We will say that τ (resp. τ, resp. τ′) simply admits σ if σ is a Jordan-Hölder constituent of Λ ⊗ \overline{F}_{ℓ} of multiplicity one.

For each of the \overline{F}_{ℓ}-representations of GL_{2}(Z_{ℓ}), U_{0}(ℓ) and Ũ_{0}(ℓ) just defined, we wish to define notions of “admittance” and “simple admittance” with respect to a continuous representation \overline{ρ} : G_{ℓ} → GL_{2}(\overline{F}_{ℓ}). Let \overline{ρ} be a fixed continuous representation G_{ℓ} → GL_{2}(\overline{F}_{ℓ}).

• The representation σ_{n,m} admits \overline{ρ} if either

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω_{2}^{1−ℓn−m(ℓ+1)} & 0 \\ 0 & ω_{2}^{ℓ−n−m(ℓ+1)} \end{matrix} )

or

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m} & ∗ \\ 0 & ω^{−n−m} \end{matrix} ) ,

which in addition we require to be peu-ramifié in the case n = 0. (Note that σ_{n,0} admits \overline{ρ} if and only if the Serre weight (see [Se2]) of \overline{ρ}^{∨} ⊗ ω is n + 2.)

• The representation σ_{n,m} simply admits \overline{ρ} if σ_{n,m} admits \overline{ρ}.

• The representation σ′_{m_{1},m_{2}} admits \overline{ρ} if either

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω_{2}^{1−ℓm_{i}−m_{j}} & 0 \\ 0 & ω_{2}^{ℓ−m_{i}−ℓm_{j}} \end{matrix} )

where {m_{i}, m_{j}} = {m_{1}, m_{2}} and m_{i} ≥ m_{j}, or

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{1}} & ∗ \\ 0 & ω^{−m_{2}} \end{matrix} ) ,

or

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{2}} & ∗ \\ 0 & ω^{−m_{1}} \end{matrix} ) .

(Note that σ′_{m_{1},m_{2}} admits barrho if and only if some irreducible constituent of Ind_{U_{0}(ℓ)}^{GL_{2}(Z_{ℓ})} σ′_{m_{1},m_{2}} admits \overline{ρ}.)

• The representation σ′_{m_{1},m_{2}} with m_{1} ≠ m_{2} simply admits \overline{ρ} if either

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{1}} & ∗ \\ 0 & ω^{−m_{2}} \end{matrix} )

or

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{2}} & ∗ \\ 0 & ω^{−m_{1}} \end{matrix} ) .

• The representation σ′_{m,m} simply admits \overline{ρ} if

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m} & ∗ \\ 0 & ω^{−m} \end{matrix} )

is très ramifié.

• The representation σ′_{\{m_{1},m_{2}\},a} with m_{1} ≠ m_{2} admits \overline{ρ} if either σ′_{m_{1},m_{2}} or σ′_{m_{2},m_{1}} admits \overline{ρ} and if (ω^{−1} det \overline{ρ})|_{W_{Q_{ℓ}}} equals the central character of σ′_{\{m_{1},m_{2}\},a}. (Note that in this case σ′_{\{m_{1},m_{2}\},a}|_{U_{0}(ℓ)} = σ′_{m_{1},m_{2}} ⊕ σ′_{m_{2},m_{1}}.)

• The representation σ′_{\{m_{1},m_{2}\},a} with m_{1} ≠ m_{2} simply admits \overline{ρ} if (ω^{−1} det \overline{ρ})|_{W_{Q_{ℓ}}} equals the central character of σ′_{{m_{1},m_{2}},a} and either

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{1}} & ∗ \\ 0 & ω^{−m_{2}} \end{matrix} ) ,

or

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m_{2}} & ∗ \\ 0 & ω^{−m_{1}} \end{matrix} ) .

• The representation σ′_{\{m\},a} admits \overline{ρ} if – σ′_{m,m} admits \overline{ρ},

– (ω^{−1} det \overline{ρ})|_{W_{Q_{ℓ}}} equals the central character of σ′_{\{m\},a},

– and, if

\overline{ρ}|_{I_{ℓ}} ∼ ( \begin{matrix}ω^{1−m} & ∗ \\ 0 & ω^{−m} \end{matrix} )

is très ramifié, then

\overline{ρ} ∼ ( \begin{matrix}∗ & ∗ \\ 0 & ω^{−m}χ \end{matrix} )

where χ is unramified and sends Frobenius to −a. (Note that σ′_{{m},a}|_{U_{0}(ℓ)} = σ′_{m,m}.)

• The representation σ′_{\{m\},a} simply admits \overline{ρ} if σ′_{\{m\},a} admits \overline{ρ}.

We remark that the definition of “σ admits the Cartier dual of \overline{ρ}” might look more natural to the reader. We are forced to adopt this version of the definition by some unfortunate choices of normalisations in [CDT].

We say that a reducible ℓ-type τ (resp. irreducible ℓ-type τ, resp. extended ℓ-type τ′ with τ′|_{I_{ℓ}} irreducible) admits a continuous representation \overline{ρ} : G_{ℓ} → GL_{2}(\overline{F}_{ℓ}) if τ (resp. τ, resp. τ′) admits an irreducible representation of GL_{2}(Z_{ℓ}) (resp. U_{0}(ℓ), resp. Ũ_{0}(ℓ)) over \overline{F}_{ℓ} which in turn admits \overline{ρ}. We say that τ (resp. τ, resp. τ′) simply admits \overline{ρ} if

• τ (resp. τ, resp. τ′) admits a unique irreducible representation σ of GL_{2}(Z_{ℓ}) (resp. U_{0}(ℓ), resp. Ũ_{0}(ℓ)) over \overline{F}_{ℓ} which admits \overline{ρ},

• τ (resp. τ, resp. τ′) simply admits σ,

• and σ simply admits \overline{ρ}.

Note that the concept of “simply admits” is strictly stronger than the concept “admits”.

The starting point for this work was the following conjecture, of which a few examples will be verified in §2.1.

Conjecture 1.3.1. Let k be a finite subfield of \overline{F}_{ℓ}, \overline{ρ} : G_{ℓ} → GL_{2}(k) a continuous representation, τ an ℓ-type and τ′ an extended ℓ-type with irreducible restriction to I_{ℓ}. Suppose that the centraliser of the image of \overline{ρ} is k and that the image of τ is not contained in the centre of GL_{2}(\overline{Q}_{ℓ}).

1. τ (resp. τ′) admits \overline{ρ} if and only if R_{V,𝒪}^{D} ≠ (0) (resp. R_{V,𝒪}^{D′} ≠ (0)), i.e. if and only if there is a finite extension K′ of Q_{ℓ} in \overline{Q}_{ℓ} and a continuous representation ρ : G_{ℓ} → GL_{2}(𝒪_{K′}) which reduces to \overline{ρ} and has type τ (resp. has extended type τ′).

2. τ (resp. τ′) simply admits \overline{ρ} if and only if τ (resp. τ′) is acceptable for \overline{ρ}.

We remark that to check if τ or τ′ simply admits \overline{ρ} is a relatively straightforward computation. On the other hand to show that τ or τ′ is acceptable for \overline{ρ} is at present a non-trivial undertaking. (The reader who doubts us might like to compare §3 with §4, §5, §6, §7, §8 and §9. All the latter sections are devoted to verifying some very special cases of this conjecture.)

1.4. The main theorems. With these definitions, we can state our two main theorems. The proofs very closely parallel the proof of Theorem 7.1.1 of [CDT].

Theorem 1.4.1. Let ℓ be an odd prime, K a finite extension of Q_{ℓ} in \overline{Q}_{ℓ} and k the residue field of K. Let

ρ : G_{Q} ⟶ GL_{2}(K)

be an odd continuous representation ramified at only finitely many primes. Assume that its reduction

\overline{ρ} : G_{Q} ⟶ GL_{2}(k)

is absolutely irreducible after restriction to Q(\sqrt{(−1)^{(ℓ−1)/2}ℓ}) and is modular. Further, suppose that • \overline{ρ}|_{G_{ℓ}} has centraliser k,

• ρ|_{G_{ℓ}} is potentially Barsotti-Tate with ℓ-type τ,

• τ admits \overline{ρ},

• and τ is weakly acceptable for \overline{ρ}. Then ρ is modular.

Proof. Note that the existence of ρ shows that τ is acceptable for \overline{ρ}. Now the proof is verbatim the proof of Theorem 7.1.1 of [CDT] (see §1.3, §1.4, §3, §4, §5 and §6 of that paper, and the corrigendum at the end of this paper), with the following exceptions.

• On page 539 one should take U_{S,ℓ} = U_{τ}, V_{S,ℓ} = ker σ_{τ} and σ_{S,l} = σ_{τ}.

• In the proof of Lemma 5.1.1 one must use Lemma 1.2.1 of this paper, in addition to the results recalled in §4 of [CDT].

• On page 546 replace “Setting T = T (\overline{ρ}) ∪ \{r\} ...” to the end of the first paragraph by the following. (Again the key component of this argument is very similar to the main idea of [Kh].)

“Set S = T (\overline{ρ}) ∪ \{r\}; U′_{S} = ∏_{p} U′_{S,p} where U′_{S,p} = U_{1}(p^{c_{p}}) if p ∈ T (\overline{ρ}) and U′_{S,p} = U_{S,p} otherwise; V′_{S} = ∏_{p} V′_{S,p} where V′_{S,p} = U_{1}(p^{c_{p}}) if p ∈ T (\overline{ρ}) and V′_{S,p} = V_{S,p} otherwise; and L′_{S} = Hom_{𝒪[U′_{S}/V′_{S}]} (M_{ℓ}, H^{1}(X_{V′_{S}}, 𝒪))[I′_{S}]. Then Γ = SL_{2}(Z) ∩ (U′_{S}GL_{2}(Z_{ℓ})) satisfies the hypotheses of Theorem 6.1.1. Furthermore

H^{1}(Y_{U′_{S}GL_{2}(Z_{ℓ})}, ℱ_{M}) ≅ H^{1}(Γ, L_{n} ⊗ k)

as a T̃′_{S}-module where M is the module for U_{S,ℓ} = GL_{2}(Z_{ℓ}) defined by the action of GL_{2}(F_{ℓ}) on L_{n}⊗k. Therefore 𝔪_{S} is in the support of H^{1}(Y_{U′_{S}GL_{2}(Z_{ℓ})}, ℱ_{M}).

We now drop the special assumption on \overline{ρ}|_{I_{ℓ}} made in the last paragraph. Twisting we see that if σ is an irreducible representation of GL_{2}(Z_{ℓ}) over \overline{F}_{ℓ} admitting \overline{ρ}|_{G_{ℓ}} then

H^{1}(Y_{U′_{S} GL_{2}(Z_{ℓ})}, ℱ_{σ^{∨}})_{𝔪_{S}} ≠ (0).

Moreover if τ is irreducible and if σ′ is an irreducible representation of U_{0}(ℓ) over \overline{F}_{ℓ} which admits \overline{ρ}|_{G_{ℓ}} then we see using the definition of admits and Lemmas 3.1.1 and 6.1.2 of [CDT] that

H^{1}(Y_{U′_{S}}, ℱ_{σ^{∨}})_{𝔪_{S}} ≅ H^{1}(Y_{U′_{S} GL_{2}(Z_{ℓ})}, ℱ_{Ind_{U_{0}(ℓ)}^{GL_{2}(Z_{ℓ})} σ^{∨}})_{𝔪_{S}} ≠ (0).

It follows from the definition of admits and Lemma 6.1.2 of [CDT] that 𝔪_{S} is in the support of H^{1}(Y_{U′_{S}}, ℱ_{Hom_{𝒪} (M_{ℓ},𝒪)} ), so L′_{S} is non-zero. Using the fact that lemma 5.1.1 holds with U′_{S} replacing U_{S} and σ_{ℓ} replacing σ_{S} and the discussion on page 541 we conclude that 𝒩_{S} is non-empty.”

Theorem 1.4.2. Let ℓ be an odd prime, K a finite extension of Q_{ℓ} in \overline{Q}_{ℓ} and k the residue field of K. Let

ρ : G_{Q} ⟶ GL_{2}(K)

be an odd continuous representation ramified at only finitely many primes. Assume that its reduction

\overline{ρ} : G_{Q} ⟶ GL_{2}(k)

is absolutely irreducible after restriction to Q(\sqrt{(−1)^{(ℓ−1)/2}ℓ}) and is modular. Further, suppose that • \overline{ρ}|_{G_{ℓ}} has centraliser k,

• ρ|_{G_{ℓ}} is potentially Barsotti-Tate with extended ℓ-type τ′,

• τ′ admits \overline{ρ},

• and τ′ is weakly acceptable for \overline{ρ}. Then ρ is modular.

Proof. The existence of ρ shows that τ′ is in fact acceptable for \overline{ρ}. Again the proof now follows very closely that of Theorem 7.1.1 of [CDT]. In this case we have to make the following changes. All references are to [CDT] unless otherwise indicated.

• On page 539 one should take U_{S,ℓ} = U_{0}(ℓ), V_{S,ℓ} = ker σ_{τ′}|_{U_{0}(ℓ)} and σ_{S,ℓ} = σ_{τ} |_{U_{0}(ℓ)}. One should also define Ũ_{S} to be the group generated by U_{S} and w_{ℓ} ∈ GL_{2}(Q_{ℓ}) and σ̃_{S} to be the extension of σ_{S} to Ũ_{S} which restricts to σ_{τ′} on Ũ_{0}(ℓ).

• In the statement of Lemma 5.1.1 one should replace Hom_{U_{S}}(σ_{S}, π^{∞}) by HomŨ_{S}(σ̃_{S}, π^{∞}).

• In the proof of Lemma 5.1.1 one must use Lemma 1.2.1 above in addition to the results recalled in §4 of [CDT].

• Because τ′ is acceptable for \overline{ρ}, we know that det τ of a Frobenius lift is ℓζ for some root of unity ζ. Thus, σ_{τ′}(ℓ^{s}) = 1 for some s > 0. Hence, σ̃_{S} factors through the finite group G̃_{S} = Ũ_{S}/V_{S}ℓ^{sZ}, where ℓ ∈ GL_{2}(Q_{ℓ}).

• In §5.3 choose M_{ℓ} so that it is invariant for the action of Ũ_{0}(ℓ)/V_{S,ℓ}ℓ^{sZ}. Also, in the definition of L_{S} replace G_{S} by G̃_{S}.

• In the proof of Lemma 5.3.1 replace U_{S} by Ũ_{S} and σ_{S} by σ̃_{S}.

• Note that w_{ℓ} acts naturally on Y_{S} and ℱ_{S}. In Lemma 6.1.3 we should replace the group H_{c}^{1}(Y_{S}, ℱ_{S}) by H_{c}^{1}(Y_{S}, ℱ_{S})^{w_{ℓ}=1} and the group H^{1}(Y_{S}, ℱ_{S}) by H^{1}(Y_{S}, ℱ_{S})^{w_{ℓ}=1}.

• Replace §6.2 with the proof of the required extension of Proposition 5.4.1 given below.

• On page 547 the isomorphism

H_{c}^{1}(Y_{S}, ℱ_{S}) ⟶ Hom_{𝒪} (H^{1}(Y_{S}, ℱ_{S}), 𝒪)

on line 6 is T̃′_{S}[w_{l}]-linear. In the next line one should not only localise at 𝔪 but restrict to the kernel of w_{ℓ} − 1. Because w_{ℓ}^{2} = 1 on H^{1}(Y_{S}, ℱ_{S})_{𝔪} we see that the natural map

H^{1}(Y_{S}, ℱ_{S})_{𝔪}^{w_{ℓ}=1} ⟶ H^{1}(Y_{S}, ℱ_{S})_{𝔪}/(w_{ℓ} − 1)

is an isomorphism, and so the map

L_{S} ⟶ Hom_{𝒪} (L_{S}, 𝒪)

is also an isomorphism.

• On page 547 the groups H^{1}(Y_{S}, ℱ_{S})_{𝔪_{S}^{(p)}} and H^{1}(Y_{S′}, ℱ_{S′})_{𝔪_{S}^{(p)}} should be replaced by their maximal subgroups on which w_{ℓ} = 1.

• On page 549 one should also define Ṽ_{0} (resp. Ṽ_{1}) to be the group generated by V_{0} (resp. V_{1}) and w_{ℓ} ∈ GL_{2}(Q_{ℓ}). Similarly define σ̃ to be σ̃_{∅} ⊗ ψ_{r′}^{−2}.

• In Lemma 6.4.1 replace V_{0} by Ṽ_{0}, V_{1} by Ṽ_{1} and σ by σ̃. In the proof of Lemma 6.4.1 also replace U_{{r,r′}} (resp. U_{S∪{r,r′}}) by Ũ_{{r,r′}} (resp. Ũ_{S∪{r,r′}}) and σ_{{r,r′}} (resp. σ_{S∪{r,r′}}) by σ_{{r,r′}} (resp. σ_{S∪{r,r′}}).

• On line 20 of page 550 M should be chosen as a model of σ̃. This is possible because ker σ̃ has finite index in Ṽ_{0}, because in turn σ_{τ′}(ℓ^{s}) = 1 for some s > 0. One should also set L_{i} = H^{1}(Y_{V_{i}}, ℱ_{M^{∨}})_{𝔪}^{w_{ℓ}=1}. On line 25, we must replace V_{i} by Ṽ_{i}.

• In the proof of Lemma 6.4.2, one must replace V_{1} by Ṽ_{1} and σ by σ̃.

• In line 2 of the proof of Lemma 6.4.3, to see that L_{1} is a direct summand of H^{1}(Y_{V_{1}}, ℱ_{M^{∨}}) as an 𝒪[Δ_{S}]- module, one needs to note that H^{1}(Y_{V_{1}}, ℱ_{M^{∨}})_{𝔪}^{w_{ℓ}=1} is a direct summand of H^{1}(Y_{V_{1}}, ℱ_{M^{∨}})_{𝔪}, because w_{ℓ}^{2} = 1 on H^{1}(Y_{V_{1}}, ℱ_{M^{∨}})_{𝔪}.

• On line 12 of page 551 replace R_{V,𝒪}^{∅,D} by R_{V,𝒪}^{∅,D′}. Proof of extension of Proposition 5.4.1 of [CDT]. Let Θ = ⨂_{p∈T (\overline{ρ})} M_{p}.

First suppose that τ′ admits σ′_{\{m_{1},m_{2}\},a} with m_{1} ≠ m_{2} and that σ′_{\{m_{1},m_{2}\},a} admits \overline{ρ}. As in the proof of Theorem 1.4.1 (especially §6.2 of [CDT] as modified above), we have

H^{1}(Y_{{r}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{m_{1},m_{2}} )^{∨}})_{𝔪′_{{r}}} ≠ (0).

On the other hand

\begin{aligned}H^{1}(Y_{\{r\}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{m_{1},m_{2}} )^{∨}})_{𝔪′_{\{r\}}} & ≅ & H^{1}(Y_{\{r\}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{m_{1},m_{2}} )^{∨}})_{𝔪′_{\{r\}}}^{w_{ℓ}^{2}=a} \\ & ≅ & H^{1}(Y_{\{r\}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{\{m_{1},m_{2}\},a})^{∨}})_{𝔪′_{\{r\}}}^{w_{ℓ}=1}. \end{aligned}

Thus, using the definition of “admits” and Lemma 6.1.2 of [CDT], we see that

H^{1}(Y_{{r}}, ℱ_{{r}})_{𝔪′_{{r}}}^{w_{ℓ}=1} ≠ (0),

so 𝒩_{∅} = 𝒩_{{r}} ≠ ∅.

Next suppose that τ′ admits σ′_{\{m\},a} which in turn admits \overline{ρ}. Assume that \overline{ρ}|_{G_{ℓ}} is irreducible or peu ramifié. By twisting we may reduce to the case m = 0. As in the proof of Theorem 1.4.1 (especially §6.2 of [CDT] as modified above), we have

H^{1}(Y_{U_{{r}}GL_{2}(Z_{ℓ})}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}} ≠ (0).

Thus

H^{1}(Y_{U_{{r}}GL_{2}(Z_{ℓ})}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}}^{w_{ℓ}^{2}=a^{2}} ≠ (0),

where ã is the Teichmüller lift of a. Using the embedding

ã + w_{ℓ} : H^{1}(Y_{U_{\{r\}}GL_{2}(Z_{ℓ})}, ℱ_{Θ^{∨}}) ⊗ \overline{Q}_{ℓ} ↪ H^{1}(Y_{\{r\}}, ℱ_{Θ^{∨}}) ⊗ \overline{Q}_{ℓ},

we deduce that

H^{1}(Y_{{r}}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}}^{w_{ℓ}=ã} ≠ (0),

and so

H^{1}(Y_{{r}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{{0},a})^{∨}})_{𝔪′_{{r}}}^{w_{ℓ}=a} ≠ (0).

Thus, using the definition of “admits” and Lemma 6.1.2 of [CDT], we see that

H^{1}(Y_{{r}}, ℱ_{{r}})_{𝔪′_{{r}}}^{w_{ℓ}=1} ≠ (0),

and so 𝒩_{∅} = 𝒩_{{r}} ≠ ∅.

Finally suppose that τ′ admits σ_{\{m\},a} which in turn admits \overline{ρ}, and that \overline{ρ}|_{G_{ℓ}} is reducible and très ramifié. By twisting we may reduce to the case m = 0. Note that \overline{ρ}_{I_{ℓ}}(Frob_{ℓ}) = −a. As in the proof of Theorem 1.4.1 (especially §6.2 of [CDT] as modified above), we have

H^{1}(Y_{{r}}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}} ≠ (0).

Suppose that π is a cuspidal automorphic representation which contributes to H^{1}(Y_{{r}}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}}, so π is a cuspidal automorphic representation of GL_{2}(A) such that π_{∞} is the lowest discrete series with trivial infinitesimal character, ρ_{π} is a lift of \overline{ρ} of type (\{r\}, 1), and hence of type (∅, 1), and dim π_{ℓ}^{U_{0}(ℓ)} = 1. As ϵ^{−1} det ρ_{π} has order prime to ℓ, we see that w_{ℓ}^{2} acts on π_{ℓ}^{U_{0}(ℓ)} by the Teichmüller lift of a^{2}. As π_{ℓ} has a U_{0}(ℓ)-fixed vector but no GL_{2}(Z_{ℓ})-fixed vector, we see that 1 + U_{ℓ}w_{ℓ}^{−1} = 0 on π_{ℓ}^{U_{0}(ℓ)}. On the other hand, the eigenvalue of U_{ℓ} on π_{ℓ}^{U_{0}(ℓ)} reduces to −a. Thus, w_{ℓ} acts on π_{ℓ}^{U_{0}(ℓ)} by the Teichmüuller lift of a, so w_{ℓ} acts on H^{1}(Y_{{r}}, ℱ_{Θ^{∨}})_{𝔪′_{{r}}} by the Teichmüller lift of a. We deduce that

H^{1}(Y_{{r}}, ℱ_{Θ^{∨}} ⊗ ℱ_{(σ′_{0,0})^{∨}})_{𝔪′_{{r}}}^{w_{ℓ}=a} ≠ (0).

Using the definition of “admits” and Lemma 6.1.2 of [CDT], we see that

H^{1}(Y_{{r}}, ℱ_{{r}})_{𝔪′_{{r}}}^{w_{ℓ}=1} ≠ (0),

so 𝒩_{∅} = 𝒩_{{r}} ≠ ∅.

2. Examples and applications.

2.1. Examples. Now we will specialise to the case ℓ = 3. Fix an element ζ ∈ GL_{2}(Z_{3}) with ζ^{3} = 1 but ζ ≠ 1. The following definitions, which concern isomorphism classes of 2-dimensional representations into GL_{2}(\overline{Q}_{3}), do not depend on this choice. We will consider the following ℓ-types. (These are in fact, up to twist, a complete list of the wildly ramified types which can arise from elliptic curves over Q_{3}, or, in the case of conductor 243, the extended types. We will not need this fact. Rather the justification for studying these particular types can be found in §2.2. More detailed information about the fixed fields of these types can be found in §6.)

• τ_{1} corresponds to the order 3 homomorphism

Z_{3}^{×} ⟶ Z_{3}[ζ]^{×}

determined by

\begin{matrix}−1 & ⟼ & 1 \\ 4 & ⟼ & ζ. \end{matrix}

• τ_{−1} corresponds to the order 3 homomorphism

Z_{3}[\sqrt{−1}]^{×} ⟶ Z_{3}[ζ]^{×}

determined by

\begin{matrix}\sqrt[4]{−1} & ⟼ & 1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−1} & ⟼ & ζ. \end{matrix}

• τ_{3} is the unique 3-type such that τ_{3}|_{I_{Q_{3}(\sqrt{3})}} corresponds to the order 6 homomorphism

Z_{3}[\sqrt{3}]^{×} ⟶ Z_{3}[ζ]^{×}

determined by

\begin{matrix}−1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + \sqrt{3} & ⟼ & ζ. \end{matrix}

• τ_{−3} is the unique 3-type such that τ_{−3}|_{I_{Q_{3}(\sqrt{−3})}} corresponds to the order 6 homomorphism

Z_{3}[\sqrt{−3}]^{×} ⟶ Z_{3}[ζ]^{×}

determined by

\begin{matrix}−1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & 1 \\ 1 + \sqrt{−3} & ⟼ & ζ. \end{matrix}

For i ∈ Z/3Z, we will also consider the unique extended 3-types τ′_{i} whose restrictions to G_{Q_{3}(\sqrt{−3})} correspond to the homomorphisms

Q_{3}(\sqrt{−3})^{×} → Q_{3}(ζ)^{×}

determined by

\begin{matrix}\sqrt{−3} & ⟼ & ζ − ζ^{−1} \\ −1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & ζ \\ 1 + \sqrt{−3} & ⟼ & ζ^{i}. \end{matrix} (2.1.1)

Subsequent sections of this paper will be devoted to checking the following special cases of Conjecture 1.3.1. Lemma 2.1.1. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3}) and

\overline{ρ}|_{I_{3}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

is très ramifié. Both τ_{1} and τ_{−1} simply admit \overline{ρ}. Theorem 2.1.2. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3}) and

\overline{ρ}|_{I_{3}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

is très ramifié. Both τ_{1} and τ_{−1} are weakly acceptable for \overline{ρ}. Lemma 2.1.3. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3})

\overline{ρ}|_{I_{3}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

is très ramifié. Both τ_{3} and τ_{−3} simply admit \overline{ρ}.

Theorem 2.1.4. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3}) and

\overline{ρ}|_{I_{3}} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

is très ramifié. Both τ_{3} and τ_{−3} are weakly acceptable for \overline{ρ}.

Lemma 2.1.5. Let i ∈ Z/3Z. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3}) and

\overline{ρ} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

is très ramifié. The extended 3-type τ′_{i} simply admits \overline{ρ}.

Theorem 2.1.6. Let i ∈ Z/3Z. Suppose that \overline{ρ} : G_{3} → GL_{2}(F_{3}) and

\overline{ρ} ∼ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

is très ramifié. Then τ′_{i} is weakly acceptable for \overline{ρ}.

We remark that in Theorems 2.1.2, 2.1.4 and 2.1.6 we could replace “weakly acceptable” by “acceptable”. This can be shown by using elliptic curves to construct explicit liftings of the desired type. For Theorems 2.1.2 and 2.1.4 the results of [M] suffice for this. For Theorem 2.1.6 a slightly more refined analysis along the lines of §2.3 is required.

We also remark that it was Lemmas 2.1.1, 2.1.3, 2.1.5, and Conjecture 1.3.1 which originally suggested to us that we try to prove Theorems 2.1.2, 2.1.4, and 2.1.6.

2.2. Applications. Conditional on the results stated in §2.1, which we will prove below, we prove the following results.

Theorem 2.2.1. Any continuous absolutely irreducible representation \overline{ρ} : G_{Q} → GL_{2}(F_{5}) with cyclotomic determinant is modular.

Proof. Choose an element \overline{ζ} ∈ GL_{2}(F_{5}) with \overline{ζ}^{3} = 1 but \overline{ζ} ≠ 1. (The following classification will be independent of the choice of \overline{ζ}.) Then up to equivalence and twisting by a quadratic character, one of the following possibilities can be attained.

1. \overline{ρ} is tamely ramified at 3.

2. \overline{ρ}|_{G_{3}} is given by the character

Q_{3}^{×} ⟶ F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}3 & ⟼ & \overline{ζ}^{i}(\overline{ζ} − \overline{ζ}^{−1}) \\ −1 & ⟼ & 1 \\ 4 & ⟼ & \overline{ζ}, \end{matrix}

where i ∈ Z/3Z.

3. \overline{ρ}|_{G_{Q_{3}(\sqrt{−1})}} is given by the character

Q_{3}(\sqrt{−1})^{×} ⟶ F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}3 & ⟼ & 2 \\ \sqrt[4]{−1} & ⟼ & 1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−1} & ⟼ & \overline{ζ}. \end{matrix}

4. \overline{ρ}|_{G_{Q_{3}(\sqrt{3})}} is given by the character

Q_{3}(\sqrt{3})^{×} ⟶ F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}\sqrt{3} & ⟼ & \overline{ζ} − \overline{ζ}^{−1} \\ −1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + \sqrt{3} & ⟼ & \overline{ζ}. \end{matrix}

5. \overline{ρ}|_{G_{Q_{3}(\sqrt{−3})}} is given by the character

Q_{3}(\sqrt{−3})^{×} ⟶ F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}\sqrt{−3} & ⟼ & \overline{ζ} − \overline{ζ}^{−1} \\ −1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & 1 \\ 1 + \sqrt{−3} & ⟼ & \overline{ζ}. \end{matrix}

6. \overline{ρ}|_{G_{Q_{3}(\sqrt{−3})}} is given by the character

Q_{3}(\sqrt{−3})^{×} → F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}\sqrt{−3} & ⟼ & \overline{ζ} − \overline{ζ}^{−1} \\ −1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & \overline{ζ} \\ 1 + \sqrt{−3} & ⟼ & \overline{ζ}^{i}, \end{matrix}

where i ∈ Z/3Z. To see that one of these cases can be attained, use the following facts, all of which are easy to verify.

• A subgroup of GL_{2}(F_{5}) with a non-trivial normal subgroup of 3-power order is, up to conjugation, contained in the normaliser of F_{5}(\overline{ζ})^{×}.

• The intersection of SL_{2}(F_{5}) with the normaliser of \overline{ζ} in GL_{2}(F_{5}) is generated by \overline{ζ} and an element α such that α^{2} = −1 and α\overline{ζ}α^{−1} = \overline{ζ}^{−1}.

• If β ∈ F_{5}(\overline{ζ})^{×}, det β = 3, and αβα^{−1} = −β, then β = ±(\overline{ζ} − \overline{ζ}^{−1}). In each case, we may choose an elliptic curve E_{1/Q_{3}} such that the representation \overline{ρ}_{E_{1},5} of G_{3} on E_{1}[5](\overline{Q}_{3})

is isomorphic to \overline{ρ}|_{G_{3}} and such that the representation \overline{ρ}_{E_{1},3} of G_{3} on E_{1}[3](\overline{Q}_{3}) has the following form (where we use the same numbering as above).

1. We place no restriction on \overline{ρ}_{E_{1},3}.

2. The restriction of \overline{ρ}_{E_{1},3} to I_{3} has the form

( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

and is très ramifié. (Use Theorem 5.3.2 of [M].)

3. The restriction of \overline{ρ}_{E_{1},3} to I_{3} has the form

( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

and is très ramifié. (Use Theorem 5.3.2 of [M].)

4. The restriction of \overline{ρ}_{E_{1},3} to I_{3} has the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

and is très ramifié. (Use §5.4 of [M].)

5. The restriction of \overline{ρ}_{E_{1},3} to I_{3} has the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

and is très ramifié. (Use §5.4 of [M].)

6. \overline{ρ}_{E_{1},3} has the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ,

is très ramifié and remains indecomposable when restricted to the splitting field of \overline{ρ}. (Use Corollary 2.3.2 below.)

In each case choose such an E_{1} and fix an isomorphism α : F_{5}^{2} →^{∼} E_{1}[5](\overline{Q}_{3}), such that the Weil pairing on E_{1}[5] corresponds to the standard alternating pairing on F_{5}^{2}, following the conventions in §1 of [SBT]. The pair (E_{1}, α) defines a Q_{3}-rational point on the smooth curve denoted X\overline{ρ} in [SBT]. We can find a 3-adic open set 𝒰 ⊂ X\overline{ρ}(Q_{3}) containing (E_{1}, α) such that if (E_{2}, β) defines a point in 𝒰 then E_{2}[3] ≅ E_{1}[3] as F_{3}[G_{3}]-modules.

Using Ekedahl’s version of the Hilbert Irreducibility Theorem (see Theorem 1.3 of [E]) and the argument of §1 of [SBT] we may find an elliptic curve E_{/Q} and an F_{5}[G_{Q}]-module isomorphism β of \overline{ρ} with E[5](\overline{Q}) such that

• under β, the standard alternating pairing on F_{5}^{2} and the Weil pairing on E[5] agree;

• the representation \overline{ρ}_{E,3} of G_{Q} on E[3](\overline{Q}) is surjective onto Aut(E[3](\overline{Q}));

• and (E, β) defines a point of 𝒰. (See also §2 of [M].)

Corresponding to the six types of \overline{ρ} considered above, Proposition B.4.2 of [CDT] ensures that the representation ρ_{E,3} of G_{Q} on the 3-adic Tate module of E is

1. either, up to quadratic twist, ordinary in the sense of Wiles [Wi] or potentially Barsotti-Tate of some tamely ramified type;

2. potentially Barsotti-Tate of type τ_{1};

3. potentially Barsotti-Tate of type τ_{−1};

4. potentially Barsotti-Tate of type τ_{3};

5. potentially Barsotti-Tate of type τ_{−3};

6. potentially Barsotti-Tate of extended type τ′_{i}. In the first case, E is modular by Theorem 7.2.1 of [CDT]. In the other cases we will simply write τ for the type/extended type. We see that \overline{ρ}_{E,3}(G_{3}) has centraliser F_{3} and the results of §2.1 show that τ admits \overline{ρ}_{E,3} and that τ is weakly acceptable for \overline{ρ}_{E,3}. Moreover \overline{ρ}_{E,3}|_{Gal(\overline{Q}/Q(\sqrt{−3}))} is absolutely irreducible and, by the Langlands-Tunnell theorem (see [Wi]), modular. Thus by Theorems 1.4.1 and 1.4.2 we see that ρ_{E,3} is modular. We deduce that E is modular, so \overline{ρ} ≅ \overline{ρ}_{E,5} is modular.

Combining this theorem with Theorem 7.2.4 of [CDT] we immediately obtain the following corollary.

Theorem 2.2.2. Every elliptic curve defined over the rational numbers is modular.

2.3. An extension of a result of Manoharmayum. The following facts follow at once from [M], particularly the classification given just before Theorem 5.4.2 of that paper. Consider elliptic curves E over Q_{3} with minimal Weierstrass equation Y^{2} = X^{3} + AX + B, where

A ≡ B + 3 ≡ 0 mod 9,

so \overline{ρ}_{E,3} has the form ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) and is très ramifié. This leaves three possibilities for the equivalence class of

\overline{ρ}_{E,3}. Fix \overline{ζ} in GL_{2}(F_{5}) with \overline{ζ}^{3} = 1 but \overline{ζ} ≠ 1. The action of G_{Q_{3}(\sqrt{−3})} on E[5](\overline{Q}_{3}) is via a representation of the form

\begin{matrix}\sqrt{−3} & ↦ & δ(\overline{ζ} − \overline{ζ}^{−1}) \\ −1 & ↦ & −1 \\ 4 & ↦ & 1 \\ 1 + \sqrt{−3} & ↦ & \overline{ζ}^{i} \\ 1 + 3\sqrt{−3} & ↦ & \overline{ζ}, \end{matrix}

for some δ = ±1 and some i ∈ Z/3Z. All nine possibilities for the pair (\overline{ρ}_{E,3}, i) satisfying these conditions can arise for some such choice of A and B.

Lemma 2.3.1. With the above notation and assumptions, we have δ = 1.

Proof. Let F = Q_{3}(\sqrt{−3}, β, α), where β^{2} = −\sqrt{−3} and

α^{3} + Aα + B = 9\sqrt{−3}.

F is a totally ramified abelian extension of Q_{3}(\sqrt{−3}) of degree 6, with uniformizer ϖ = α/β. The change of coordinates Y ↦ ϖ^{15}Y, X ↦ ϖ^{10}X + α shows that E has good reduction over F, and the reduction is isomorphic to

Y^{2} = X^{3} − X − 1.

The arithmetic Frobenius of W_{F} therefore has trace 3 on E[5]. Since

N_{F/Q_{3}(\sqrt{−3})}(ϖ) ≡ \sqrt{−3}(1 − 3\sqrt{−3}) mod 9\sqrt{−3}

we conclude that

tr δ(\overline{ζ} − \overline{ζ}^{−1})\overline{ζ}^{−1} = 3,

so δ = 1.

Twisting by quadratic characters we immediately deduce the following corollary.

Corollary 2.3.2. Let \overline{ρ}_{3} : G_{3} → GL_{3}(F_{3}) have the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) or ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )

and be très ramifié. Let \overline{ρ}_{5} : G_{3} → GL_{2}(F_{5}) have cyclotomic determinant and restriction to G_{Q_{3}(\sqrt{−3})} given by a character

Q_{3}(\sqrt{−3})^{×} ⟶ F_{5}(\overline{ζ})^{×}

determined by

\begin{matrix}\sqrt{−3} & ↦ & (\overline{ζ} − \overline{ζ}^{−1}) \\ −1 & ↦ & −1 \\ 4 & ↦ & 1 \\ 1 + \sqrt{−3} & ↦ & \overline{ζ}^{i} \\ 1 + 3\sqrt{−3} & ↦ & \overline{ζ}, \end{matrix}

for some i ∈ Z/3Z. There is an elliptic curve E_{/Q_{3}}, with E[3](\overline{Q}_{3}) ∼ \overline{ρ}_{3} and E[5](\overline{Q}_{3}) ∼ \overline{ρ}_{5}. In particular, the action of I_{3} on T_{5}E factors through a finite group and so E has potentially good reduction.

3. Admittance.

In this section we will check Lemmas 2.1.1, 2.1.3, and 2.1.5. We freely use the terminology introduced in §1.2 and §1.3.

3.1. The case of τ_{1}. In this case σ_{τ_{1}} is the induction from U_{0}(9) to GL_{2}(Z_{3}) of a character of order 3. Its reduction modulo a prime above 3 has the same Jordan-Hölder constituents as the reduction modulo 3 of Ind_{U_{0}(9)}^{GL_{2}(Z_{3})} 1. Using Brauer characters, we see that the reduction modulo 3 of Ind_{U_{0}(9)}^{U_{0}(3)} 1 has Jordan-Hölder constituents σ′_{0,0}, σ′_{0,0} and σ′_{1,1}. Thus, τ_{1} admits σ_{0,0}, σ_{2,0}, σ_{0,1} and σ_{2,1}, the latter two simply. Lemma 2.1.1 follows in this case.

3.2. The case of τ_{−1}. Let U denote the subgroup of GL_{2}(Z_{3}) consisting of matrices

( \begin{matrix}a & b \\ c & d \end{matrix} )

with a ≡ d mod 3 and b + c ≡ 0 mod 3, so σ_{τ_{−1}} is the induction from U to GL_{2}(Z_{3}) of a character of order 3. Upon reduction modulo a prime above 3 this will have the same Jordan-Hölder constituents as the reduction modulo 3 of Ind_{U}^{GL_{2}(Z_{3})} 1. If ψ denotes the non-trivial character of F_{3}^{×} and ϕ a character of F_{9}^{×} of order 4, then this latter induction splits up as the sum of the representations of GL_{2}(Z_{3}) → → GL_{2}(F_{3}) denoted 1, sp_{ψ} and Θ(ϕ) in §3.1 of [CDT]. By Lemma 3.1.1 of [CDT] we see that τ_{−1} admits σ_{0,0}, σ_{2,1} and σ_{0,1}, the latter two simply. Lemma 2.1.1 follows in this case.

3.3. The case of τ_{±3}. Let U denote the subgroup of GL_{2}(Z_{3}) consisting of matrices

( \begin{matrix}a & b \\ c & d \end{matrix} )

with a ≡ d mod 3 and c ≡ 0 mod 3. Then σ_{τ_{±3}} is the induction from U to U_{0}(3) of a character of order 3. Upon reduction modulo a prime above 3 this will have the same Jordan-Hölder constituents as the reduction modulo 3 of Ind_{U}^{U_{0}(3)} 1. Thus, τ_{±3} simply admits σ′_{0,0} and σ′_{1,1}. Lemma 2.1.3 follows.

3.4. The case of τ′_{i}. Let χ be the character of Q_{3}(\sqrt{−3})^{×} as in (2.1.1). Let ψ be a character of Q_{3} with kernel Z_{3} and which sends 1/3 to ζ. If x ∈ (3\sqrt{−3})Z_{3}[\sqrt{−3}] we have

χ(1 + x) = ψ(tr_{Q_{3}(\sqrt{−3})/Q_{3}}(−x\sqrt{−3}/54)).

We deduce that if χ′′ is the character used to define σ_{τ′_{i}} in §1.2, then χ′′(\sqrt{−3}) = (ζ − ζ^{−1})^{−1}. Let U denote the subgroup of GL_{2}(Z_{3}) consisting of matrices

( \begin{matrix}a & b \\ 3c & d \end{matrix} )

with a ≡ d mod 3 and b+c ≡ 0 mod 3. Let Ũ be the group generated by w_{3} (see (1.2.1)) and U, so σ_{τ′_{i}} is the representation of Ũ_{0}(3) induced from a character of Ũ which sends w_{3} to 1 and has order 3 when restricted to U. Thus, the Jordan-Hölder constituents of the reduction of σ_{τ′_{i}} modulo a prime above 3 are the same as

the Jordan-Hölder constituents of the reduction modulo 3 of IndŨ_{0}(3) ; Ũ 1.

Let V denote the subgroup of GL_{2}(Z_{3}) consisting of matrices

( \begin{matrix}a & b \\ 3c & d \end{matrix} )

with a ≡ d mod 3. Let Ṽ be the group generated by w_{3} and V, and let ν denote the character of Ṽ/V which sends w_{3} to −1. We have

IndṼ ; Ũ (1) ∼ 1 ⊕ IndṼ ; V Q_{3}^{×} η

where η is a non-trivial character of V/U = (V Q_{3}^{×})/(UQ_{3}^{×}). The reduction modulo a prime above 3 of this (3-dimensional) representation has the same Jordan-Hölder constituents as the reduction modulo 3 of 1 ⊕ 1 ⊕ ν. Thus, τ′_{i} admits σ′_{{0},1}, σ′_{{1},1}, σ′_{{0},−1} and σ′_{{1},−1}, the latter two simply. Lemma 2.1.5 follows.

4. New deformation problems.

In this section we begin the proof of Theorems 2.1.2, 2.1.4 and 2.1.6. One could approach this directly by using the results of [Br2] to attempt to describe R_{V,𝒪}^{D} (resp. R_{V,𝒪}^{D′} ). At least one of us (R.T.) thinks that such an approach holds out more promise of attacking the non-acceptable case, and another of us (C.B.) has indeed made several computations along these lines. However in the present case it seems to be easier to proceed less directly.

To this end we will use ad hoc arguments to define deformation problems, which will be represented by 𝒪-algebras S such that

• dim_{k} 𝔪_{S}/(℘_{K}, 𝔪_{S}^{2}) ≤ 1

• and the map R_{V,𝒪} → → R_{V,𝒪}^{D} (resp. R_{V,𝒪}^{D′} ) factors through S. An important advantage of this approach is that to calculate 𝔪_{S}/(℘_{K}, 𝔪_{S}^{2}) one need only work in the category of finite flat group schemes killed by a prime. Breuil modules (see section §5) for finite flat group schemes killed by an odd prime are significantly simpler than the general case (of prime power torsion). This is particularly true when we also use descent data. On the other hand, to suitably define the new deformation problems is rather delicate. That is what we will do in this section.

4.1. Some generalities on group schemes. In this section, and in §4.2, ℓ will again be an arbitrary rational prime. Moreover R will denote a complete discrete valuation ring with fraction field F′ of characteristic zero and perfect residue field k of characteristic ℓ. We will let Γ denote a finite group of continuous automorphisms of R and we will let F_{0} denote subfield of F′ consisting of elements fixed by Γ. Thus F′/F_{0} will be finite and Galois with group Γ. In our applications of these results it suffices to consider the case where F_{0} is a finite extension of Q_{3} (although we will occasionally pass to the completion of the maximal unramified extension of F).

Lemma 4.1.1. Let 𝒢 be a finite flat R-group scheme. Scheme theoretic closure gives a bijection between subgroup schemes of 𝒢 × F′ and finite flat closed subgroup schemes of 𝒢.

(See for instance §1.1 of [Co].)

Lemma 4.1.2. Let 𝒢_{1} and 𝒢_{2} be finite flat group schemes over R which have local-local closed fibre. Suppose that 𝒢_{1} and 𝒢_{2} are the only finite flat R-group schemes with local-local closed fibre which have generic fibres 𝒢_{1} × F′ and 𝒢_{2} × F′ respectively. Suppose also that we have an exact sequence of finite flat R-group schemes

(0) ⟶ 𝒢_{1} ⟶ 𝒢 ⟶ 𝒢_{2} ⟶ (0).

Then 𝒢 is the unique finite flat R-group scheme with local-local closed fibre and with generic fibre 𝒢 × F.

Proof. Let 𝒢_{+} and 𝒢_{−} denote the maximal and minimal local-local models for 𝒢 × F. The proof that these exist follows the of Proposition 2.2.2 of [Ra] and uses the fact that the Cartier dual of a local-local finite flat group scheme is local-local. We must show that the canonical map 𝒢_{+} → 𝒢_{−} is an isomorphism. The scheme-theoretic closure of 𝒢_{1} × F in 𝒢_{±} must be isomorphic to 𝒢_{1} (by uniqueness), so we have closed immersions 𝒢_{1} ↪ 𝒢_{±} extending 𝒢_{1} × F ↪ 𝒢_{±} × F. Similarly 𝒢_{±}/𝒢_{1} must be isomorphic to 𝒢_{2}. This gives a commutative diagram with exact rows

0 → 𝒢_{1} → 𝒢_{+} → 𝒢_{2} → 0
↓ ↓ ↓
0 → 𝒢_{1} → 𝒢_{−} → 𝒢_{2} → 0

The vertical maps 𝒢_{1} → 𝒢_{1} and 𝒢_{2} → 𝒢_{2} induce isomorphisms on the generic fibre and hence are isomorphisms. This is because some power of them is the identity on the generic fibre and hence is the identity. Working in the abelian category of fppf abelian sheaves over Spec R, the middle map must also be an isomorphism.

When 𝒢 has ℓ-power order, we will let D(𝒢) denote the classical (contravariant) Dieudonné module of 𝒢 × k. It is a W (k)-module equipped with a Frobenius operator F and a Verschiebung operator V. We have FV = VF = ℓ and for all x ∈ W (k), Fx = (Frob_{ℓ} x)F and Vx = (Frob_{ℓ}^{−1} x)V.

If 𝒢 is a finite flat R-group scheme, then by descent data for 𝒢 over F_{0} we mean a collection {[g]} of group scheme isomorphisms

[g] : 𝒢 ⟶^{∼} ^{g}𝒢

for g ∈ Gal(F′/F_{0}) such that for all g, h ∈ Gal(F′/F_{0}) we have

[gh] = (^{g}[h]) ∘ [g].

Note that this is not descent data in the sense of Grothendieck, since R/R^{Γ} might be ramified. However, Spec F′/ Spec F_{0} is etale, so by étale descent we obtain a finite flat group scheme (𝒢, {[g]})_{F_{0}} over F_{0} together with an isomorphism

(𝒢, {[g]})_{F_{0}} ×_{F_{0}} F′ ≅ 𝒢 ×_{R} F′

compatible with descent data. We also obtain a natural left action of Gal(F′/F_{0}) on the Dieudonné module D(𝒢), semilinear with respect to the W (k)-module structure and commuting with F and V. We refer to the pair (𝒢, {[g]}) as an R-group scheme with descent data relative to F_{0}. One defines morphisms of such objects to be morphisms of R-group schemes which commute with the descent data. By a closed finite flat subgroup scheme with descent data we mean a closed finite flat subgroup scheme such that the descent data on the ambient scheme takes the subscheme to itself. Quotients by such subobjects are defined in the obvious way. Thus we obtain an additive category with a notion of short exact sequence. Suppose that G id a finite flat F_{0}-group scheme. By a model with descent data (or simply model) for G over R we shall mean a triple (𝒢, {[g]}, i), where (𝒢, {[g]}) is an R-group scheme with descent data relative to F_{0} and where i : (𝒢, {[g]})_{F_{0}} →^{∼} G. Sometimes we will suppress i from the notation. It is easy to check that isomorphism classes of models with descent data for G over R form a sublattice of the lattice of models for G_{/F′} over R. The following lemma follows without difficulty from Lemma 4.1.1.

Lemma 4.1.3. Let F′/F_{0} be a finite Galois extension as above, and let (𝒢, {[g]}) be a finite flat R-group scheme with descent data relative to F_{0}. Base change from F_{0} to F′, followed by scheme theoretic closure, gives a bijection between subgroup schemes of (𝒢, {[g]})_{F_{0}} and closed finite flat subgroup schemes with descent data in (𝒢, {[g]}).

We will let ℱℱ_{F′} denote the category of finite flat group schemes over R and ℱ𝒟_{F′/F_{0}} the category of finite flat group schemes over R with descent data over F_{0}. Let W (k)[F, V][Gal(F′/F_{0})] denote the (noncommutative) W (k)-algebra generated by elements F, V and [g] for g ∈ Gal(F′/F_{0}) satisfying

• [gh] = [g][h] for all g, h ∈ Gal(F′/F_{0});

• [g]F = F[g] and [g]V = V[g] for all g ∈ Gal(F′/F_{0});

• FV = VF = ℓ;

• [g]x = (gx)[g] for all x ∈ W (k) and g ∈ Gal(F′/F_{0});

• Fx = (Frob_{ℓ} x)F and Vx = (Frob_{ℓ}^{−1} x)V for all x ∈ W (k). If ℐ is a two-sided ideal in W (k)[F, V][Gal(F′/F_{0})], we will let ℱ𝒟_{F′/F_{0},ℐ} denote the full subcategory of ℱ𝒟_{F′/F_{0}} consisting of objects (𝒢, {[g]}) such that ℐ acts trivially on D(𝒢). If (𝒢, {[g]}) is an object of ℱ𝒟_{F′/F_{0},ℐ} and if (ℋ, {[g]}) ⊂ (𝒢, {[g]}) is a closed finite flat subgroup scheme then (ℋ, {[g]}) and (𝒢, {[g]})/(ℋ, {[g]}) are again objects of ℱ𝒟_{F′/F_{0},ℐ}.

Lemma 4.1.4. For ℐ a two-sided ideal of the ring W (k)[F, V][Gal(F′/F_{0})], choose objects (𝒢_{1}, {[g]}) and (𝒢_{2}, {[g]}) in ℱ𝒟_{F′/F_{0},ℐ} so that (𝒢_{1}, {[g]})_{F_{0}} ≅ (𝒢_{2}, {[g]})_{F_{0}}. Let G denote the base change of this F_{0}-group scheme to F′, so G has canonical descent data relative to F′/F_{0}. Then the sup and inf of 𝒢_{1} and 𝒢_{2} in the lattice of integral models for G are stable under the descent data on G and with this descent data are objects of ℱ𝒟_{F′/F_{0},ℐ}.

Proof. By uniqueness of the inf and sup, they are stable under the descent data on the generic fibre. It follows from Raynaud’s construction of the inf and sup (Proposition 2.2.2 of [Ra]) in terms of subgroupschemes and quotients of 𝒢_{1} × 𝒢_{2} that the sup and inf are objects of ℱ𝒟_{F′/F_{0},ℐ}.

Corollary 4.1.5. Let ℐ be a two-sided ideal of the ring W (k)[F, V][Gal(F′/F_{0})]. Let

(0) ⟶ G_{1} ⟶ G ⟶ G_{2} ⟶ (0) (4.1.1)

be an exact sequence of finite flat group schemes over F_{0}. Let (𝒢_{1}, {[g]}) and (𝒢_{2}, {[g]}) be objects of ℱ𝒟_{F′/F_{0},ℐ} such that (𝒢_{1}, {[g]})_{F_{0}} ≅ G_{1} and (𝒢_{2}, {[g]})_{F_{0}} ≅ G_{2}. Suppose that for all objects (𝒢, {[g]}) of ℱ𝒟_{F′/F_{0},ℐ} with (𝒢, {[g]})_{F_{0}} ≅ G, the filtration on (𝒢, {[g]}) induced by the filtration on G has subobject isomorphic to (𝒢_{1}, {[g]}) and quotient isomorphic to (𝒢_{2}, {[g]}) (without any assumed compatibility with (4.1.1) ). Then there is at most model for G in ℱ𝒟_{F′/F_{0},ℐ}.

Proof. By Lemma 4.1.4, it suffices to prove that if (𝒢_{+}, {[g]}, i_{+}) and (𝒢_{−}, {[g]}, i_{−}) are two such models with a morphism between them, then the morphism between them must be an isomorphism. In such a case we have a commutative diagram with exact rows

0 → 𝒢_{1} → 𝒢_{+} → 𝒢_{2} → 0
↓ ↓ ↓
0 → 𝒢_{1} → 𝒢_{−} → 𝒢_{2} → 0

The vertical maps 𝒢_{1} → 𝒢_{1} and 𝒢_{2} → 𝒢_{2} induce isomorphisms on the generic fibre and hence are isomorphisms. This is because some power of them is the identity on the generic fibre and hence is the identity. Working in the abelian category of fppf abelian sheaves over Spec R, the middle map must also be an isomorphism.

4.2. Filtrations. We keep the notation and assumptions of the previous section. Let Σ be a finite non-empty set of objects (𝒢_{i}, {[g]}) of ℱ𝒟_{F′/F_{0},(ℐ,ℓ)}. (Note the ℓ in the subscript (ℐ, ℓ).) Suppose that

(4.2.1) Hom((𝒢_{i}, \{[g]\}), (𝒢_{j}, \{[g]\})) = Hom((𝒢_{i}, \{[g]\})_{F_{0}}, (𝒢_{j}, \{[g]\})_{F_{0}}) = \{ \begin{matrix}0 & if i ≠ j \\ finite field & if i = j \end{matrix}

(in particular, the objects in Σ are non-zero and pairwise non-isomorphic). By a Σ-filtration on a finite flat F_{0}- group scheme G we mean an increasing filtration Fil^{j} G such that for all j the graded piece Fil^{j} G/ Fil^{j−1} G is isomorphic to (𝒢_{i(j)}, {[g]})_{F_{0}} for a (unique) (𝒢_{i(j)}, {[g]}) ∈ Σ. The following lemma is proved by the standard Jordan-Hölder argument.

Lemma 4.2.1. If G is a finite flat F_{0}-group scheme which admits a Σ-filtration and if H is a quotient or subobject of G which admits a Σ-filtration, then any Σ-filtration of H can be extended to a Σ-filtration of G. In addition, all Σ-filtrations of G have the same length and the same set of successive quotients (with multiplicities).

We say that an object (𝒢, {[g]}) of ℱ𝒟_{F′/F_{0},ℐ} is weakly filtered by Σ if there is some increasing filtration Fil^{j}(𝒢, {[g]}) of (𝒢, {[g]}) by (closed) subobjects such that for all j, the graded piece

Fil^{j}(𝒢, {[g]})/ Fil^{j−1}(𝒢, {[g]})

is isomorphic to an element of Σ. We say that an object (𝒢, {[g]}) of ℱ𝒟_{F′/F_{0},ℐ} is strongly filtered by Σ if (𝒢, {[g]}) is weakly filtered by Σ and if for every Σ-filtration of (𝒢, {[g]})_{F_{0}} the corresponding filtration of (𝒢, {[g]}) satisfies

Fil^{j}(𝒢, {[g]})/ Fil^{j−1}(𝒢, {[g]})

is isomorphic to an element of Σ for all j. The following lemma follows at once from the definitions and from Lemma 4.2.1.

Lemma 4.2.2. 1. If (𝒢, {[g]}) and (𝒢′, {[g]}) are objects of ℱ𝒟_{F′/F_{0},ℐ} which are weakly filtered by Σ, then

(𝒢, {[g]}) × (𝒢′, {[g]}) is also weakly filtered by Σ.

2. Let (𝒢, {[g]}) and (𝒢′, {[g]}) be objects of ℱ𝒟_{F′/F_{0},ℐ} with (𝒢′, {[g]}) a closed subobject or quotient of (𝒢, {[g]}). Suppose that (𝒢, {[g]}) is strongly filtered by Σ and that (𝒢′, {[g]})_{F_{0}} admits a Σ-filtration. Then (𝒢′, {[g]}) is strongly filtered by Σ.

If any object of ℱ𝒟_{F′/F_{0},ℐ} which is weakly filtered by Σ is strongly filtered by Σ, then we will let ℱ𝒟_{F′/F_{0},ℐ,Σ} denote the full subcategory of ℱ𝒟_{F′/F_{0},ℐ} consisting of objects which are weakly (and therefore strongly) filtered by Σ.

Lemma 4.2.3. Suppose that any object of ℱ𝒟_{F′/F_{0},ℐ} which is weakly filtered by Σ is strongly filtered by Σ. Let G be a finite flat F_{0}-group scheme. If (𝒢_{1}, {[g]}) and (𝒢_{2}, {[g]}) are two objects of ℱ𝒟_{F′/F_{0},ℐ,Σ} with isomorphisms

i_{j} : G ⟶^{∼} (𝒢_{j}, {[g]})_{F_{0}}

for j = 1, 2, then there is a unique isomorphism

ϕ : (𝒢_{1}, {[g]}) ⟶^{∼} (𝒢_{2}, {[g]})

such that on the generic fibre i_{2} = ϕ ∘ i_{1}.

Proof. It follows from Raynaud’s construction of sup and inf that the sup and inf of ((𝒢_{1}, {[g]}), i_{1}) and ((𝒢_{2}, {[g]}), i_{2}) are again objects of ℱ𝒟_{F′/F_{0},ℐ,Σ}. Thus we may suppose that there exists a map ϕ : (𝒢_{1}, {[g]}) → (𝒢_{2}, {[g]}) such that on the generic fibre i_{2} = ϕ ∘ i_{1}. We will argue by induction on the rank of G that ϕ is an isomorphism.

If (𝒢_{1}, {[g]}) is isomorphic to an element of Σ then the result follows by our assumption on Σ.

If (𝒢_{1}, {[g]}) is not isomorphic to an element of Σ then choose an exact sequence

(0) ⟶ (𝒢_{11}, {g}) ⟶ (𝒢_{1}, {g}) ⟶ (𝒢_{12}, {g}) ⟶ (0),

where (𝒢_{11}, {g}) and (𝒢_{12}, {g}) are weakly filtered by Σ. Let (𝒢_{21}, {[g]}) denote the closed subobject of (𝒢_{2}, {[g]}) corresponding to (𝒢_{11}, {[g]})_{F_{0}} and define (𝒢_{22}, {[g]}) = (𝒢_{2}, {[g]})/(𝒢_{21}, {[g]}). Then we have a commutative diagram with exact rows

0 → 𝒢_{11} → 𝒢_{1} → 𝒢_{12} → 0
↓ ↓ ↓
0 → 𝒢_{21} → 𝒢_{2} → 𝒢_{22} → 0

compatible with descent data, where the central vertical arrow is ϕ and where by inductive hypothesis the outside vertical arrows are isomorphisms. Working in the abelian category of fppf abelian sheaves over Spec R, we see that ϕ is an isomorphism.

The following lemma and its corollary give criteria for the equivalence of the notions of being weakly filtered by Σ and of being strongly filtered by Σ.

Lemma 4.2.4. Fix ℐ and Σ as above. Suppose that for any pair of (possibly equal) elements (𝒢′, {[g]}) and (𝒢′′, {[g]}) in Σ, the natural map

Ext_{ℱ𝒟 _{F′/F_{0},(ℐ,ℓ)}}^{1} ((𝒢′′, {[g]}), (𝒢′, {[g]})) ⟶ Ext_{F_{ℓ}[G_{F_{0}} ]}^{1}((𝒢′′, {[g]})_{F_{0}}, (𝒢′, {[g]})_{F_{0}})

is injective. (Here (ℐ, ℓ) denotes the two-sided ideal generated by ℐ and ℓ.) Then any object (𝒢, {[g]}) of ℱ𝒟_{F′/F_{0},ℐ} which is weakly filtered by Σ is also strongly filtered by Σ.

Proof. For brevity, we say ‘weakly/strongly filtered’ rather than ‘weakly/strongly filtered by Σ’ since the data Σ is fixed for the entire proof. Also, we omit the specification of descent data from the notation, but it should not be forgotten.

Suppose 𝒢 is weakly filtered. In order to prove that 𝒢 is strongly filtered, we induct on the length of a Σ-filtration of 𝒢_{F_{0}}, this length being well-defined by Lemma 4.2.1. The case of length ≤ 1 is clear. Otherwise, by the definition of being weakly filtered, there is a short exact sequence of finite flat R-group schemes (with descent data relative to F_{0})

(0) ⟶ 𝒢′ ⟶ 𝒢 ⟶ 𝒢′′ ⟶ (0)

with 𝒢′ ∈ Σ and 𝒢′′ weakly filtered (and hence, by inductive hypothesis, strongly filtered). Let ℋ be any closed subgroupscheme of 𝒢 (with compatible descent data relative to F_{0}) such that ℋ_{F_{0}} ≃ 𝒢_{i_{0},F_{0}} for some 𝒢_{i_{0}} ∈ Σ and such that (𝒢/ℋ)_{F_{0}} admits a Σ-filtration. We need to prove (in the category of finite flat group schemes with descent data relative to F′/F_{0}) that • ℋ ≃ 𝒢_{i_{0}},

• and 𝒢/ℋ is weakly filtered.

If the composite map

ℋ ↪ 𝒢 → 𝒢′′

is zero, then ℋ = 𝒢′ as closed subgroupschemes of 𝒢 (with descent data) and likewise 𝒢/ℋ = 𝒢′′ so we are done. The interesting case is when the composite map is non-zero. The map 𝒢_{i_{0},F_{0}} ≅ ℋ_{F_{0}} → 𝒢′′_{F_{0}} is then nonzero and therefore must be a closed immersion by the assumption (4.2.1) on Σ and a devissage with respect to a Σ-filtration of 𝒢′′_{F_{0}}. We conclude that the map of generic fiber étale group schemes ℋ × F′ → 𝒢′′ × F′ is a closed immersion.

Taking scheme-theoretic closures, we obtain a closed subgroupscheme ℋ′′ ↪ 𝒢′′ (with unique compatible descent data over F_{0}) fitting into a commutative diagram of group schemes with descent data

ℋ ⟶^{α} ℋ′′
↓  ↓
(0) ⟶ 𝒢′ ⟶ 𝒢 ⟶ 𝒢′′ ⟶ (0)

in which the lower row is short exact, the vertical maps are closed immersions, and the top map ℋ → ℋ′′ induces an isomorphism on generic fibers. By Lemma 4.2.1 we may extend ℋ_{F_{0}} ↪ 𝒢′′_{F_{0}} to a Σ-filtration on 𝒢′′_{F_{0}} and so, because 𝒢′′ is strongly filtered by induction, we may extend ℋ′′ ↪ 𝒢′′ to a Σ-filtration. In particular ℋ′′ is isomorphic to an object in Σ and 𝒢′′/ℋ′′ is strongly filtered.

Pulling back the short exact sequence

0 → 𝒢′ → 𝒢 → 𝒢′′ → 0

by ℋ′′ → 𝒢′′, we get a diagram

ℋ
↓
0 → 𝒢′ → 𝒢 ×_{𝒢′′} ℋ′′ → ℋ′′ → 0

in which the row is a short exact sequence of fppf abelian sheaves and all of the terms are finite flat group schemes (for the middle, this follows from the flatness of 𝒢 → 𝒢′′). Thus, this bottom row is a short exact sequence of finite flat group schemes (with descent data). As ℋ_{F_{0}} →^{∼} ℋ′′_{F_{0}},

0 → 𝒢′_{F_{0}} → (𝒢 ×_{𝒢′′} ℋ′′)_{F_{0}} → ℋ′′_{F_{0}} → 0

is split. In particular (𝒢 ×_{𝒢′′} ℋ′′)_{F_{0}} and hence 𝒢 ×_{𝒢′′} ℋ′′ are killed by ℓ. By the hypothesis of the lemma

0 → 𝒢′ → 𝒢 ×_{𝒢′′} ℋ′′ → ℋ′′ → 0

is also split, i.e. we have an isomorphism

𝒢 ×_{𝒢′′} ℋ′′ ≅ 𝒢′ ×_{R} ℋ′′

such that 𝒢′ ↪ 𝒢 ×_{𝒢′′} ℋ′′ corresponds to injection to the first factor of 𝒢′ ×_{R} ℋ′′ and 𝒢 ×_{𝒢′′} ℋ′′ → → ℋ′′ corresponds to projection onto the second factor. By our hypotheses on Σ we can find a morphism ϕ : ℋ′′ → 𝒢′ extending

ℋ′′_{F_{0}} ⟵^{∼} ℋ_{F_{0}} ↪ 𝒢′_{F_{0}} × ℋ′′_{F_{0}} → →^{pr} 𝒢′_{F_{0}}.

Then our closed immersion ℋ ↪ 𝒢′ ×_{R} ℋ′′ factors as

ℋ ⟶ ℋ′′ ⟶^{ϕ×1} 𝒢′ ×_{R} ℋ′′.

As ℋ → 𝒢′ ×_{R} ℋ′′ is a closed immersion α : ℋ → ℋ′′ must be a closed immersion and hence an isomorphism. Thus ℋ is isomorphic to an object in Σ.

Now we turn to the proof that 𝒢/ℋ is weakly filtered. Since α : ℋ → ℋ′′ is an isomorphism, it is clear that the natural map

ℋ ×_{R} 𝒢′ ⟶ 𝒢 ×_{𝒢′′} ℋ′′

is an isomorphism, and hence that

ℋ ×_{R} 𝒢′ ⟶ 𝒢

is a closed immersion. Thus, the finite flat group scheme 𝒢/(ℋ × 𝒢′) makes sense and the natural map

𝒢/(ℋ × 𝒢′) → 𝒢′′/ℋ′′

is an isomorphism (as one sees by using the universal properties of quotients to construct an inverse map). We therefore arrive at a short exact sequence

0 → 𝒢′ → 𝒢/ℋ → 𝒢′′/ℋ′′ → 0

(compatible with descent data). Since 𝒢′′/ℋ′′ is strongly filtered, as we noted above, and 𝒢′ ∈ Σ, it follows that 𝒢/ℋ is weakly filtered.

Corollary 4.2.5. Fix ℐ and Σ as above. Suppose that Σ = {(𝒢, {[g]})} is a singleton. Suppose also that we have a short exact sequence

(0) ⟶ (𝒢_{1}, {[g]}) ⟶ (𝒢, {[g]}) ⟶ (𝒢_{2}, {[g]}) ⟶ (0)

in ℱ𝒟_{F′/F_{0},ℐ}, where for any i, j (possibly equal)

Hom((𝒢_{i}, \{[g]\}), (𝒢_{j}, \{[g]\})) = Hom((𝒢_{i}, \{[g]\})_{F_{0}}, (𝒢_{j}, \{[g]\})_{F_{0}}) = \{ \begin{matrix}0 & if i ≠ j \\ finite field & if i = j \end{matrix}

and the natural map

Ext_{ℱ𝒟 _{F′/F_{0},(ℐ,ℓ)}}^{1} ((𝒢_{i}, {[g]}), (𝒢_{j}, {[g]})) ⟶ Ext_{F_{ℓ}[G_{F_{0}} ]}^{1}((𝒢_{i}, {[g]})_{F_{0}}, (𝒢_{j}, {[g]})_{F_{0}}) (4.2.2)

is injective. Then any object (ℋ, {[g]}) of ℱ𝒟_{F′/F_{0},ℐ} which is weakly filtered by Σ is also strongly filtered by Σ.

Proof. As (ℋ, {[g]}) is weakly filtered by Σ, it is weakly filtered by {(𝒢_{1}, {[g]}), (𝒢_{2}, {[g]})}, and so by Lemma 4.2.4 is strongly filtered by {(𝒢_{1}, {[g]}), (𝒢_{2}, {[g]})}. Any Σ-filtration of (ℋ, {[g]})_{F_{0}} extends to a {(𝒢_{1}, {[g]}), (𝒢_{2}, {[g]})} filtration of (ℋ, {[g]})_{F_{0}}, which in turn gives rise to a {(𝒢_{1}, {[g]}), (𝒢_{2}, {[g]})}- filtration of (ℋ, {[g]}). By the injectivity of (4.2.2) we see that this yields a Σ-filtration of (ℋ, {[g]}) that induces to our chosen Σ-filtration of (ℋ, {[g]})_{F_{0}}.

4.3. Generalities on deformation theory. Again in this section ℓ will denote an arbitrary rational prime. We will let K denote a finite extension of Q_{ℓ}. We will also let 𝒪 denote the ring of integers K, ℘_{K} its maximal ideal and k its residue field. Note that k has a different meaning from the previous two sections. Let V be a two dimensional k-vector space and \overline{ρ} : G_{ℓ} → Aut_{k}(V) a continuous representation. Suppose also that the centraliser of G_{ℓ} in End_{k}(V) is k. Also let ψ : G_{ℓ} → 𝒪^{×} denote a continuous character such that (ψ mod ℘_{K}) ≅ det \overline{ρ}. Let 𝒮(\overline{ρ}) denote the full subcategory of the category of finite length (discrete) 𝒪-modules with a continuous 𝒪-linear action of G_{ℓ} consisting of objects which admit a finite filtration so that each graded piece is isomorphic to V. Because End_{k[G_{ℓ}]}(V) = k, it follows from the usual Jordan-Hölder argument that 𝒮(\overline{ρ}) is an abelian category.

Let 𝒮 be a full subcategory of 𝒮(\overline{ρ}) stable under isomorphisms and which is closed under finite products, 𝒮(\overline{ρ})-subobjects and 𝒮(\overline{ρ})-quotients, and which contains V. We will consider the following set-valued functors on the category of complete noetherian local 𝒪-algebras with finite residue field.

• 𝒟_{V,𝒪} (R) is the set of conjugacy classes of continuous representations ρ : G_{ℓ} → GL_{2}(R) such that ρ mod 𝔪_{R} is conjugate to \overline{ρ}.

• 𝒟_{V,𝒪}^{ψ} (R) is the set of conjugacy classes of continuous representations ρ : G_{ℓ} → GL_{2}(R) such that ρ mod 𝔪_{R} is conjugate to \overline{ρ} and det ρ = ψ.

• 𝒟_{V,𝒪}^{𝒮} (R) is the set of conjugacy classes of continuous representations ρ : G_{ℓ} → GL_{2}(R) such that ρ mod 𝔪_{R} is conjugate to \overline{ρ} and such that for each open ideal 𝔞 ⊂ R the action ρ makes (R/𝔞)^{2} into an object of 𝒮.

• 𝒟_{V,𝒪}^{ψ,𝒮} (R) is the set of conjugacy classes of continuous representations ρ : G_{ℓ} → GL_{2}(R) such that ρ mod 𝔪_{R} is conjugate to \overline{ρ}, such that det ρ = ψ, and such that for each open ideal 𝔞 ⊂ R the action ρ makes (R/𝔞)^{2} into an object of 𝒮.

Each of these deformation problems is representable by objects which we will denote R_{V,𝒪} , R_{V,𝒪}^{ψ} , R_{V,𝒪}^{𝒮} and R_{V,𝒪}^{ψ,𝒮} , respectively.

Recall that the following sets are in natural (k-linear) bijection with each other.

• (𝔪_{R_{V,𝒪}} /(℘_{K}, 𝔪_{R_{V,𝒪}}^{2} ))^{∨}.

• The set of deformations of \overline{ρ} to k[ε]/(ε^{2}).

• Ext_{k[G_{ℓ}]}^{1}(V, V).

• H^{1}(G_{ℓ}, ad \overline{ρ}). This restricts to an isomorphism

(𝔪_{R_{V,𝒪}^{ψ}} /(℘_{K}, 𝔪_{R_{V,𝒪}^{ψ}}^{2} ))^{∨} ≅ H^{1}(G_{ℓ}, ad^{0} \overline{ρ}),

and also restricts to bijections between • (𝔪_{R_{V,𝒪}^{𝒮}} /(℘_{K}, 𝔪_{R_{V,𝒪}^{𝒮}}^{2} ))^{∨},

• the set of deformations of \overline{ρ} to k[ε]/(ε^{2}) which make (k[ε]/(ε^{2}))^{2} into an object of 𝒮,

• Ext_{k[G_{ℓ}],𝒮}^{1}(V, V), i.e. Ext^{1} in the category of discrete k[G_{ℓ}]-modules which are also objects of 𝒮.

• the subgroup H_{𝒮}^{1}(G_{ℓ}, ad \overline{ρ}) ⊂ H^{1}(G_{ℓ}, ad \overline{ρ}) corresponding to Ext_{k[G_{ℓ}],𝒮}^{1}(V, V).

We will set H_{𝒮}^{1}(G_{ℓ}, ad^{0} \overline{ρ}) = H_{𝒮}^{1}(G_{ℓ}, ad \overline{ρ}) ∩ H^{1}(G_{ℓ}, ad^{0} \overline{ρ}), so that we get an isomorphism

(𝔪_{R_{V,𝒪}^{ψ,𝒮}} /(℘_{K}, 𝔪_{R_{V,𝒪}^{ψ,𝒮}}^{2} ))^{∨} ≅ H_{𝒮}^{1}(G_{ℓ}, ad^{0} \overline{ρ}).

4.4. Reduction steps for Theorem 2.1.2. We now begin the proof of Theorem 2.1.2. Making an unramified twist we may suppose that \overline{ρ} has the form

( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} ) .

We may also suppose that 𝒪 = Z_{3}.

Let F_{1} = F′_{1} denote a totally ramified cubic Galois extension of Q_{3}. Let F′_{−1} denote the unique cubic extension of Q_{3}(\sqrt{−1}) such that F′_{−1}/Q_{3} is Galois but not abelian, and let F_{−1} denote a cubic subfield of F′_{−1}, so F′_{−1}/F_{−1} is unramified.

Let 𝒮_{±1} denote the full subcategory 𝒮(\overline{ρ}) consisting of objects X for which there exists a finite flat 𝒪_{F_{±1}}- group scheme (𝒢, \{[g]\}) with descent data over Q_{3} such that X ≅ (𝒢, \{[g]\})_{Q_{3}}(\overline{Q}_{3}) as a Z_{3}[G_{3}]-module. By Lemma 4.1.3 we see that 𝒮_{±1} is closed under finite products, subobjects and quotients. Using Tate’s theorem on the uniqueness of extensions of 3-divisible groups from F′_{±1} to 𝒪_{F′_{±1}} (Theorem 4 of [T]), we see that the

map R_{V,Z_{3}} → → R_{V,Z_{3}}^{τ_{±1}} factors through R_{V,Z_{3}}^{ϵ,𝒮_{±1}}. Thus, Theorem 2.1.4 follows from the following result which we will prove in §7.

Theorem 4.4.1. dim H_{𝒮_{±1}}^{1}(G_{3}, ad^{0} \overline{ρ}) ≤ 1.

4.5. Reduction steps for Theorem 2.1.4. We now begin the proof of Theorem 2.1.4. Making an unramified twist, we may suppose that \overline{ρ} has the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) .

We may also suppose that 𝒪 = Z_{3}.

Let F′_{±3} denote the degree 12 abelian extension of Q_{3}(\sqrt{±3}) with norm subgroup in Q_{3}(\sqrt{±3})^{×} topologically generated by ±3, 4 and 1 + 3\sqrt{±3}. Note that F′_{±3}/Q_{3} is Galois. We have an isomorphism

Gal(F′_{±3}/Q_{3}(\sqrt{±3})) ≅ C_{2} × C_{2} × C_{3}.

Let γ_{4}^{2} ∈ I_{F′_{±3}/Q_{3}(\sqrt{±3})} be the unique element of order 2. (In later applications this will be the square of an element of order 4 in Gal(F′_{±3}/Q_{3}).) We also let F_{±3} denote the fixed field of a Frobenius lift of order 2, so F_{±3}/Q_{3} is totally ramified.

We will let ℐ_{±3} denote the two sided ideal of W (F_{9})[F, V][Gal(F′_{±3}/Q_{3})] generated by

• F + V = 0

• and [γ_{4}^{2}] + 1.

Let 𝒮_{±3} denote the full subcategory of 𝒮(\overline{ρ}) consisting of objects X for which we can find an object (𝒢, \{[g]\}) of ℱ𝒟_{F′_{±3}/Q_{3},ℐ_{±3}} such that X ≅ (𝒢, \{[g]\})_{Q_{3}}(\overline{Q}_{3}) as a Z_{3}[G_{3}]-module. By Lemma 4.1.3, we see that 𝒮_{±3} is closed under finite products, subobjects and quotients.

Now choose a continuous map of rings f : R_{V,Z_{3}}^{τ_{±3}} → \overline{Q}_{3}. Let K denote the finite extension of Q_{3} generated by the image of f, so that we get a continuous representation ρ : G_{3} ⟶ GL_{2}(𝒪_{K}). Let G be the corresponding 3-divisible group over Q_{3}. By Tate’s theorem (Theorem 4 of [T]), the base change of G to F′_{±3} has a unique extension to a 3-divisible group 𝒢 over 𝒪_{F′_{±3}}. By the uniqueness of this extension, it is also equipped with descent data {[g]} relative to F′_{±3}/Q_{3} and with an action of 𝒪_{K}, compatible with the canonical structure on the generic fibre.

Let γ̃_{2} ∈ Gal(Q_{3}(\sqrt{±3})^{ab}/Q_{3}(\sqrt{±3})) correspond to \sqrt{±3}. We will use the notation of Appendix B of [CDT] (in particular WD and D′(𝒢)), except that we will write F and F′ in place of ϕ and ϕ′. Then

• WD(ρ)(γ_{4}^{2}) = −1,

• WD(ρ)(γ_{2}^{2}), but not WD(ρ)(γ̃_{2}), is a scalar,

• and det WD(ρ)(γ̃_{2}) = 3. Thus WD(ρ)(γ_{2}^{2}) = −3. Hence on D′(𝒢) ⊗ Q_{3} we have

• [γ_{4}^{2}] = WD(ρ)(γ_{4}^{2}) = −1,

• and (F′)^{2} = [γ_{2}^{2}] WD(ρ)(γ_{2}^{−2}) = −1/3. We conclude that on D(𝒢) we have

• [γ_{4}^{2}] = −1,

• F^{2} = −3,

• and so F = −V. In particular ℐ_{±3} acts trivially on D(𝒢) and for all m ≥ 1 the map R_{V,Z_{3}}^{τ_{±3}} → 𝒪_{K}/(3^{m}) factors through R_{V,Z_{3}}^{ϵ,𝒮_{±3}}. Hence, the map R_{V,Z_{3}} → → R_{V,Z_{3}}^{τ_{±3}} factors through R_{V,Z_{3}}^{ϵ,𝒮_{±3}} and Theorem 2.1.4 follows from the following result which we will prove in §8.

Theorem 4.5.1. dim H_{𝒮_{±3}}^{1}(G_{3}, ad^{0} \overline{ρ}) ≤ 1.

4.6. Reduction steps for Theorem 2.1.6. We now begin the proof of Theorem 2.1.6. We may suppose that 𝒪 = Z_{3}.

Let F′_{i} denote the degree 12 abelian extension of Q_{3}(\sqrt{−3}) with norms the subgroup of Q_{3}(\sqrt{−3})^{×} topologically generated by −3, 4, 1 + 9\sqrt{−3} and 1 + (1 − 3ĩ)\sqrt{−3}, where ĩ is the unique lift of i to Z with 0 ≤ ĩ < 3. Note that F′_{i}/Q_{3} is Galois. We identify

Gal(F′_{i}/Q_{3}(\sqrt{−3})) ≅ ⟨γ_{2}⟩ × ⟨γ_{3}⟩ × ⟨γ_{4}^{2}⟩,

where γ_{2} corresponds to \sqrt{−3} and has order 2, γ_{3} corresponds to 1 − 3\sqrt{−3} and has order 3, and γ_{4}^{2} corresponds to −1 and has order 2. We also let F_{i} denote the fixed field of {1, γ_{2}}, so F_{i}/Q_{3} is totally ramified.

We will let ℐ_{i} denote the two sided ideal of W (F_{9})[F, V][Gal(F′_{i}/Q_{3})] generated by

• F + V,

• [γ_{4}^{2}] + 1,

• and ([γ_{3}] − [γ_{3}^{−1}])[γ_{2}] − F. We remark that the ideal ℐ_{i} is unchanged if we change our choice of \sqrt{−3}.

In §9 we will prove the following result (and explain the unusual looking notation).

Theorem 4.6.1. There are objects (𝒢, {[g]})_{(2,6)}, (𝒢, {[g]})_{(6,10)}, (𝒢, {[g]})_{(2,10)} and (𝒢, {[g]})_{(6,6)} in the category ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i}} with the following properties.

1. For (r, s) = (2, 6), (6, 10), (2, 10) and (6, 6), we have \overline{ρ} ≅ ((𝒢, \{[g]\})_{(r,s)})_{Q_{3}}(\overline{Q}_{3}) as G_{3}-modules.

2. For (r, s) = (2, 6), (6, 10), (2, 10) and (6, 6) there is a short exact sequence in ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i}},

(0) ⟶ (𝒢_{1}, {[g]})_{(r,s)} ⟶ (𝒢, {[g]})_{(r,s)} ⟶ (𝒢_{2}, {[g]})_{(r,s)} ⟶ (0),

such that (𝒢_{1}, {[g]})_{(r,s)} and (𝒢_{2}, {[g]})_{(r,s)} have order 3 and for all a, b (possibly equal) the natural map

Ext_{ℱ𝒟 _{F′/Q_{3},(ℐ_{i},3)}}^{1} ((𝒢_{a}, {[g]})_{(r,s)}, (𝒢_{b}, {[g]}))_{(r,s)} ⟶ Ext_{F_{3}[G_{Q_{3}} ]}^{1}((𝒢_{a}, {[g]})_{(r,s),Q_{3}}, (𝒢_{b}, {[g]})_{(r,s),Q_{3}})

is injective.

3. If k/F_{3} is a finite field extension and if (𝒢, {[g]}) is an object of ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i}} with an action of k such that (𝒢, \{[g]\})_{Q_{3}}(\overline{Q}_{3}) is isomorphic to \overline{ρ} ⊗ k, then for some (r, s) = (2, 6), (6, 10), (2, 10) or (6, 6) the object (𝒢, {[g]}) of ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i}} is weakly filtered by {(𝒢_{1}, {[g]})_{(r,s)}, (𝒢_{2}, {[g]})_{(r,s)}}.

4. For (r, s) = (2, 6), (6, 10) and (2, 10) we have F = 0 on D(𝒢_{(r,s)}), while F ≠ 0 on D(𝒢_{(6,6)}).

Note that for all a, b (possibly equal), we must have

Hom((𝒢_{a}, \{[g]\})_{(r,s)}, (𝒢_{b}, \{[g]\})_{(r,s)}) = Hom((𝒢_{a}, \{[g]\})_{(r,s),Q_{3}}, (𝒢_{b}, \{[g]\})_{(r,s),Q_{3}}) = \{ \begin{matrix}0 & if a ≠ b \\ F_{3} & if a = b \end{matrix}

For (r, s) = (2, 6), (6, 10), (2, 10) and (6, 6), we let 𝒮_{i,(r,s)} denote the full subcategory of 𝒮(\overline{ρ}) consisting of objects X which are isomorphic to (ℋ, {[g]})_{Q_{3}} for some object (ℋ, {[g]}) of ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i},{(𝒢,{[g]})_{(r,s)}}} . By Corollary 4.2.5 and Theorem 4.6.1 we see that 𝒮_{i,(r,s)} is closed under finite products, 𝒮(\overline{ρ})-subobjects and 𝒮(\overline{ρ})-quotients. In §9 we will also prove the following two results.

Theorem 4.6.2. For (r, s) = (2, 6), (6, 10), (2, 10) and (6, 6) we have

dim H_{𝒮_{i,(r,s)}}^{1}(G_{3}, ad^{0} \overline{ρ}) ≤ 1.

Theorem 4.6.3. For (r, s) = (2, 6), (6, 10) and (2, 10) and for any N ≥ 1 there exists a continuous representation

ρ_{N} : G_{Q_{3}} ⟶ GL_{2}(F_{3}[[T]]/(T^{N}))

such that • det ρ_{N} = ϵ,

• ρ_{N} ≅ (𝒢_{N}, \{[g]\})_{Q_{3}}(\overline{Q}_{3}) for some object (𝒢_{N}, \{[g]\}) of ℱ𝒟_{F′_{i}/Q_{3},(ℐ_{i},F),\{(𝒢,\{[g]\})_{(r,s)}\}} (where (ℐ_{i}, F) denotes the two sided ideal of W (F_{9})[F, V][Gal(F′_{i}/Q_{3})] generated by ℐ_{i} and F),

• and ρ mod (T^{2}) ≅ \overline{ρ̸} ⊗ k[[T]]/(T^{2}).

(We are not asserting that ρ_{N} and 𝒢_{N} are independent of the choice of (r, s), though in fact we believe that ρ_{N} is independent of this choice.)

From these results we can easily draw the following consequence.

Corollary 4.6.4. For (r, s) = (2, 6), (6, 10) and (2, 10) we have

R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}} ≅ F_{3}[[T]].

Proof. By Theorems 4.6.2 and 4.6.3 we see that R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}}/(3) ≅ F_{3}[[T]] and that if R is an Artinian

quotient of R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}}/(3) corresponding to a (necessarily unique, see Lemma 4.2.3) object (𝒢, {[g]}) of ℱ𝒟_{F′_{i}/Q_{3},(ℐ_{i},3),{(𝒢,{[g]})_{(r,s)}}} then F = 0 on D(𝒢).

Now suppose R is any Artinian quotient of R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}} which corresponds to an object (𝒢, {[g]}) of the category ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i},{(𝒢,{[g]})_{(r,s)}}} . Let G = (𝒢, {[g]})_{Q_{3}} and consider the exact sequences

(0) ⟶ G[3] ⟶ G ⟶ 3G ⟶ (0)

and

(0) ⟶ 3G ⟶ G ⟶ G/3G ⟶ (0).

By Lemma 4.2.3, we have exact sequences

(𝒢, {[g]}) ⟶ (𝒦, {[g]}) ⟶ (0)

and

(0) ⟶ (𝒦, {[g]}) ⟶ (𝒢, {[g]}) ⟶ (ℋ, {[g]}) ⟶ (0)

in ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i},{(𝒢,{[g]})_{(r,s)}}} such that the composite

(𝒢, {[g]}) → → (𝒦, {[g]}) ↪ (𝒢, {[g]})

is multiplication by 3. In particular we have exact sequences

(0) ⟶ D(𝒦) ⟶ D(𝒢)

and

(0) ⟶ D(ℋ) ⟶ D(𝒢) ⟶ D(𝒦) ⟶ (0),

such that the composite

D(𝒢) → → D(𝒦) ↪ D(𝒢)

is multiplication by 3. As F = −V = 0 on D(ℋ) we see that F and V factor through maps D(𝒦) → D(𝒢), i.e. we can write F = 3F′ and V = 3V′ for some endomorphisms F′ and V′ of D(𝒢). Thus 3 = 9F′V′ equals zero on D(𝒢)/9D(𝒢) and so D(𝒦) = 0. We conclude that 𝒦 = (0), so that 3G = (0) and 3R = (0).

Thus

R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}} = R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}}/(3) = F_{3}[[T]].

We now modify the argument in §4.4. Consider a continuous ring homomorphism f : R_{V,Z_{3}}^{τ′_{i}} → \overline{Q}_{3}. Let K denote the finite extension of Q_{3} generated by the image of f. Then we get a continuous representation ρ : G_{3} ⟶ GL_{2}(𝒪_{K}). Let G be the corresponding 3-divisible group over Q_{3}. By Tate’s theorem (Theorem 4 of [T]) G has a unique extension to a 3-divisible group 𝒢 over 𝒪_{F′_{i}}. By the uniqueness of this extension, 𝒢 comes equipped with descent data {[g]} relative to F′_{i}/Q_{3} and with an action of 𝒪_{K}, compatible with the canonical structure on the generic fibre.

Let γ̃_{2} ∈ Gal(Q_{3}(\sqrt{−3})^{ab}/Q_{3}(\sqrt{−3})) correspond to \sqrt{−3}. We will use the notation of Appendix B of [CDT] (in particular WD and D′(𝒢)), except that we will write F and F′ in place of ϕ and ϕ′. Then

• WD(ρ)(γ_{4}^{2}) = −1,

• WD(ρ)(γ_{2}^{2}) = −3,

• and WD(ρ)(γ̃_{2})(WD(ρ)(γ_{3}) − WD(ρ)(γ_{3})^{−1}) = 3. Thus on D′(𝒢) ⊗ Q_{3} we have

• [γ_{4}^{2}] = WD(ρ)(γ_{4}^{2}) = −1,

• (F′)^{2} = [γ_{2}^{2}] WD(ρ)(γ_{2}^{−2}) = −1/3,

• and [γ_{2}]([γ_{3}] − [γ_{3}^{−1}]) = 3F′. We conclude that on D(𝒢) we have

• [γ_{4}^{2}] = −1,

• F^{2} = −3,

• and [γ_{2}]([γ_{3}^{−1}] − [γ_{3}]) = 3F^{−1}. Hence also

• F = −V

• and [γ_{2}]([γ_{3}] − [γ_{3}^{−1}]) = F.

In particular ℐ_{i} acts trivially on D(𝒢).

Thus (𝒢[℘_{K}], \{[g]\}) is an object of ℱ𝒟_{F′_{i}/Q_{3},ℐ_{i}} such that (𝒢[℘_{K}], \{[g]\})_{Q_{3}} corresponds to \overline{ρ} ⊗ 𝒪_{K}/℘_{K}. By Theorem 4.6.1 we see that (𝒢[℘_{K}], {[g]}) is weakly filtered by {(𝒢_{1}, {[g]})_{(r,s)}, (𝒢_{2}, {[g]})_{(r,s)}} for some (r, s) = (2, 6), (6, 10), (2, 10) or (6, 6). We will prove (r, s) = (6, 6). By Theorem 4.6.1 and Lemma 4.2.4, (𝒢[℘_{K}], \{[g]\}) is strongly filtered by \{(𝒢_{1}, \{[g]\})_{(r,s)}, (𝒢_{2}, \{[g]\})_{(r,s)}\}. As (𝒢[℘_{K}], \{[g]\})_{Q_{3}} is filtered by \overline{ρ}, using Theorem 4.6.1, we see that (𝒢[℘_{K}], {[g]}) is weakly filtered by (𝒢, {[g]})_{(r,s)}. For all m ≥ 1 we have

(𝒢[℘_{K}^{m}]/𝒢[℘_{K}^{m−1}], {[g]}) ⟶^{∼} (𝒢[℘_{K}], {[g]}),

so for all m ≥ 1 the object (𝒢[℘_{K}^{m}], {[g]}) is also weakly and hence strongly filtered by (𝒢, {[g]})_{(r,s)} for the same (r, s). Thus, for all m ≥ 1, the map R_{V,Z_{3}} → 𝒪_{K′}/(3^{m}) factors through R_{V,Z_{3}}^{ϵ,𝒮_{i,(r,s)}}. By Corollary 4.6.4

we see that (r, s) = (6, 6), so the map R_{V,Z_{3}} → → R_{V,Z_{3}}^{τ′_{i}} factors through R_{V,Z_{3}}^{ϵ,𝒮_{i,(6,6)}} and Theorem 2.1.6 follows from Theorems 4.6.2.

4.7. Some Galois cohomology. In this section we will begin the proofs of Theorems 4.4.1, 4.5.1 and 4.6.2. We will let 𝒮 denote one of the categories 𝒮_{±1}, 𝒮_{±3} or 𝒮_{i,(r,s)}. We will let χ = ω in the cases 𝒮_{±1} and χ = 1 otherwise. In all cases

\overline{ρ} ∼ ( \begin{matrix}χω & ∗ \\ 0 & χ \end{matrix} )

is très ramifié.

The maps ω ⊗ χ ↪ \overline{ρ} and \overline{ρ} → → χ induce a commutative diagram with exact rows and columns

(0) ↓

Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, ω ⊗ χ)

↓

Ext_{F_{3}[G_{3}]}^{1}(\overline{ρ}, \overline{ρ}) ⟶ Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, \overline{ρ})
↓  ↓
(0) ⟶ Ext_{F_{3}[G_{3}]}^{1}(χ, χ) ⟶ Ext_{F_{3}[G_{3}]}^{1}(\overline{ρ}, χ) ⟶ Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, χ).

We will let θ_{0} denote the composite map

Ext_{F_{3}[G_{3}]}^{1}(\overline{ρ}, \overline{ρ}) ⟶ Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, χ);

and θ_{1} (resp. θ_{ω}) the induced mapping

ker θ_{0} ⟶ Ext_{F_{3}[G_{3}]}^{1}(χ, χ)

(resp.

ker θ_{0} ⟶ Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, ω ⊗ χ)).

We will also let \overline{θ}_{1} (resp. \overline{θ}_{ω}) the induced mapping

ker θ_{0} ⟶ Ext_{F_{3}[G_{3}]}^{1}(χ, χ) ⟶ Ext_{F_{3}[I_{3}]}^{1}(χ, χ)

(resp.

ker θ_{0} ⟶ Ext_{F_{3}[G_{3}]}^{1}(ω ⊗ χ, ω ⊗ χ) ⟶ Ext_{F_{3}[I_{3}]}^{1}(ω ⊗ χ, ω ⊗ χ)).

If we reinterpret our Ext-groups as cohomology groups and use the isomorphism \overline{ρ}^{∨} ∼ \overline{ρ} ⊗ ω, our diagram becomes

(0) ↓

H^{1}(G_{3}, F_{3})

↓

H^{1}(G_{3}, ad \overline{ρ})  ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ ω ⊗ χ)
↓  ↓
(0) ⟶ H^{1}(G_{3}, F_{3}) ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ ω ⊗ χ) ⟶  H^{1}(G_{3}, ω).

Fix a basis of F_{3}^{2} so that \overline{ρ} takes the form

( \begin{matrix}ω ⊗ χ & ∗ \\ 0 & χ \end{matrix} ) .

Then any extension of \overline{ρ} by \overline{ρ} in characteristic 3 may be represented by a matrix

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

where the cocycle

ϕ = ( \begin{matrix}ϕ_{11} & ϕ_{12} \\ ϕ_{21} & ϕ_{22} \end{matrix} ) ∈ Z^{1}(G_{3}, ad \overline{ρ})

represents the class of this extension in Ext_{F_{3}[G_{3}]}^{1}(\overline{ρ}, \overline{ρ}) ≅ H^{1}(G_{3}, ad \overline{ρ}). Moreover

• θ_{0}([ϕ]) = [ϕ_{21}] ∈ H^{1}(G_{3}, ω),

• if ϕ_{21} = 0 then θ_{1}([ϕ]) = [ϕ_{22}] ∈ H^{1}(G_{3}, F_{3}) and θ_{ω}([ϕ]) = [ϕ_{11}] ∈ H^{1}(G_{3}, F_{3}),

• and [ϕ] ∈ H^{1}(G_{3}, ad^{0} \overline{ρ}) if and only if 0 = [ϕ_{11} + ϕ_{22}] ∈ H^{1}(G_{3}, F_{3}). In particular we have θ_{1} = −θ_{ω} on H^{1}(G_{3}, ad^{0} \overline{ρ}) ∩ ker θ_{0}.

We have an exact sequence

(0) ⟶ \overline{ρ} ⊗ χ ⟶ ad^{0} \overline{ρ} ⟶ ω ⟶ (0)

where the first map sends

( \begin{matrix}x \\ y \end{matrix} ) ⟼ ( \begin{matrix}−y/2 & x \\ 0 & y/2 \end{matrix} )

and the second map sends

( \begin{matrix}a & b \\ c & −a \end{matrix} ) ⟼ c.

Thus we get an exact sequence

(0) ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ χ) ⟶ H^{1}(G_{3}, ad^{0} \overline{ρ}) ⟶^{θ_{0}} H^{1}(G_{3}, ω)

and so we may identify H^{1}(G_{3}, ad^{0} \overline{ρ}) ∩ ker θ_{0} with H^{1}(G_{3}, \overline{ρ} ⊗ χ). We also have an exact sequence

(0) ⟶ ω ⟶ \overline{ρ} ⊗ χ ⟶ 1 ⟶ (0), (4.7.1)

which gives rise to an exact sequence

(0) ⟶ F_{3} ⟶ H^{1}(G_{3}, ω) ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ χ) ⟶ H^{1}(G_{3}, F_{3}) ⟶ H^{2}(G_{3}, ω).

If we identify H^{1}(G_{3}, \overline{ρ} ⊗ χ) with H^{1}(G_{3}, ad^{0} \overline{ρ}) ∩ ker θ_{0} then the latter map H^{1}(G_{3}, \overline{ρ} ⊗ χ) ⟶ H^{1}(G_{3}, F_{3}) is identified with θ_{ω} = −θ_{1}.

Lemma 4.7.1. The sequence

(0) ⟶ F_{3} ⟶ H^{1}(G_{3}, ω) ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ χ) ⟶ H^{1}(I_{3}, F_{3})

is exact.

Proof. The key point is that \overline{ρ} is très ramifié (compare with Proposition 6.1 of [Di1]). It suffices to show that the composite

H^{1}(G_{F_{3}}, F_{3}) ⟶ H^{1}(G_{3}, F_{3}) ⟶ H^{2}(G_{3}, ω)

is injective. Suppose that x ∈ H^{1}(G_{3}, F_{3}) maps to zero in H^{2}(G_{3}, ω), then by Tate duality x is annihilated by the image of the map H^{0}(G_{3}, F_{3}) → H^{1}(G_{3}, ω) coming from the short exact sequence

(0) ⟶ ω ⟶ (\overline{ρ} ⊗ χ)^{∨} ⊗ ω ⟶ 1 ⟶ (0)

Cartier dual to (4.7.1). As (\overline{ρ} ⊗ χ)^{∨} ⊗ ω is très ramifié we see that the image of

H^{0}(G_{3}, F_{3}) ⟶ H^{1}(G_{3}, ω) ≅ Q_{3}^{×}/(Q_{3}^{×})^{3}

is not contained in Z_{3}^{×}/(Z_{3}^{×})^{3}. Thus

x ∈ Hom(Q_{3}^{×}/Z_{3}^{×}, F_{3}) ≅ H^{1}(G_{F_{3}}, F_{3}) ⊂ H^{1}(G_{3}, F_{3}) ≅ Hom(Q_{3}^{×}, F_{3})

must be zero (see Proposition 3 of §1 of chapter XIV of [Se1]).

Corollary 4.7.2. The maps

\overline{θ}_{1} : H^{1}(G_{3}, ad^{0} \overline{ρ}) ∩ ker θ_{0} ⟶ H^{1}(I_{3}, F_{3})

and

\overline{θ}_{ω} : H^{1}(G_{3}, ad^{0} \overline{ρ}) ∩ ker θ_{0} ⟶ H^{1}(I_{3}, F_{3})

have the same kernel and this has dimension 1 over F_{3}.

Theorems 4.4.1, 4.5.1 and 4.6.2 now follow from the following results, which we will prove later. One advantage of these new formulations is that, with one exception, they refer only to Ext_{𝒮}^{1}(\overline{ρ}, \overline{ρ}) and make no mention of the determinant or ad^{0} \overline{ρ}, concepts which we found tricky to translate into the language of integral models.

Theorem 4.7.3. 1. θ_{0} : Ext_{𝒮_{±1}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω) is the zero map.

2. \overline{θ}_{ω} : Ext_{𝒮_{−1}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3}) is the zero map.

3. \overline{θ}_{ω} : H_{𝒮_{1}}^{1}(G_{3}, ad^{0} \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3}) is the zero map.

Theorem 4.7.4. 1. θ_{0} : Ext_{𝒮_{±3}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω) is the zero map.

2. \overline{θ}_{ω} : Ext_{𝒮_{±3}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3}) is the zero map.

Theorem 4.7.5. Suppose that i ∈ Z/3Z and (r, s) = (2, 6), (6, 10), (2, 10) or (6, 6).

1. θ_{0} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω) is the zero map.

2. Either \overline{θ}_{ω} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3}) or \overline{θ}_{1} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3}) is the zero map.

The deduction of Theorems 4.4.1, 4.5.1 and 4.6.2 from these results is immediate.

5. Breuil Modules.

In this section we recall some results from [Br2] (see also the summary [Br1]) and give some slight extensions of them. Three of the authors apologise to the fourth for the title of this section, but they find that the term “Breuil module” is much more convenient than “filtered ϕ_{1}-module”.

Throughout this section, ℓ will be an odd rational prime and R will be a complete discrete valuation ring with fraction field F′ of characteristic zero and perfect residue field k of characteristic ℓ.

5.1. Basic properties of Breuil modules. We will fix a choice of uniformizer π of R and let

E_{π}(u) = u^{e} − ℓG_{π}(u)

be the Eisenstein polynomial which is the minimal polynomial of π over the fraction field of W (k), so G_{π}(u) ∈ W (k)[u] is a polynomial with unit constant term G_{π}(0) ∈ W (k)^{×} (and degree at most e − 1). The ℓ^{th} power map on k[u]/u^{eℓ} is denoted ϕ, and we define

c_{π} = −ϕ(G_{π}(u)) ∈ (k[u]/u^{eℓ})^{×}. (5.1.1)

It is very important to keep in mind that these definitions, as well as many of the definitions below, depend on the choice of the uniformizer π.

The category of ℓ-torsion Breuil modules (or “ℓ-torsion Breuil modules over R”, or simply “Breuil modules” or “Breuil modules over R”) is defined to be the category of triples (ℳ, ℳ_{1}, ϕ_{1}) where

• ℳ is a finite free k[u]/u^{eℓ}-module,

• ℳ_{1} is a k[u]/u^{eℓ}-submodule of ℳ containing u^{e} ℳ,

• ϕ_{1} : ℳ_{1} → ℳ is ϕ-semilinear and has image whose k[u]/u^{eℓ}-span is all of ℳ. We define the rank of (ℳ, ℳ_{1}, ϕ_{1}) to be the rank of ℳ over k[u]/u^{eℓ}. Breuil modules form an additive category (not abelian in general) in the obvious manner and this category does not depend on the choice of π. It is denoted \underline{ϕ_{1}−mod}_{R} or \underline{ϕ_{1}−mod}_{F′}. The induced ϕ-semilinear map of k-vector spaces

\overline{ϕ}_{1} : ℳ_{1} /u ℳ_{1} → ℳ /u ℳ

is bijective (see 2.1.2.1 of [Br2]). In particular, a map of Breuil modules

(ℳ, ℳ_{1}, ϕ_{1}) → (ℳ′, ℳ′_{1}, ϕ′_{1})

is an isomorphism if and only if the map ℳ → ℳ′ on underlying k[u]/u^{eℓ}-modules is an isomorphism.

Lemma 5.1.1. Suppose that

0 → ℳ′ → ℳ → ℳ′′ → 0

is a complex of Breuil modules. The following are equivalent.

1. The underlying sequence of k[u]/u^{eℓ}-modules is exact.

2. The underlying sequence of k[u]/u^{eℓ}-modules is exact as is the sequence

0 → ℳ′_{1} → ℳ_{1} → ℳ′′_{1} → 0.

3. The complex of vector spaces

0 → ℳ′ /u → ℳ /u → ℳ′′ /u → 0

is exact.

Proof. The second statement clearly implies the first. The first implies the third as Breuil modules are free over k[u]/u^{eℓ}. It remains to show that the third condition implies the second. Using Nakayama’s lemma and the freeness of Breuil modules we see that

0 → ℳ′ → ℳ → ℳ′′ → 0

is an exact sequence of k[u]/u^{eℓ}-modules. Using the bijectivity of \overline{ϕ}_{1}, we see that the natural map

f_{1} : ℳ_{1} → ℳ′′_{1}

is surjective modulo u and therefore is surjective. It remains to check that the inclusion of k[u]/u^{eℓ}-modules ℳ′_{1} ⊆ ker(f_{1}) is an equality. Since f_{1} is compatible with f : ℳ → ℳ′′ via the inclusions ℳ_{1} ⊆ ℳ, ℳ′′_{1} ⊆ ℳ′′ and also via the maps ϕ_{1} and ϕ′′_{1}, it is obvious that ker(f_{1}) ⊆ ker(f) = ℳ′ and that ϕ_{1}(ker(f_{1})) ⊆ ℳ′. Since ker(f_{1}) contains ℳ′_{1}, which in turn contains u^{e} ℳ′, we see that (ℳ′, ker(f_{1}), ϕ_{1}) is a Breuil module! Then (ℳ′, ℳ′_{1}, ϕ′_{1}) → (ℳ′, ker(f_{1}), ϕ_{1}) defined via the identity map on ℳ′ is a map of Breuil modules which is an isomorphism on underlying k[u]/u^{eℓ}-modules, so it must be an isomorphism of Breuil modules. This forces ker(f_{1}) = ℳ′_{1}.

When the equivalent conditions of this lemma are met we call the sequence of Breuil modules

0 → ℳ′ → ℳ → ℳ′′ → 0

exact.

For any Breuil module (ℳ, ℳ_{1}, ϕ_{1}), we define the Frobenius endomorphism ϕ : ℳ → ℳ by

ϕ(m) = \frac{1}{c_{π}} ϕ_{1}(u^{e}m), (5.1.2)

where c_{π} is defined as in (5.1.1). Note that this depends on our choice of uniformiser.

We let N : W (k)[[u]] → W (k)[[u]] denote the unique continuous W (k)-linear derivation satisfying Nu = u, i.e. N = u\frac{d}{du}. This operator “extends” to any Breuil module. More precisely, we have the following lemma.

Lemma 5.1.2. Let ℳ be an object of \underline{ϕ_{1}−mod}_{R}. There is a unique additive operator N : ℳ → ℳ satisfying the three conditions:

1. N(sx) = N(s)x + sN(x), s ∈ k[u]/u^{eℓ}, x ∈ ℳ,

2. N ∘ ϕ_{1} = ϕ ∘ N on ℳ_{1},

3. N(ℳ) ⊂ uℳ. Moreover, any morphism of Breuil modules ℳ → ℳ′ automatically commutes with N.

Proof. Let’s start with unicity. Recall we have an isomorphism k[u]/u^{eℓ} ⊗_{k[u^{ℓ}]/u^{eℓ}} ϕ_{1}(ℳ_{1}) →^{∼} ℳ ([Br2], 2.1.2.1). Suppose there are two N and N′ satisfying (1), (2) and (3) above, so Δ = N − N′ is k[u]/u^{eℓ}-linear and satisfies Δϕ_{1} = ϕΔ and Δ(ℳ) ⊂ uℳ. Thus,

Δϕ_{1}(ℳ_{1}) = ϕΔ(ℳ_{1}) ⊂ ϕ(uℳ) ⊂ u^{ℓ}ℳ,

so Δ(ℳ) = Δ(k[u]/u^{eℓ} ⊗_{k[u^{ℓ}]/u^{eℓ}} ϕ_{1}(ℳ_{1})) ⊂ u^{ℓ}ℳ. Iterating Δϕ_{1}(ℳ_{1}) ⊂ ϕΔ(ℳ) ⊂ u^{ℓ^{2}}ℳ so Δ(ℳ) ⊂ u^{ℓ^{2}}ℳ, and so on. As u^{eℓ} = 0, we get Δ = 0. For the existence, let N_{0} = N ⊗ 1 on

k[u]/u^{eℓ} ⊗_{k[u^{ℓ}]/u^{eℓ}} ϕ_{1}(ℳ_{1}) ≃ ℳ,

and note N_{0} satisfies N_{0}(sx) = N(s)x + sN_{0}(x). Call a derivation of ℳ any additive operators satisfying this relation and define successive derivations of ℳ by the formula

N_{j+1}(s ⊗ ϕ_{1}(x)) = N(s) ⊗ ϕ_{1}(x) + sϕ(N_{j}(x)),

for j ≥ 0. Note that N_{j+1} is well defined by the following observations.

• N(u^{ℓ}s) = u^{ℓ}N(s) and N_{j}(ux) = ux + uN_{j}(x) imply that N_{j+1}(u^{ℓ}s ⊗ ϕ_{1}(x)) = N_{j+1}(s ⊗ ϕ_{1}(ux)).

• If ϕ_{1}(x) = 0 then x ∈ u^{e}ℳ (see (1) of Lemma 2.1.2.1 of [Br1]) and so N_{j}(x) ∈ u^{e}ℳ and ϕ(N_{j}(x)) = 0.

As N_{0}(ℳ) ⊂ uℳ, we have (N_{j+1} − N_{j})(ℳ) ⊂ u^{ℓ^{j+1}}ℳ, so N_{j} = N_{j+1} for j ≫ 0. This N_{j} satisfies (1), (2) and (3).

The reason for introducing Breuil modules (and putting the factor c_{π}^{−1} in the definition of ϕ) is the following theorem.

Theorem 5.1.3. 1. Given the choice of uniformiser π for R there is a functor ℳ_{π} from finite flat R-group

schemes which are killed by ℓ to \underline{ϕ_{1}−mod}_{R} and a quasi-inverse functor 𝒢_{π}.

2. If 𝒢 is a finite flat R-group scheme killed by ℓ, then 𝒢 has rank ℓ^{r} if and only if ℳ_{π}(𝒢) has rank r.

3. If 𝒢 is a finite flat R-group scheme killed by ℓ, then there is a canonical k-linear isomorphism

D(𝒢) ⊗_{k,Frob_{ℓ}} k ≅ ℳ_{π}(𝒢)/uℳ_{π}(𝒢).

Under this identification, F ⊗ Frob_{ℓ} corresponds to ϕ and V ⊗ Frob_{ℓ}^{−1} corresponds to the composite

V_{ℳ} : ℳ /u ℳ →^{\overline{ϕ}_{1}^{−1}} ℳ_{1} /u ℳ_{1} → ℳ /u ℳ .

4. If

0 → 𝒢′ → 𝒢 → 𝒢′′ → 0

is a diagram of finite flat group schemes over R which are killed by ℓ and if

0 → ℳ_{π}(𝒢′′) → ℳ_{π}(𝒢) → ℳ_{π}(𝒢′) → 0

is the corresponding diagram of Breuil modules, then the diagram of finite flat group schemes is a short exact sequence if and only if the diagram of Breuil modules is a short exact sequence.

Proof. See §2.1.1, Proposition 2.1.2.2, Theorem 3.3.7, Theorem 4.2.1.6 and the proof of Theorem 3.3.5 of [Br2]. In 3.3.5 of [Br2] it is shown that ℳ(𝒢)/u ℳ(𝒢) can be k-linearly identified with the crystalline Dieudonné module of 𝒢 × k. In 4.2.14 of [BBM] the crystalline Dieudonné module of 𝒢 × k is identified with D(𝒢) ⊗_{k,Frob_{ℓ}} k. The equivalence of the two notions of exactness can be deduced from the compatibility of ℳ_{π} with Dieudonné theory, from Lemma 5.1.1, and from the fact that a complex of finite flat group schemes over R is exact if and only if its special fibre is exact (see for example Proposition 1.1 of [deJ]).

5.2. Examples. For 0 ≤ r ≤ e an integer and for a ∈ k^{×}, define a Breuil module ℳ(r, a) by • ℳ(r, a) = (k[u]/u^{eℓ})e,

• ℳ(r, a)_{1} = (k[u]/u^{eℓ})u^{r}e,

• ϕ_{1}(u^{r}e) = ae. It is easy to check that ϕ_{1} is well defined (and uniquely determined by the given conditions). We will refer to e as the standard generator of ℳ(r, a) and write 𝒢(r, a) for 𝒢_{π}(ℳ(r, a)). The following lemma is easy to check.

Lemma 5.2.1. 1. Any Breuil module of rank 1 over k[u]/u^{eℓ} is isomorphic to some ℳ(r, a).

2. There is a non-zero morphism ℳ(r, a) → ℳ(r′, a′) if and only if r′ ≥ r, r′ ≡ r mod ℓ − 1 and a/a′ ∈ (k^{×})^{ℓ−1}. All such morphisms are then of the form e ↦ bu^{ℓ(r′−r)/(ℓ−1)}e′, where b^{ℓ−1} = a/a′.

3. The modules ℳ(r, a) and ℳ(r′, a′) are isomorphic if and only if r = r′ and a/a′ ∈ (k^{×})^{ℓ−1}, or equivalently if and only if there are non-zero morphisms ℳ(r, a) → ℳ(r′, a′) and ℳ(r′, a′) → ℳ(r, a).

4. If we order the ℳ(r, a) by setting ℳ(r, a) ≥ ℳ(r′, a′) if there is a non-zero morphism ℳ(r′, a′) → ℳ(r, a), then the set of isomorphism classes of ℳ_{π}(𝒢)’s as 𝒢 runs over models of a fixed finite flat F′-group scheme G of order ℓ is well ordered.

5. On ℳ(r, a) we have Ne = 0, so N ∘ ϕ_{1} = 0.

6. 𝒢(r, a) is étale (resp. multiplicative) if and only if r = e (resp. r = 0).

7. 𝒢(0, 1) ≅ μ_{ℓ} and 𝒢(e, −G_{π}(0)) ≅ Z/ℓZ.

8. The Cartier dual of 𝒢(r, −G_{π}(0)) is 𝒢(e − r, 1).

Proof. The first three parts are easy computations. For the fourth part note that two finite flat group schemes 𝒢 and 𝒢′ of order ℓ over R have isomorphic generic fibres if and only if there is a non-zero morphism 𝒢 → 𝒢′ or 𝒢′ → 𝒢. The fifth part is another easy computation and the sixth part follows on computing the Dieudonné module using Theorem 5.1.3.

By 3.1.2 of [Br2] we see that the affine R-algebra of the group scheme attached to ℳ(r, a) is

R[X]/(X^{ℓ} + \frac{π^{e−r}ã}{G_{π}(π)}X),

where ã is a lift of a to W (k). This has constant generic fiber if and only if −π^{e−r}ã/G_{π}(π) ∈ F′ is an (ℓ − 1)^{th} power. This occurs if and only if r ≡ e mod ℓ − 1 and −a/G_{π}(0) ∈ k is a (ℓ − 1)^{th} power. Thus ℳ(e, −G_{π}(0)) corresponds to the étale group scheme Z/ℓZ over R.

Next, we show that the group scheme 𝒢 corresponding to the Breuil module ℳ(0, 1) is isomorphic to μ_{ℓ}. By using the relation between Breuil modules and Dieudonné modules (see Theorem 5.1.3) we see that the Dieudonné module of the closed fiber of 𝒢 is isomorphic to the Dieudonné module of the closed fiber of μ_{ℓ}. This forces 𝒢 →^{∼} μ_{ℓ}, since we may consider Cartier duals and observe that a finite flat R-group scheme 𝒢 is étale if and only if its special fibre is étale, and then §18.5.15 of book IV_{4} of [EGA] may be used.

This establishes the seventh part. The final part follows from parts four and seven.

Now suppose that 0 ≤ r, s ≤ e are integers and choose a, b ∈ k^{×}, f ∈ u^{max(0,r+s−e)}k[u]/u^{eℓ}. We can define an extension class

(0) ⟶ ℳ(s, b) ⟶ ℳ(s, b; r, a; f) ⟶ ℳ(r, a) ⟶ (0)

in \underline{ϕ_{1}−mod}_{R} by

• ℳ(s, b; r, a; f) = (k[u]/u^{eℓ})e ⊕ (k[u]/u^{eℓ})e′,

• ℳ(s, b; r, a; f)_{1} = ⟨u^{s}e, u^{r}e′ + fe⟩,

• ϕ_{1}(u^{s}e) = be and ϕ_{1}(u^{r}e′ + fe) = ae′,

• the standard generator of ℳ(s, b) maps to e,

• e maps to 0 and e′ maps to the standard generator in ℳ(r, a). The following lemma is also easy to check.

Lemma 5.2.2. 1. Any extension of ℳ(r, a) by ℳ(s, b) in \underline{ϕ_{1}−mod}_{R} is isomorphic to ℳ(s, b; r, a; f) for

some f ∈ u^{max(0,r+s−e)}k[u]/u^{eℓ}.

2. Two such extensions ℳ(s, b; r, a; f) and ℳ(s, b; r, a; f′) are isomorphic as extension classes if and only if

f′ − f = u^{s}h − (b/a)u^{r}h^{ℓ}

for some h ∈ k[u]/u^{eℓ}, in which case one such isomorphism fixes e and sends e′ to e′ − (b/a)h^{ℓ}e.

We remark that f ∈ u^{max(0,r+s−e)}k[u]/u^{eℓ} is required so that ℳ(s, b; r, a; f)_{1} ⊃ u^{e} ℳ(s, b; r, a; f). We will write 𝒢(r, a; s, b; f) for 𝒢_{π}(ℳ(r, a; s, b; f)).

We will also need some slight extensions of these results to allow for coefficients. To this end let k′/F_{ℓ} be a finite extension linearly disjoint from k and write k′k for the field k′ ⊗_{F_{ℓ}} k. For 0 ≤ r ≤ e an integer and for a ∈ (k′k)^{×}, define a Breuil module with an action of k′, ℳ(k′; r, a) by

• ℳ(k′; r, a) = ((k′k)[u]/u^{eℓ})e,

• ℳ(k′; r, a)_{1} = ((k′k)[u]/u^{eℓ})u^{r}e,

• ϕ_{1}(u^{r}e) = ae. We will let ϕ denote the automorphism of k′k[u], which is the identity on k′ and which raises elements of k[u] to the ℓ^{th} power. The following lemma is easy to check.

Lemma 5.2.3. 1. Any Breuil module with an action of k′ which is free of rank [k′ : k] over k[u]/u^{eℓ} is

isomorphic to some ℳ(k′; r, a).

2. There is a non-zero morphism ℳ(k′; r, a) → ℳ(k′; r′, a′) if and only if r′ ≥ r, r′ ≡ r mod ℓ − 1 and a/a′ ∈ ϕ(b)/b for some b ∈ (k′k)^{×}. All such morphisms are then of the form e ↦ b′u^{ℓ(r′−r)/(ℓ−1)}e′, where b ∈ (k′k)^{×} and ϕ(b′)/b′ = a/a′.

3. The modules ℳ(k′; r, a) and ℳ(k′; r′, a′) are isomorphic if and only if r = r′ and a/a′ ∈ ((k′k)^{×})^{ϕ−1}.

4. On ℳ(k′; r, a) we have Ne = 0 and so N ∘ ϕ_{1} = 0.

5. 𝒢_{π}(ℳ(k′; r, a)) is étale (resp. multiplicative) if and only if r = e (resp.r = 0).

Now choose 0 ≤ r, s ≤ e integers, a, b ∈ (k′k)^{×}, and f ∈ u^{max(0,r+s−e)}(k′k)[u]/u^{eℓ}. We define an extension class

(0) ⟶ ℳ(k′; s, b) ⟶ ℳ(k′; s, b; r, a; f) ⟶ ℳ(k′; r, a) ⟶ (0)

in \underline{ϕ_{1}−mod}_{R} with an action of k′ by

• ℳ(k′; s, b; r, a; f) = ((k′k)[u]/u^{eℓ})e ⊕ ((k′k)[u]/u^{eℓ})e′,

• ℳ(k′; s, b; r, a; f)_{1} = ⟨u^{s}e, u^{r}e′ + fe⟩,

• ϕ_{1}(u^{s}e) = be and ϕ_{1}(u^{r}e′ + fe) = ae′,

• the standard generators of ℳ(k′; s, b) maps to e,

• e maps to 0 and e′ to the standard generators in ℳ(k′; r, a). Then the following lemma is easy to check.

Lemma 5.2.4. 1. Any extension of ℳ(k′; r, a) by ℳ(k′; s, b) in \underline{ϕ_{1}−mod}_{R} with a compatible action of k′

is isomorphic to ℳ(k′; s, b; r, a; f) for some f ∈ u^{max(0,r+s−e)}(k′k)[u]/u^{eℓ}.

2. Two such extensions ℳ(k′; s, b; r, a; f) and ℳ(k′; s, b; r, a; f′) are isomorphic (as extensions) if and only if

f′ − f = u^{s}h − (b/a)u^{r}ϕ(h)

for some h ∈ (k′k)[u]/u^{eℓ}, in which case one such isomorphism fixes e and sends e′ to e′ − (b/a)ϕ(h)e.

We will write 𝒢(k′; r, a) and 𝒢(k′; r, a; s, b; f) for 𝒢_{π}(ℳ(k′; r, a)) and 𝒢_{π}(ℳ(k′; r, a; s, b; f)) respectively.

5.3. Relationship to syntomic sheaves. Let us first recall some of the notations of [Br1] and [Br2]. Let Spf(R)_{syn} be the small ℓ-adic formal syntomic site over R, S the ℓ-adic completion of W (k)[u, \frac{u^{ie}}{i!} ]_{i∈N}, S_{n} = S/ℓ^{n}S, E_{n} = Spec(S_{n}) and for any 𝔛 ∈ Spf(R)_{syn}:

𝒪_{n,π}^{cris}(𝔛) = H^{0}((𝔛_{n}/E_{n})_{cris}, 𝒪_{𝔛_{n}/E_{n}})

where 𝔛_{n} = 𝔛 ×_{R} R/ℓ^{n} is viewed over E_{n} via the thickening (Spec(R/ℓ^{n}) ↪ E_{n}, u ↦ π). It turns out 𝒪_{n,π}^{cris} is the sheaf of S_{n}-modules on Spf(R)_{syn} associated to the presheaf (c.f. the proof of Lemma 2.3.2 in [Br2]):

𝔛 ↦ (W_{n}(k)[u] ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(𝔛_{1}, 𝒪_{𝔛_{1}})))^{DP} = ((W_{n}(k)[u]/u^{eℓ^{n}}) ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(𝔛_{1}, 𝒪_{𝔛_{1}})))^{DP} . (5.3.1)

Here, the subscript “ϕ^{n}” means we twist by the n^{th} power of the Frobenius when sending W_{n}(k) to W_{n}(k)[u] and the exponent “DP” means we take the divided power envelope with respect to the kernel of the canonical map:

\begin{matrix}W_{n}(k)[u] ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(𝔛_{1}, 𝒪_{𝔛_{1}})) & → & Γ(𝔛_{n}, 𝒪_{𝔛_{n}}) \\ s(u) ⊗ (w_{0}, ..., w_{n−1}) & ↦ & s(π)(ŵ_{0}^{ℓ^{n}} + ℓŵ_{1}^{ℓ^{n−1}} + ... + ℓ^{n−1}ŵ_{n−1}^{ℓ}) \end{matrix}

where ŵ_{i} is a local lifting of w_{i}, these divided powers being compatible with the usual divided powers γ_{i}(ℓx) = \frac{ℓ^{i}}{i!} x^{i}. Note that the latter map induces a canonical surjection of sheaves of S_{n}-modules on Spf(R)_{syn}:

𝒪_{n,π}^{cris} → 𝒪_{n}

where 𝒪_{n}(𝔛) = Γ(𝔛_{n}, 𝒪_{𝔛_{n}}). We denote by 𝒥_{n,π}^{cris} the kernel of this surjection. For any n, let ϕ : S_{n} → S_{n} be the unique lifting of Frobenius on W_{n}(k) such that ϕ(u) = u^{ℓ} and ϕ(\frac{u^{ie}}{i!} ) = \frac{u^{ieℓ}}{i!} . The sheaf 𝒪_{n,π}^{cris} is equipped with the crystalline Frobenius ϕ, which is also induced by the map s(u) ⊗ (w_{0}, ..., w_{n−1}) ↦ ϕ(s(u)) ⊗ (w_{0}^{ℓ}, ..., w_{n−1}^{ℓ}) on the above presheaf (5.3.1). (Here ϕ on W_{n}(k)[u] is Frobenius on W_{n}(k) and takes u to u^{ℓ}.) Since ℓ divides ϕ(x) − x^{ℓ}, we get ϕ(𝒥_{n,π}^{cris}) ⊂ ℓ𝒪_{n,π}^{cris} for all n, so we can define an S_{1}-linear ϕ_{1} = \frac{ϕ}{ℓ}|_{𝒥_{n,π}^{cris}} by the usual “flatness” trick (see §2.3 of [Br2]). Let N : S_{n} → S_{n} be the unique W_{n}(k)-linear derivation such that N(u) = u and N(γ_{i}(u^{e})) = eu^{e}γ_{i−1}(u^{e}) = ieγ_{i}(u^{e}). Finally define:

N : 𝒪_{n,π}^{cris} → 𝒪_{n,π}^{cris}

to be the unique W_{n}(k)-linear morphism of sheaves which on the presheaf (5.3.1) is given by N(γ_{i}(s ⊗ w)) = (N(s) ⊗ w)γ_{i−1}(s ⊗ w). Note that N ∘ ϕ = ℓϕ ∘ N, so N ∘ ϕ_{1} = ϕ ∘ N on 𝒥_{n,π}^{cris}.

Let 𝒢 be a finite flat group scheme over R, which is killed by ℓ. Viewing 𝒢 as a formal scheme over R, it is an object in Spf(R)_{syn}. Viewing it as a sheaf of groups on Spf(R)_{syn}, its associated Breuil module is defined as:

1. ℳ_{π}(𝒢) = Hom_{sheaves of groups}(𝒢, 𝒪_{1,π}^{cris}) ⊗_{S_{1}} k[u]/u^{eℓ},

2. ℳ_{π}(𝒢)_{1} = image of Hom_{sheaves of groups}(𝒢, 𝒥_{1,π}^{cris}) ⊗_{S_{1}} k[u]/u^{eℓ} in ℳ_{π}(𝒢),

3. ϕ_{1} is induced by ϕ_{1} ⊗ ϕ, where the S_{1}-module structures are induced by the compatible S_{1} actions on 𝒪_{1,π}^{cris} and 𝒥_{1,π}^{cris} (see §3.2 and §2.1.2.2 of [Br2]). Here S_{1} → k[u]/u^{eℓ} is the surjection that sends u^{i} to u^{i} if 0 ≤ i < eℓ and γ_{i}(u^{e}) = \frac{u^{ie}}{i!} to 0 if i ≥ ℓ.

We record for future reference the following straightforward observation.

Lemma 5.3.1. If we denote by Δ (resp. pr_{i}, i ∈ {1, 2}) the coproduct (resp. the two projections)

𝒢 ×_{Spec(R)} 𝒢 → 𝒢,

then for any sheaf of commutative groups ℱ on Spf(R)_{syn} we have:

Hom_{sheaves of groups}(𝒢, ℱ) = {x ∈ ℱ(𝒢) | (Δ^{∗} − pr_{1}^{∗} − pr_{2}^{∗})(x) = 0}.

The operator N on 𝒪_{1,π}^{cris} induces an operator N on Hom_{sheaves of groups}(𝒢, 𝒪_{1,π}^{cris}), hence on ℳ_{π}(𝒢).

Lemma 5.3.2. The above operator N on ℳ_{π}(𝒢) coincides with the operator N defined in Lemma 5.1.2.

Proof. By unicity in Lemma 5.1.2, we only have to prove that N satisfies N(ℳ_{π}(𝒢)) ⊂ uℳ_{π}(𝒢), since the other conditions are automatically satisfied. It’s enough to prove that N(ϕ_{1}(x)) = (ϕ ∘ N)(x) ∈ uℳ_{π}(𝒢) for any x ∈ ℳ_{π}(𝒢)_{1}. But u^{eℓ−ℓ}ϕ ∘ N = 0 on 𝒪_{1,π}^{cris} because it is so on (k[u] ⊗ Γ(𝔛_{1}, 𝒪_{𝔛_{1}}))^{DP}. Thus one also has u^{eℓ−ℓ}ϕ ∘ N = 0 on Hom_{groups}(𝒢, 𝒪_{1,π}^{cris}), hence on ℳ_{π}(𝒢). This implies ϕ ∘ N(ℳ_{π}(𝒢)) ⊂ u^{ℓ}ℳ_{π}(𝒢) ⊂ uℳ_{π}(𝒢) since ℳ_{π}(𝒢) is free over k[u]/u^{eℓ}.

5.4. Base change. In this section we will examine the relationship of the functor ℳ_{π} with two instances of base change. First we consider unramified base change.

Let k′ be a perfect field of characteristic ℓ which is an extension of k and R′ = R ⊗_{W (k)} W (k′). Choose π′ = π ⊗ 1 as uniformizer in R′. If 𝔛 ∈ Spf(R)_{syn}, let 𝔛′ = Spf(R′) ×_{Spf(R)} 𝔛 and define:

𝒪_{n,π}^{cris′}(𝔛) = 𝒪_{n,π′}^{cris}(𝔛′) and 𝒥_{n,π}^{cris′}(𝔛) = 𝒥_{n,π′}^{cris}(𝔛′).

As in the proof of 2.3.2 of [Br2], we have that 𝒪_{n,π}^{cris′} is the sheaf on Spf(R)_{syn} associated to the presheaf:

\begin{aligned}𝔛 & ↦ & (W_{n}(k′)[u] ⊗_{ϕ^{n},W_{n}(k′)} W_{n}(Γ(𝔛′_{1}, 𝒪_{𝔛′_{1}})))^{DP} \\ & = & (W_{n}(k′)[u] ⊗_{ϕ^{n},W_{n}(k′)} W_{n}(k′ ⊗_{k} Γ(𝔛_{1}, 𝒪_{𝔛_{1}})))^{DP} . \end{aligned}

Define S′_{n} as S_{n} but with k′ instead of k. There is a canonical isomorphism of sheaves:

𝒪_{n,π}^{cris} ⊗_{S_{n}} S′_{n} = 𝒪_{n,π}^{cris} ⊗_{W_{n}(k)} W_{n}(k′) →^{∼} 𝒪_{n,π}^{cris′}

coming from the obvious isomorphism:

(W_{n}(k′)[u]/u^{eℓ^{n}}) ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(𝔛_{1}, 𝒪_{𝔛_{1}})) →^{∼} (W_{n}(k′)[u]/u^{eℓ^{n}}) ⊗_{ϕ^{n},W_{n}(k′)} W_{n}(k′ ⊗_{k} Γ(𝔛_{1}, 𝒪_{𝔛_{1}}))

and one easily sees it induces an isomorphism 𝒥_{n,π}^{cris} ⊗_{W_{n}(k)} W_{n}(k′) →^{∼} 𝒥_{n,π}^{cris′}. Moreover, we have the following obvious lemma.

Lemma 5.4.1. The diagram of sheaves on Spf(R)_{syn}:

𝒥_{n,π}^{cris} ⊗_{W_{n}(k)} W_{n}(k′) →^{∼} 𝒥_{n,π}^{cris′}
↓ ϕ_{1}⊗ϕ  ϕ_{1} ↓
𝒪_{n,π}^{cris} ⊗_{W_{n}(k)} W_{n}(k′) →^{∼} 𝒪_{n,π}^{cris′}

is commutative.

Using the identification from §5.3, Lemma 5.3.1 and Lemma 5.4.1 (for n = 1), together with obvious functorialities, we obtain after tensoring by k[u]/u^{eℓ} the following corollary.

Corollary 5.4.2. Let 𝒢 be a finite flat group scheme over R, which is killed by ℓ. Let k′/k be an extension of fields with k′ perfect and let π′ = π ⊗ 1, a uniformiser for R′ = R ⊗_{W (k)} W (k′). Then there is a canonical isomorphism in the category \underline{ϕ_{1}−mod}_{R}

(ℳ_{π}(𝒢) ⊗_{k} k′, ℳ_{π}(𝒢)_{1} ⊗_{k} k′, ϕ_{1} ⊗ ϕ) →^{∼} (ℳ_{π′}(𝒢′), ℳ_{π′}(𝒢′)_{1}, ϕ_{1})

compatible with composites of such residue field extensions.

We will now turn to the case of base change by a continuous automorphism g : R →^{∼} R. For any s = ∑ w_{i}u^{i} ∈ W (k)[[u]], let ^{(g)}s = ∑ g(w_{i})u^{i} and ^{(ϕ)}s = ∑ ϕ(w_{i})u^{i}, where g and ϕ act on W (k) through their action on k. Choose H_{g}(u) ∈ W (k)[[u]] such that g(π) = πH_{g}(π). Notice that H_{g}(u) ∈ W (k)[[u]]^{×}. Define ĝ: W (k)[[u]] →^{∼} W (k)[[u]] by ĝ(∑ w_{i}u^{i}) = ∑ g(w_{i})u^{i}H_{g}(u)^{i}.

Lemma 5.4.3. There is a unique element _{g}t(u) ∈ W (k)[[u]] such that, if _{g}ϕ is defined by _{g}ϕ(∑ w_{i}u^{i}) = ∑ ϕ(w_{i})(u^{ℓ}(1 + ℓ_{g}t(u)))^{i}, one has ĝ ∘_{g}ϕ = ϕ ∘ ĝ.

Proof. One has to solve in W (k)[[u]]:

1 + ℓ_{g}^{(g)}t(uH_{g}(u)) = (H_{g}(u)^{−1})^{ℓ}^{(ϕ)}H_{g}(u^{ℓ})

(where the two sides clearly belong to 1 + ℓW (k)[[u]]). As H_{g}(u) ∈ W (k)[[u]]^{×}, there is a unique K_{g} ∈ uW (k)[[u]]^{×} such that K_{g}(u)H_{g}(K_{g}(u)) = u, so we have

1 + ℓ_{g}^{(g)}t(u) = (H_{g}(K_{g}(u))^{−1})^{ℓ}^{(ϕ)}H_{g}(K_{g}(u)^{ℓ}).

For any object ℳ of \underline{ϕ_{1}−mod}_{R}, define _{g}ϕ_{1} : ℳ_{1} → ℳ by the following formula:

gϕ_{1}(x) = ϕ_{1}(x) +_{g}t(u)N(ϕ_{1}(x)) (5.4.1)

where N is as in Lemma 5.1.2.

For any 𝔛 ∈ Spf(R)_{syn}, let ^{g}𝔛 = Spf(R) ×_{g^{∗},Spf(R)} 𝔛 and define:

𝒪_{n,π}^{cris,(g)}(𝔛) = 𝒪_{n,π}^{cris}(^{g}𝔛) and 𝒥_{n,π}^{cris,(g)}(𝔛) = 𝒥_{n,π}^{cris}(^{g}𝔛).

Then 𝒪_{n,π}^{cris,(g)} is the sheaf on Spf(R)_{syn} associated to the presheaf:

\begin{aligned}𝔛 & ↦ & (W_{n}(k)[u] ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(^{g}𝔛_{1}, 𝒪_{^{g}𝔛}_{1})))^{DP} \\ & = & (W_{n}(k)[u] ⊗_{ϕ^{n},W_{n}(k)} W_{n}(R ⊗_{g,R} Γ(𝔛_{1}, 𝒪_{𝔛_{1}})))^{DP} . \end{aligned}

Let ĝ: S_{n} → S_{n} be the unique ring isomorphism such that ĝ(w_{i} \frac{u^{ei+j}}{i!} ) = g(w_{i})\frac{u^{ei+j}}{i!} H_{g}(u)^{ei+j} for 0 ≤ j < e, i ≥ 0. There is a canonical isomorphism of sheaves:

𝒪_{n,π}^{cris} ⊗_{S_{n},ĝ} S_{n} →^{∼} 𝒪_{n,π}^{cris,(g)}

coming from the obvious ĝ-semi-linear isomorphism:

\begin{matrix}(W_{n}(k)[u]/u^{eℓ^{n}}) ⊗_{ϕ^{n},W_{n}(k)} W_{n}(Γ(𝔛_{1}, 𝒪_{𝔛_{1}})) & →^{∼} & (W_{n}(k)[u]/u^{eℓ^{n}}) ⊗_{ϕ^{n},W_{n}(k)} W_{n}(R ⊗_{g,R} Γ(𝔛_{1}, 𝒪_{𝔛_{1}})) \\ s ⊗ (w_{0}, ..., w_{n−1}) & ↦ & ĝ(s) ⊗ (1 ⊗ w_{0}, ..., 1 ⊗ w_{n−1}) \end{matrix}

and one easily sees it induces an isomorphism 𝒥_{n,π}^{cris} ⊗_{S_{n},ĝ} S_{n} →^{∼} 𝒥_{n,π}^{cris,(g)}.

Define _{g}ϕ : S_{n} → S_{n} as in Lemma 5.4.3 and define:

gϕ : 𝒪_{n,π}^{cris} → 𝒪_{n,π}^{cris}

to be the unique morphism of sheaves which is induced by _{g}ϕ(γ_{i}(s ⊗ w)) = γ_{i}(_{g}ϕ(s) ⊗ ϕ(w)) on the presheaf (5.3.1) (see §5.3 and note that this is well defined). Since _{g}ϕ(𝒥_{n,π}^{cris}) ⊂ ℓ𝒪_{n,π}^{cris}, we can define _{g}ϕ_{1} = \frac{_{g}ϕ}{ℓ}|_{𝒥_{n,π}^{cris}}.

Lemma 5.4.4. The diagram of sheaves on Spf(R)_{syn}:

𝒥_{n,π}^{cris} ⊗_{S_{n},ĝ} S_{n} →^{∼} 𝒥_{n,π}^{cris,(g)}
↓ _{g}ϕ_{1}⊗ϕ  ϕ_{1} ↓
𝒪_{n,π}^{cris} ⊗_{S_{n},ĝ} S_{n} →^{∼} 𝒪_{n,π}^{cris,(g)}

is commutative. Moreover we have on 𝒥_{n,π}^{cris}:

gϕ_{1} = ∑_{i=0}^{∞}(\frac{log(1 + ℓ_{g}t(u))}{ℓ})^{i} \frac{N^{i}}{i!} ∘ ϕ_{1}

where N is defined as in §5.3.

Proof. By working modulo ℓ^{n+1}, i.e. with 𝒥_{n+1,π}^{cris} and_{g}ϕ, and looking on the above presheaves, it is completely straightforward.

Let 𝒢 be a finite flat group scheme over R which is killed by ℓ. Note that thanks to Lemma 5.3.2 and the formula for _{g}ϕ_{1} in Lemma 5.4.4, the operator ℳ_{π}(𝒢)_{1} → ℳ_{π}(𝒢) induced by the map _{g}ϕ_{1} : 𝒥_{n,π}^{cris} → 𝒪_{n,π}^{cris} is precisely the operator denoted_{g}ϕ_{1} earlier in this section (see (5.4.1)). Using this, together with Lemma 5.3.1, Lemma 5.4.4 (for n = 1) and obvious functorialities, we obtain, after tensoring by k[u]/u^{eℓ}, the following corollary.

Corollary 5.4.5. 1. Let 𝒢 be a finite flat group scheme over R, which is killed by ℓ. Then there is a canonical isomorphism in the category \underline{ϕ_{1}−mod}_{R}:

(ℳ_{π}(𝒢) ⊗_{k[u]/u^{eℓ},ĝ} k[u]/u^{eℓ}, ℳ_{π}(𝒢)_{1} ⊗_{k[u]/u^{eℓ},ĝ} k[u]/u^{eℓ},_{g}ϕ_{1} ⊗ ϕ) →^{∼} (ℳ_{π}(^{g}𝒢), ℳ_{π}(^{g}𝒢)_{1}, ϕ_{1}).

2. If f : 𝒢 → 𝒢′ is a morphism of finite flat R-group schemes killed by ℓ and ℳ_{π}(f) is the corresponding morphism in \underline{ϕ_{1}−mod}_{R}, then ℳ_{π}(f) also commutes with the _{g}ϕ_{1} and there is a commutative diagram in \underline{ϕ_{1}−mod}_{R}:

ℳ_{π}(𝒢′) ⊗_{k[u]/u^{eℓ},ĝ} (k[u]/u^{eℓ}) ⟶^{ℳ_{π}(f)⊗1} ℳ_{π}(𝒢) ⊗_{k[u]/u^{eℓ},ĝ} (k[u]/u^{eℓ})
≀ ↓  ↓ ≀
ℳ_{π}(^{g}𝒢′)  ⟶^{ℳ_{π}(^{g}f)}  ℳ_{π}(^{g}𝒢).

3. If g_{1}, g_{2} are two continuous automorphisms of R and if we choose the unique H_{g_{2}g_{1}} ∈ W (k)[[u]] such that ĝ_{2}g_{1} = ĝ_{2} ∘ ĝ_{1} on W (k)[[u]], then on

(ℳ_{π}(𝒢) ⊗_{k[u]/u^{eℓ},ĝ_{1}} k[u]/u^{eℓ}) ⊗_{k[u]/u^{eℓ},ĝ_{2}} k[u]/u^{eℓ} ≃ ℳ_{π}(𝒢) ⊗_{k[u]/u^{eℓ},ĝ_{2}g_{1}} k[u]/u^{eℓ},

one has _{g_{2}}(_{g_{1}}ϕ_{1} ⊗ ϕ) ⊗ ϕ = _{g_{2}g_{1}}ϕ_{1} ⊗ ϕ.

Corollary 5.4.6. Let 𝒢 be a finite flat group scheme over R, which is killed by ℓ. To give an morphism of schemes [g] : 𝒢 → 𝒢 such that the diagram of schemes

𝒢  ⟶^{[g]}  𝒢
↓  ↓
Spec(R) ⟶^{Spec(g)} Spec(R)

is commutative and the induced morphism 𝒢 → Spec(R) ×_{g,Spec(R)} 𝒢 is an morphism of group schemes over R, is equivalent to giving an additive map ĝ: ℳ_{π}(𝒢) → ℳ_{π}(𝒢) such that both of the following hold.

1. For all s ∈ k[u]/u^{ep} and x ∈ ℳ_{π}(𝒢), ĝ(sx) = ĝ(s)ĝ(x).

2. ĝ(ℳ_{π}(𝒢)_{1}) ⊂ ℳ_{π}(𝒢)_{1} and ϕ_{1} ∘ ĝ = ĝ ∘ ϕ_{1} + ĝ(_{g}t(u))ĝ ∘ N ∘ ϕ_{1} with _{g}t as in Lemma 5.4.3 and N as in Lemma 5.1.2.

Proof. Note that the last condition is equivalent to ϕ_{1} ∘ ĝ = ĝ ∘_{g}ϕ_{1}. The first two conditions are equivalent to giving a morphism ĝ : ℳ_{π}(^{g}𝒢) → ℳ_{π}(𝒢) in \underline{ϕ_{1}−mod}_{R}, which is equivalent to the last two by Corollary 5.4.5.

Finally we make some computations that concern the dependence of the above compatibilities on the choice of H_{g}(u). Let f(u) be an element of (k[u]/u^{eℓ})_{1} = u^{e}(k[u]/u^{eℓ}) and define, for any ℳ in \underline{ϕ_{1}−mod}_{R}, the additive map 1_{f} : ϕ_{1}(ℳ_{1}) → ℳ via

1_{f} = 1 + ( ∑_{i=1}^{ℓ−1} \frac{(−1)^{i−1}}{i} f(u)^{i})N

where N is as in Lemma 5.1.2. Using k[u]/u^{eℓ} ⊗_{k[u^{ℓ}]/u^{eℓ}} ϕ_{1}(ℳ_{1}) ≃ ℳ, we extend 1_{f} to all of ℳ by the formula:

1_{f}(u^{i}x) = u^{i}(1 + f(u))^{i}1_{f}(x)

for x ∈ ϕ_{1}(ℳ_{1}). If x ∈ ℳ_{1}, one checks that:

1_{f}(ϕ_{1}(u^{i}x)) = u^{iℓ}1_{f}(ϕ_{1}(x)) = 1_{f}(u^{iℓ}ϕ_{1}(x))

so 1_{f} is well defined. Moreover, it is clear that 1_{f}(ℳ_{1}) ⊂ ℳ_{1}. Let

1_{f} : 𝒪_{1,π}^{cris} →^{∼} 𝒪_{1,π}^{cris}

be the unique isomorphism of sheaves coming from the semi-linear isomorphism of presheaves:

\begin{matrix}(k[u]/u^{eℓ}) ⊗_{ϕ,k} Γ(𝔛_{1}, 𝒪_{𝔛_{1}}) & →^{∼} & (k[u]/u^{eℓ}) ⊗_{ϕ,k} (Γ(𝔛_{1}, 𝒪_{𝔛_{1}}) \\ s(u) ⊗ (w_{0}, ..., w_{n−1}) & ↦ & s(u(1 + f(u))) ⊗ (w_{0}, ..., w_{n−1}) \end{matrix}

(see 5.3.1).

Let 𝒢 be a finite flat group scheme over R killed by ℓ and recall that

ℳ_{π}(𝒢) = Hom_{sheaves of groups}(𝒢, 𝒪_{1,π}^{cris}) ⊗ k[u]/u^{eℓ}.

Lemma 5.4.7. The operator 1_{f} on ℳ_{π}(𝒢) is induced by the operator 1_{f} on 𝒪_{1,π}^{cris}.

Proof. One can check that the operator 1_{f} on 𝒪_{1,π}^{cris} satisfies 1_{f} ∘ ϕ_{1} = ϕ_{1} + log(1 + f)N ∘ ϕ_{1} where N is defined as in §5.3 and log(1 + f) is the usual expansion of log in S_{1}, which makes sense because of the assumption that u^{e}|f and because of the divided powers γ_{i}(u^{e}) = \frac{u^{ei}}{i!} . After tensoring with k[u]/u^{eℓ}, we get

1_{f} = 1 + (∑_{i=1}^{ℓ−1} \frac{(−1)^{i−1}}{i} f(u)^{i})N on ϕ_{1}(ℳ_{π}(𝒢)_{1}) which clearly implies the two 1_{f}’s are the same.

Let g = 1 and choose H_{g}(u) = 1 + f(u) for some f ∈ E_{π}(u)W (k)[[u]]. Recall from Corollary 5.4.5 that we have a canonical isomorphism ℳ_{π}(𝒢) ⊗_{k[u]/u^{eℓ},ĝ} (k[u]/u^{eℓ}) →^{∼} ℳ_{π}(^{g}𝒢).

Lemma 5.4.8. The map 1_{f} is the composite ℳ_{π}(𝒢) →^{∼} ℳ_{π}(^{g}𝒢) →^{∼} ℳ_{π}(𝒢) where the first map is the one in Corollary 5.4.5 and the second comes from the obvious isomorphism 𝒢 →^{∼} ^{g}𝒢. In other words, once H_{g}(u) = 1+f(u) has been chosen, 1_{f} : ℳ_{π}(𝒢) → ℳ_{π}(𝒢) is the map corresponding to the identity 1_{𝒢} : 𝒢 → 𝒢 under the equivalence of Corollary 5.4.6.

The proof is straightforward by looking at the usual presheaves and using Lemma 5.4.7. We remark that 1_{f} is not necessarily the identity even though 1_{𝒢} is. However, with f = 0, 1_{f} is the identity.

5.5. Reformulation. In this section, we will reformulate Corollary 5.4.6.

Lemma 5.5.1. There is a unique element t_{g}(u) ∈ W (k)[[u]] such that if ϕ_{g} is defined by ϕ_{g}(∑ w_{i}u^{i}) = ∑ ϕ(w_{i})(u^{ℓ}(1 + ℓt_{g}(u))^{i}, one has ĝ ∘ ϕ = ϕ_{g} ∘ ĝ.

Proof. One has to solve in W (k)[[u]]:

u^{ℓ}H_{g}(u)^{ℓ} = u^{ℓ}(1 + ℓt_{g}(u))^{(ϕ)}H_{g}(u^{ℓ}(1 + ℓt_{g}(u))).

As H_{g}(u) ∈ W (k)[[u]]^{×}, there is a unique L_{g} ∈ uW (k)[[u]]^{×} such that L_{g}(uH_{g}(u)) = u. Applying L_{g} to u = K_{g}(u)H_{g}(K_{g}(u)) (cf. the proof of Lemma 5.4.3), we get L_{g}(u) = K_{g}(u). We must solve:

1 + ℓt_{g}(u) = \frac{^{(ϕ)}K_{g}(u^{ℓ}H_{g}(u)^{ℓ})}{u^{ℓ}}.

Lemma 5.5.2. There is a unique λ_{g}(u) ∈ 1 + uW (k)[[u]] such that if N_{g} = λ_{g}N, then N_{g} ∘ ĝ = ĝ ∘ N. Similarly, there is a unique _{g}λ(u) ∈ 1 + uW (k)[[u]] such that if _{g}N = _{g}λN, then ĝ ∘_{g}N = N ∘ ĝ. Moreover, N_{g} ∘ ϕ_{g} = ℓϕ_{g} ∘ N_{g} and _{g}N ∘_{g}ϕ = ℓ_{g}ϕ ∘_{g}N.

Proof. Since N is a derivation, so is λN for any λ ∈ W (k)[[u]]. One has to solve λ_{g}(u)N(uH_{g}(u)) = uH_{g}(u) and _{g}^{(g)}λ(uH_{g}(u)) = 1 + \frac{N(H_{g}(u))}{H_{g}(u)}, which amounts to:

λ_{g}(u) = (1 + \frac{N(H_{g}(u))}{H_{g}(u)})^{−1},
_{g}^{(g)}λ(u) = 1 + \frac{N(H_{g})(K_{g}(u))}{H_{g}(K_{g}(u))},

where K_{g} is as in the proof of Lemma 5.4.3. The commutation relations with the Frobenius follow from N ∘ ϕ = ℓϕ ∘ N, ϕ_{g} ∘ ĝ = ĝ ∘ ϕ, N_{g} ∘ ĝ = ĝ ∘ N, ĝ ∘_{g}ϕ = ϕ ∘ ĝ, ĝ ∘_{g}N = N ∘ ĝ and the fact ĝ is bijective on W (k)[[u]].

We also denote by _{g}N = _{g}λN and N_{g} = λ_{g}N the corresponding derivations on k[u]/u^{eℓ}. For any object ℳ of \underline{ϕ_{1}−mod}_{R}, define ϕ_{1,g} : ℳ_{1} → ℳ by the formula:

ϕ_{1,g}(x) = ϕ_{1}(x) + t_{g}(u)N(ϕ_{1}(x))

where N is as in Lemma 5.1.2, and we recall that we defined _{g}ϕ_{1} in (5.4.1). One checks that ϕ_{1,g}(u^{e}) = gϕ_{1}(u^{e}) = ϕ_{1}(u^{e}) = c_{π} (see (5.1.1)). Note that we also have ϕ_{1,g} ∘ĝ = ĝ∘_{g}ϕ_{1}, N_{g} ∘ĝ = ĝ∘N, ĝ∘_{g}ϕ_{1} = ϕ_{1} ∘ĝ, ĝ ∘_{g}N = N ∘ ĝ in k[u]/u^{eℓ}.

Lemma 5.5.3. Let ℳ be an object of \underline{ϕ_{1}−mod}_{R}, then there is a unique operator N_{g} : ℳ → ℳ satisfying the three conditions:

1. N_{g}(sx) = N_{g}(s)x + sN_{g}(x), s ∈ k[u]/u^{eℓ}, x ∈ ℳ,

2. N_{g}ϕ_{1,g}(x) = ϕ_{g}N_{g}(x), x ∈ ℳ_{1} where ϕ_{g}(y) = \frac{1}{c_{π}} ϕ_{1,g}(u^{e}y) if y ∈ ℳ,

3. N_{g}(ℳ) ⊂ uℳ. The same statement holds for _{g}N, _{g}ϕ, and _{g}ϕ_{1}.

Proof. The proof is the same as for Lemma 5.1.2, using the fact we still have isomorphisms

k[u]/u^{eℓ} ⊗_{k[u^{ℓ}]/u^{eℓ}} ϕ_{1,g}(ℳ_{1}) →^{∼} ℳ

(resp. with _{g}ϕ_{1} replacing ϕ_{1,g}).

Lemma 5.5.4. For ℳ an object of \underline{ϕ_{1}−mod}_{R}, N_{g} = λ_{g}N and _{g}N = _{g}λN where N_{g}, _{g}N are as in Lemma 5.5.3, λ_{g}, _{g}λ as in Lemma 5.5.2 and N as in Lemma 5.1.2.

Proof. By unicity of N_{g}, one has to check λ_{g}N satisfies the three conditions of Lemma 5.5.3. The first and last are obvious. Note that Nϕ_{1}(u^{e}x) = ϕN(u^{e}x) = 0 so ϕ_{1,g}(u^{e}x) = ϕ_{1}(u^{e}x), which implies ϕ = ϕ_{g} on ℳ (ϕ_{g} is as in Lemma 5.5.3). One computes:

(λ_{g}(u)N) ∘ ϕ_{1,g} = λ_{g}(u)(1 + N(t_{g}(u)))ϕ ∘ N
ϕ_{g} ∘ (λ_{g}(u)N) = ^{(ϕ)}λ_{g}(u^{ℓ})ϕ ∘ N.

But the equality N_{g} ∘ ϕ_{g}(u) = ℓϕ_{g} ∘ N_{g}(u) in W (k)[[u]] (from Lemma 5.5.2) yields

λ_{g}(u)(1 + N(t_{g}(u))) −^{(ϕ)}λ_{g}(u^{ℓ}) ∈ ℓW (k)[[u]].

We thus get (λ_{g}N) ∘ ϕ_{1,g} = ϕ_{g} ∘ (λ_{g}N) hence condition (2). For _{g}N, the proof is completely similar.

Lemma 5.5.5. Let ℳ be an object of \underline{ϕ_{1}−mod}_{R} and ĝ : ℳ → ℳ be an additive map such that for all s ∈ k[u]/u^{ep} and x ∈ ℳ, ĝ(sx) = ĝ(s)ĝ(x) and ĝ(ℳ_{1}) ⊂ ℳ_{1}. If ĝ ∘ ϕ_{1} = ϕ_{1,g} ∘ ĝ, then ĝ ∘ N = N_{g} ∘ ĝ. Similarly, if ϕ_{1} ∘ ĝ = ĝ ∘_{g}ϕ_{1}, then N ∘ ĝ = ĝ ∘_{g}N.

Proof. We prove the first case, the other one being the same. As in the proof of Lemma 5.1.2, we define N_{g,0}, N_{g,1},..., with N_{g} = N_{g,i} for i large enough, using k[u]/u^{eℓ} ⊗_{k[u^{p}]/u^{eℓ}} ϕ_{1,g}(ℳ_{1}) →^{∼} ℳ. It is enough to show ĝ ∘ N_{i} = N_{g,i} ∘ ĝ for all i. Suppose ĝ ∘ N_{i−1} = N_{g,i−1} ∘ ĝ and let s ∈ k[u]/u^{eℓ} and x ∈ ℳ_{1}, then:

N_{g,i}ĝ(sϕ_{1}(x)) = N_{g,i}(ĝ(s)ϕ_{1,g}(ĝ(x)))
= N_{g}(ĝ(s))ϕ_{1,g}(ĝ(x)) + ĝ(s)ϕ_{1,g}N_{g,i−1}(ĝ(x))
= ĝ(N(s)ϕ_{1}(x)) + ĝ(s)ϕ_{1,g}ĝ(N_{i−1}(x))
= ĝ(N(s)ϕ_{1}(x)) + ĝ(sϕ_{1}(N_{i−1}(x)))
= ĝN_{i}(sϕ_{1}(x)),

so ĝ ∘ N_{i} = N_{g,i} ∘ ĝ by linearity. One easily checks by a similar computation that N_{g,0} ∘ ĝ = ĝ ∘ N_{0}, hence the result follows by induction.

Lemma 5.5.6. Let ℳ be an object of \underline{ϕ_{1}−mod}_{R} and ĝ : ℳ → ℳ an additive map such that for all s ∈ k[u]/u^{ep} and x ∈ ℳ, ĝ(sx) = ĝ(s)ĝ(x) and ĝ(ℳ_{1}) ⊂ ℳ_{1}. Then the following two conditions are equivalent:

1. ϕ_{1} ∘ ĝ = ĝ ∘ ϕ_{1} + ĝ(_{g}t(u))ĝ ∘ N ∘ ϕ_{1}

2. and ĝ ∘ ϕ_{1} = ϕ_{1} ∘ ĝ + t_{g}(u)N ∘ ϕ_{1} ∘ ĝ.

Proof. One has to show ϕ_{1} ∘ ĝ = ĝ ∘_{g}ϕ_{1} is equivalent to ĝ ∘ ϕ_{1} = ϕ_{1,g} ∘ ĝ. We prove (1) ⇒ (2), the other case being the same. On ℳ, we have ĝ ∘ ϕ = ϕ ∘ ĝ, because ϕ = ϕ_{g} = _{g}ϕ, as in the proof of Lemma 5.5.4. By Lemmas 5.5.4 and 5.5.5, we have ĝ ∘ N = ĝ(_{g}λ^{−1})N ∘ ĝ. Thus we get from (1), using Nϕ_{1} = ϕN,

ĝ ∘ ϕ_{1} = ϕ_{1} ∘ ĝ − ĝ(_{g}t(u))ĝ(_{g}λ(u)^{−1})^{ℓ}N ∘ ϕ_{1} ∘ ĝ.

Playing the same game over W_{2}(k)[[u]] with the relation ϕ ∘ ĝ = ĝ ∘ ϕ + ĝ(_{g}t(u))ĝ ∘ N ∘ ϕ, which is easily checked to hold in W_{2}(k)[[u]], we again end up with ĝ∘ ϕ = ϕ ∘ ĝ− ĝ(_{g}t(u))ĝ(_{g}λ(u)^{−1})^{ℓ}N ∘ ϕ ∘ ĝ in W_{2}(k)[[u]]. But we also have in W_{2}(k)[[u]] the equality:

ĝ ∘ ϕ = ϕ ∘ ĝ + t_{g}(u)N ∘ ϕ ∘ ĝ.

Thus −ĝ(_{g}t(u))ĝ(_{g}λ(u)^{−1})^{ℓ} = t_{g}(u) in k[u]/u^{eℓ}, so relation (2) holds.

We can now derive the variant of Corollary 5.4.6 which we will use.

Corollary 5.5.7. Let 𝒢 be a finite flat R-group scheme killed by ℓ. Let g : R →^{∼} R be a continuous automorphism, choose H_{g}(u) ∈ W (k)[[u]] such that g(π) = πH_{g}(π) and define ĝ : k[u]/u^{eℓ} → k[u]/u^{eℓ} by ĝ(Σw_{i}u^{i}) = Σg(w_{i})u^{i}H_{g}(u)^{i}. To give an morphism of schemes [g] : 𝒢 → 𝒢 such that the diagram of schemes

𝒢  ⟶^{[g]}  𝒢
↓  ↓
Spec(R) ⟶^{Spec(g)} Spec(R)

is commutative and the induced morphism 𝒢 → Spec(R) ×_{g,Spec(R)} 𝒢 is an morphism of group schemes over R, is equivalent to giving an additive map ĝ: ℳ_{π}(𝒢) → ℳ_{π}(𝒢) such that both of the following hold.

1. For all s ∈ k[u]/u^{ep} and x ∈ ℳ_{π}(𝒢), ĝ(sx) = ĝ(s)ĝ(x).

2. ĝ(ℳ_{π}(𝒢)_{1}) ⊂ ℳ_{π}(𝒢)_{1} and ĝ∘ϕ_{1} = (1+t_{g}(u)N)∘ϕ_{1} ∘ĝ, with t_{g} as in Lemma 5.5.1 and N as in Lemma

5.1.2. Moreover, [g] is an isomorphism if and only if ĝ is. Assume these are isomorphisms. Choose H_{g^{−1}} such that ĝ^{−1}(u) = g^{−1}(u) on W (k)[[u]], i.e. H_{g^{−1}}(u) = g^{−1}(u)/u. Then the map ĝ^{−1} that corresponds to [g]^{−1} is equal to g^{−1}. Also, if g_{1}, g_{2} are two automorphisms of R and if we choose H_{g_{1}}, H_{g_{2}} as above, then [g_{1}] ∘ [g_{2}] corresponds to ĝ_{2} ∘ ĝ_{1} provided we choose H_{g_{2}g_{1}} such that ĝ_{2}(ĝ_{1}(u)) = uH_{g_{2}g_{1}}(u).

Proof. The equivalence is clear thanks to Corollary 5.4.6 and Lemma 5.5.6. The fact that [g_{1}] ∘ [g_{2}] corresponds to ĝ_{2} ∘ ĝ_{1} is automatic using Corollary 5.4.5 and the functor 𝒢 ↦ ℳ_{π}(𝒢). Applying this to g_{1} = g and g_{2} = g^{−1}, we see that 1_{𝒢} = [g] ∘ [g]^{−1} corresponds to ĝ^{−1} ∘ ĝ. But by Lemma 5.4.8, 1_{𝒢} corresponds to 1_{f} with f defined by (ĝ^{−1} ∘ ĝ)(u) = u(1 + f) in W (k)[[u]]. We see that f = 0 and that 1_{f} is the identity on ℳ_{π}(𝒢). Thus ĝ^{−1} = g^{−1} on ℳ_{π}(𝒢).

5.6. Descent data. Assume now that R is endowed with a continuous left action of a finite group Γ such that Γ becomes the Galois group of the fraction field F′ of R over some subfield. For each g ∈ Γ, choose H_{g}(u) ∈ W (k)[u] so that g(π) = πH_{g}(π), with the one condition that H_{1}(u) = 1. Recall from Lemma 5.5.1 that this uniquely determines elements t_{g}(u) ∈ W (k)[[u]] such that

u^{ℓ}H_{g}(u)^{ℓ} = u^{ℓ}(1 + ℓt_{g}(u))^{(ϕ)}H_{g}(u^{ℓ}(1 + ℓt_{g}(u))).

Moreover for any pair g_{1}, g_{2} ∈ Γ, there is obviously a unique f_{g_{1},g_{2}}(u) ∈ E_{π}(u)W (k)[[u]] such that

ĝ_{1} ∘ ĝ_{2}(u) = { \atop g_{1} ∘ g_{2}}(u(1 + f_{g_{1},g_{2}}(u))).

If ℳ is an object of \underline{ϕ_{1}−mod}_{R}, then we will denote by 1_{g_{1},g_{2}} the unique k-linear map ℳ → ℳ such that for x ∈ ℳ_{1} we have

• 1_{g_{1},g_{2}}(ϕ_{1}(x)) = (1 + (∑_{i=1}^{ℓ−1} \frac{(−1)^{i−1}}{i} f_{g_{1},g_{2}}(u)^{i})N)(ϕ_{1}(x)), where N is as in Lemma 5.1.2;

• and 1_{g_{1},g_{2}}(u^{i}ϕ_{1}(x)) = u^{i}(1 + f_{g_{1},g_{2}}(u))^{i}1_{g_{1},g_{2}}(ϕ_{1}(x)). (See §5.4 where we denoted 1_{g_{1},g_{2}} by 1_{f_{g_{1},g_{2}}} .)

Suppose that 𝒢 is a finite flat R-group scheme. Recall that by descent data on 𝒢 for Γ we mean isomorphisms of finite flat group schemes

[g] : 𝒢 →^{∼} ^{g}𝒢

for g ∈ Γ, such that

[gh] = (^{g}[h]) ∘ [g]

for all g, h ∈ Γ. Equivalently we may think of [g] as a map of schemes 𝒢 → 𝒢 over g^{∗} : Spec R → Spec R which induces an isomorphism of group schemes 𝒢 → ^{g}𝒢. In this picture the compatibility condition simply becomes

[gh] = [h][g].

Theorem 5.6.1. Suppose that 𝒢 is a finite flat R-group scheme killed by ℓ.

1. To give descent data on 𝒢 relative to Γ is equivalent to giving additive bijections ĝ : ℳ_{π}(𝒢) → ℳ_{π}(𝒢)

for all g ∈ Γ so that ĝ takes ℳ_{π}(𝒢)_{1} into ℳ_{π}(𝒢)_{1} and:

• ĝ(wu^{i}m) = g(w)(uH_{g}(u))^{i}ĝ(m) for m ∈ ℳ_{π}(𝒢), w ∈ k,

• ĝ ∘ ϕ_{1} = (1 + t_{g}(u)N) ∘ ϕ_{1} ∘ ĝ on ℳ_{π}(𝒢)_{1},

• ˆ1_{Γ} = 1 and ĝ_{1} ∘ ĝ_{2} = ĝ_{1}g_{2} ∘ 1_{g_{1},g_{2}}.

2. The above equivalence is functorial in 𝒢 and is compatible with classical Dieudonné theory in the following sense: if the action {ĝ}_{g∈Γ} on ℳ_{π}(𝒢) corresponds to descent data {[g]} on 𝒢, then the gsemilinear map D([g]) induced on the contravariant Dieudonné module D(𝒢) and the g-semilinear map ĝ mod u induced on ℳ_{π}(𝒢)/uℳ_{π}(𝒢) are compatible via the isomorphism of Theorem 5.1.3.

Proof. Part (1) is a consequence of Corollary 5.5.7, Lemma 5.4.8 and the choice H_{1} = 1. The functoriality in (2) follows from Corollary 5.4.5, and the last statement there comes from ^{g}(𝒢 × k) ≅ ^{g}𝒢×k, the functoriality of the isomorphism in Theorem 5.1.3, and the reduction modulo u of Corollary 5.4.5.

Suppose that π ∈ R^{Γ}. Then we may take H_{g}(u) = 1 for all g ∈ Γ. With this choice we see that t_{g_{1}} = 0, f_{g_{1},g_{2}} = 0 and 1_{g_{1},g_{2}} = 1 for all g_{1}, g_{2} ∈ Γ. In this case to give bijections ĝ: ℳ_{π}(𝒢) → ℳ_{π}(𝒢) as in the lemma is equivalent to giving an R-semilinear Γ-action on ℳ_{π}(𝒢) which commutes with u and ϕ_{1} and preserves ℳ_{π}(𝒢)_{1}. Thus (ℳ_{π}(𝒢)^{Γ}, ℳ_{π}(𝒢)_{1}^{Γ}, ϕ_{1}) is a Breuil module over R^{Γ} from which we can recover ℳ_{π}(𝒢) by tensoring with W (k) over W (k^{Γ}). In other words, étale descent for group schemes translates in the obvious manner for Breuil modules if we choose π correctly.

To build an action of Γ on 𝒢 using Theorem 5.6.1, the conditions ĝ_{1} ∘ ĝ_{2} = ĝ_{1}g_{2} ∘ 1_{g_{1},g_{2}} are not very convenient to check in practice since there are too many of them. It is useful to have the following variant.

Choose d ∈ Z_{>0} and a group surjection θ : Γ_{d} → Γ, where Γ_{d} is the free group on d generators γ_{1}, ..., γ_{d}. The group Γ_{d} still acts on R (via its quotient Γ) and for each i ∈ {1, ..., d}, choose elements H_{γ_{i}}(u) ∈ W (k)[[u]] such that πH_{γ_{i}}(π) = γ_{i}(π). This determines isomorphisms γ̂_{i} on W (k)[[u]] and k[u]/u^{eℓ} and, by composition, isomorphisms γ̂ for all γ ∈ Γ_{d}. Note that if γ ∈ ker(θ), then H_{γ}(u) = u(1 + f_{γ}(u)) for some f_{γ} ∈ E_{π}(u)W (k)[[u]]. For such γ, denote by 1_{γ} the unique k-linear on any object ℳ of \underline{ϕ_{1}−mod}_{R} such that for x ∈ ℳ_{1} we have

• 1_{γ} = 1 + ( ∑_{i=1}^{ℓ−1} \frac{(−1)^{i−1}}{i} f_{γ}(u)^{i})N on the image of ϕ_{1},

• and 1_{γ}(u^{i}ϕ_{1}(x)) = u^{i}(1 + f_{γ}(u))^{i}1_{γ}(ϕ_{1}(x)), where N is as in Lemma 5.1.2. (See §5.4, where we denoted 1_{γ} by 1_{f_{γ}}.) Let ℛ be a subset of ker(θ) such that ker(θ) is the smallest normal subgroup of Γ_{d} containing ℛ.

Corollary 5.6.2. With the above notation, to give descent data on 𝒢 for Γ is equivalent to giving additive bijections γ̂_{j} : ℳ_{π}(𝒢) → ℳ_{π}(𝒢) for j ∈ {1, ..., d} so that γ̂_{j} takes ℳ_{π}(𝒢)_{1} into ℳ_{π}(𝒢)_{1} and:

• γ̂_{j}(wu^{i}m) = γ_{j}(w)(uH_{γ_{j}}(u))^{i}γ̂_{j}(m) for m ∈ ℳ_{π}(𝒢), w ∈ k,

• γ̂_{j} ∘ ϕ_{1} = (1 + t_{γ_{j}}(u)N) ∘ ϕ_{1} ∘ γ̂_{j} on ℳ_{π}(𝒢)_{1},

• if γ = γ_{i_{1}}^{n_{1}} · · · · · γ_{i_{m}}^{n_{m}} ∈ ℛ, where i_{j} ∈ {1, …, d}, n_{j} ∈ Z, and i_{j} ≠ i_{j+1} for 1 ≤ j < m, and if we define γ̂ = γ_{i_{1}}^{n_{1}} ∘ ⋯ ∘ γ_{i_{m}}^{n_{m}}, then γ̂ = 1_{γ}.

Proof. Straightforward from Corollary 5.5.7 and Lemma 5.4.8.

We define a category \underline{ϕ_{1}DD}_{F′/(F′)^{Γ}} of Breuil modules with descent data for Γ in the obvious way. This category is additive but not necessarily abelian. We call a complex in \underline{ϕ_{1}DD}_{F′/(F′)^{Γ}} exact if the underlying complex in \underline{ϕ_{1}−mod}_{R} is exact. In the natural way, we extend ℳ_{π} to a functor from ℱ𝒟_{F′/(F′)^{Γ}} to \underline{ϕ_{1}DD}_{F′/(F′)^{Γ}}.

5.7. Another example. In this section we will determine the possible descent data on a rank one Breuil module. Let Γ be as in section §5.6.

Lemma 5.7.1. Let Γ_{1} be the subgroup of γ ∈ Γ such that γ(π)/π ≡ 1 mod (π). Suppose that 𝒢 is a finite flat R-group scheme of order ℓ and that its generic fibre admits descent data over (F′)^{Γ}. Then there is unique descent data on 𝒢 over (F′)^{Γ} extending any choice of descent data on 𝒢×F′ over (F′)^{Γ}. If ℳ_{π}(𝒢) ≅ ℳ(r, a) and if γ ∈ Γ_{1} then

γ̂(e) = H_{γ}(u)^{−rℓ/(ℓ−1)}e,

where H_{γ}(u)^{−rℓ/(ℓ−1)} denotes the unique (ℓ − 1)^{th} root of H_{γ}(u)^{−rℓ} in k[u]/u^{eℓ} with constant term 1.

We remark that since Aut(ℳ(r, a)) = (Z/ℓZ)^{×} by consideration of the geometric generic fibre, the choice of isomorphism ℳ_{π}(𝒢) ≅ ℳ(r, a) does not matter.

Proof. We first claim two such finite flat group schemes 𝒢 and 𝒢′ have isomorphic generic fibres if and only if there is a non-zero morphism 𝒢 → 𝒢′ or 𝒢′ → 𝒢. By Lemma 5.2.1 we see that if G is a finite flat F′- group scheme then the lattice of models for G over R is well ordered. Suppose all the integral models are 𝒢_{1} < ... < 𝒢_{n}. For γ ∈ Γ, any isomorphism [γ] : G →^{∼} ^{γ}G must then induce isomorphisms [γ] : 𝒢_{i} →^{∼} ^{γ}𝒢_{i} for all i = 1, ..., n. The first part of the lemma follows.

Let ℳ = ℳ(r, a), so ℳ is a free k[u]/u^{eℓ}-module of rank 1 with the usual basis element e. The submodule ℳ_{1} is spanned by u^{r}e and ϕ_{1}(u^{r}e) = ae. From Theorem 5.2.1, we have N ∘ ϕ_{1} = 0, which implies that

γ̂ ∘ ϕ_{1} = ϕ_{1} ∘ γ̂.

For γ ∈ Γ_{1}, H_{γ}(0) ≡ 1 mod ℓ. Clearly

γ̂ : cu^{i}e ↦ cu^{i}H_{γ}(u)^{i}γ̂(e)

is a bijection if and only if γ̂(e) = ξ_{γ}e for some unit ξ_{γ} ∈ (k[u]/u^{eℓ})^{×}. Evaluating γ̂ ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂ on the element u^{r}e ∈ ℳ_{1}, we get

ξ_{γ} = H_{γ}(u)^{rℓ}ξ_{γ}^{ℓ}

in k[u]/u^{eℓ}. Thus,

ξ_{γ} = ϵ_{γ}H_{γ}^{−rℓ/(ℓ−1)}

for some unit ϵ_{γ} ∈ F_{ℓ}^{×}.

In order to show that ϵ_{γ} = 1, we need the action of γ on ℳ /u ℳ to be the identity. Since Breuil module descent data always induces a k-linear action of the inertia group on the k-vector space ℳ /u ℳ and in this case dim_{k} ℳ /u ℳ = 1, the action of the ℓ-group Γ_{1} on ℳ /u ℳ must be trivial.

6. Some local fields.

In order to apply the methods of §5, we need some more explicit information about the fields F′ introduced in §4. In this section we will collect this essentially elementary information. In each case we will give an explicit description of the Galois group Gal(F′/Q_{3}). This is needed to carry out the delicate Breuil module calculations in subsequent sections. We will also specify a uniformiser π of F′ and partially calculate the following polynomials and power series (depending on our choice of π).

• G(u) ∈ W (k_{F′})[u] a polynomial of degree at most e(F′/Q_{3}) − 1 such that π has minimal polynomial u^{e(F′/Q_{3})} − 3G(u) over Q_{3}.

• c_{π} ≡ −G(u)^{3} mod (3, u^{3e(F′/Q_{3})}).

• For γ ∈ Gal(F′/Q_{3}), the unique polynomial H_{γ}(u) ∈ W (k_{F′})[u] of degree at most e(F′/Q_{3}) − 1 such that γ(π)/π = H_{γ}(π).

• In some cases power series t_{γ} and f_{γ,γ′} as in §5.6.

6.1. The case of F′_{1}. Recall that tau_{1} corresponds to the order 3 homomorphism

Z_{3}^{×} ⟶ GL_{2}(\overline{Q}_{3})

is determined by

\begin{matrix}−1 & ⟼ & 1 \\ 4 & ⟼ & ζ, \end{matrix}

where det ζ = 1 and ζ^{3} = 1 but ζ ≠ 1. Recall also that F′_{1} = F_{1} is any totally ramified cubic Galois extension of Q_{3}. We may take F′_{1} = F_{1} = Q_{3}[π], where π is a root of X^{3} − 3X^{2} + 3. One may check that the other roots of X^{3} − 3X^{2} + 3 are π^{2} − 2π and 3 + π − π^{2}, so Gal(F′_{1}/Q_{3}) is generated by one element γ_{3}, which sends π to π^{2} − 2π and satisfies γ_{3}^{3} = 1. Also, π is a uniformiser for F′_{1}, so

• G(u) = u^{2} − 1,

• c_{π} ≡ 1 − u^{6} mod (3, u^{9}),

• H_{γ_{3}}(u) = u − 2.

6.2. The case of F′_{−1}. Recall that tau_{−1} corresponds to the order 3 homomorphism

Z_{3}[\sqrt{−1}]^{×} ⟶ GL_{2}(\overline{Q}_{3})

determined by

\begin{matrix}\sqrt[4]{−1} & ⟼ & 1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−1} & ⟼ & ζ, \end{matrix}

where det ζ = 1 and ζ^{3} = 1 but ζ ≠ 1. Recall also that F′_{−1}/Q_{3}(\sqrt{−1}) is the unique cubic extension such that F′_{−1}/Q_{3} is Galois but not abelian and that F_{−1} is any cubic subfield. We may take F_{−1} = Q_{3}(π) and F′_{−1} = F_{−1}(\sqrt{−1}), where π is a root of X^{3}−3X^{2}+6. The other roots of X^{3}−3X^{2}+6 are (\sqrt{−1}π^{2}−π+3(1−\sqrt{−1}))/2 and (−\sqrt{−1}π^{2} − π + 3(1 + \sqrt{−1}))/2. Thus, Gal(F′_{−1}/Q_{3}) is generated by two elements γ_{2} and γ_{3} defined by

• γ_{2}(π) = π,

• γ_{2}(\sqrt{−1}) = −\sqrt{−1},

• γ_{3}(π) = (\sqrt{−1}π^{2} − π + 3(1 − \sqrt{−1}))/2

• and γ_{3}(\sqrt{−1}) = \sqrt{−1}.

We have γ_{2}^{2} = γ_{3}^{3} = 1 and γ_{2}γ_{3} = γ_{3}^{2}γ_{2}, and π is a uniformiser for F′_{−1}. Thus • G(u) = u^{2} − 2,

• c_{π} ≡ −1 − u^{6} mod (3, u^{9}),

• H_{γ_{2}}(u) = 1,

• H_{γ_{3}}(u) = ((\sqrt{−1} − 1)u^{2} + (3 − \sqrt{−1})u − 2)/4.

6.3. The case of F′_{3}. Recall that τ_{3} is the unique 3-type such that τ_{3}|_{I_{Q_{3}(\sqrt{3})}} corresponds to the order 6 homomorphism

Z_{3}[\sqrt{3}]^{×} ⟶ GL_{2}(\overline{Q}_{3})

determined by

\begin{matrix}−1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + \sqrt{3} & ⟼ & ζ, \end{matrix}

where det ζ = 1 and ζ^{3} = 1 but ζ ≠ 1. Recall also that F′_{3} is the degree 12 abelian extension of Q_{3}(\sqrt{3}) with norm subgroup in Q_{3}(\sqrt{3})^{×} topologically generated by 3, 4 and 1 + 3\sqrt{3}. We also let γ_{4}^{2} denote the unique element of I_{F′_{3}/Q_{3}(\sqrt{3})} of order 3 and we let F_{3} denote the fixed field of some Frobenius lift of order 2.

We claim that F′_{3} = Q_{3}(\sqrt{3})(\sqrt{−1}, α, β) where α is a root of X^{3}−3X+3 and β a root of X^{2}−\sqrt{3}. To verify this, set F′′ = Q_{3}(\sqrt{3})(\sqrt{−1}, α, β). We must check that F′′/Q_{3}(\sqrt{3}) is abelian and that N_{F′′/Q_{3}(\sqrt{3})}(F′′)^{×} contains 3, 4, and 1+3\sqrt{3}. To see that F′′/Q_{3}(\sqrt{3}) is abelian, note that if α is one root of X^{3}−3X+3 then the other roots are (2\sqrt{3}α^{2}−(−3\sqrt{3}+\sqrt{−5})α−4\sqrt{3})/2\sqrt{−5} and (−2\sqrt{3}α^{2}−(3\sqrt{3}+\sqrt{−5})α+4\sqrt{3})/2\sqrt{−5} (where for definiteness we choose \sqrt{−5} ∈ 1 + 3Z_{3}). Note that N_{F′′/Q_{3}(\sqrt{3})}(α/β) = 3 and N_{F′′/Q_{3}(\sqrt{3})}(1 + α) = 5^{4}.

Note that Gal(F′_{3}/Q_{3}(\sqrt{3})) is generated by three commuting elements γ_{2}, γ_{4}^{2} and γ_{3} of respective orders 2, 2 and 3. They may be defined by

• γ_{2}\sqrt{−1} = −\sqrt{−1}, γ_{2}β = β and γ_{2}α = α;

• γ_{4}^{2}\sqrt{−1} = \sqrt{−1}, γ_{4}^{2}β = −β and γ_{4}^{2}α = α;

• γ_{3}\sqrt{−1} = \sqrt{−1}, γ_{3}β = β and γ_{3}α = (−2\sqrt{3}α^{2} − (3\sqrt{3} + \sqrt{−5})α + 4\sqrt{3})/2\sqrt{−5}. Choose an element γ ∈ I_{F′_{3}/Q_{3}} − I_{F′_{3}/Q_{3}(\sqrt{3})}. Then γ^{2} ∈ ⟨γ_{4}^{2}, γ_{3}⟩. As γγ_{3}γ^{−1} = γ_{3}^{2} we may alter our choice of γ so that γ^{2} ∈ ⟨γ_{4}^{2}⟩. As γ\sqrt{3} = −\sqrt{3} we see that γβ = ±\sqrt{−1}β, so γ^{2} = γ_{4}^{2}. We will rename γ as γ_{4} and suppose it chosen so that γ_{4}β = \sqrt{−1}β. Thus, Gal(F′_{3}/Q_{3}) is generated by elements γ_{2}, γ_{3} and γ_{4} satisfying

• γ_{2}^{2} = γ_{3}^{3} = γ_{4}^{4} = 1,

• γ_{2}γ_{3} = γ_{3}γ_{2},

• γ_{4}γ_{2} = γ_{2}γ_{4},

• and γ_{4}γ_{3} = γ_{3}^{2}γ_{4}. The element γ_{4}^{2} is the unique element of I_{F′_{3}/Q_{3}(\sqrt{3})} of order 2 and hence coincides with our previous definition. The element γ_{2} is a Frobenius lift of order 2 and so we may take F_{3} to be its fixed field, i.e. F_{3} = Q_{3}(π), where π = α/β is a uniformiser for F′_{3}. (We are not asserting that γ_{2} equals the element denoted γ̃_{2} in section 4.) One can check that

γ_{3}(π)/π ≡ 1 + π^{2} mod π^{4}.

Note also that ⟨γ_{2}, γ_{4}⟩ provides a lifting of tame inertia.

We conclude

• G(0) = 1,

• c_{π} ≡ −1 mod (3, u),

• H_{γ_{2}}(u) = 1,

• H_{γ_{4}}(u) = −\sqrt{−1},

• H_{γ_{3}}(u) ≡ 1 + u^{2} mod (3, u^{4}),

• t_{g} = f_{g,g′} = 0 for g, g′ ∈ ⟨γ_{2}, γ_{4}⟩.

6.4. The case of F′_{−3}. Recall that τ_{−3} is the unique 3-type such that τ_{−3}|_{I_{Q_{3}(\sqrt{−3})}} corresponds to the order 6 homomorphism

Z_{3}[\sqrt{−3}]^{×} ⟶ GL_{2}(\overline{Q}_{3})

determined by

\begin{matrix}−1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & 1 \\ 1 + \sqrt{−3} & ⟼ & ζ, \end{matrix}

where det ζ = 1 and ζ^{3} = 1 but ζ ≠ 1. Recall also that F′_{−3} is the degree 12 abelian extension of Q_{3}(\sqrt{−3}) with norm subgroup in Q_{3}(\sqrt{−3})^{×} topologically generated by −3, 4 and 1 + 3\sqrt{−3}. We also let γ_{4}^{2} denote the unique element of I_{F′_{−3}/Q_{3}(\sqrt{−3})} of order 3 and we let F_{−3} denote the fixed field of some Frobenius lift of order 2.

We claim that F′_{−3} = Q_{3}(\sqrt{−3})(\sqrt{−1}, α, β) where α is a root of X^{3} − 4 and β a root of X^{2} + \sqrt{−3}. To verify this, set F′′ = Q_{3}(\sqrt{−3})(\sqrt{−1}, α, β). Then F′′/Q_{3}(\sqrt{−3}) is abelian and so we must check that N_{F′′/Q_{3}(\sqrt{−3})}(F′′)^{×} contains −3, 4, and 1 + 3\sqrt{−3}. But note that we have the identities N_{F′′/Q_{3}(\sqrt{−3})}((α − 1)/β) = −3, N_{F′′/Q_{3}(\sqrt{−3})}(α) = 4^{4} and N_{F′′/Q_{3}(\sqrt{−3})}(1 − β) = (1 + \sqrt{−3})^{6}.

Note that Gal(F′_{3}/Q_{3}(\sqrt{−3})) is generated by three commuting elements γ_{2}, γ_{4}^{2} and γ_{3} of respective orders 2, 2 and 3. They may be defined by

• γ_{2}\sqrt{−1} = −\sqrt{−1}, γ_{2}β = β and γ_{2}α = α;

• γ_{4}^{2}\sqrt{−1} = \sqrt{−1}, γ_{4}^{2}β = −β and γ_{4}^{2}α = α;

• γ_{3}\sqrt{−1} = \sqrt{−1}, γ_{3}β = β and γ_{3}α = (−1 − \sqrt{−3})α/2. Choose an element γ ∈ I_{F′_{−3}/Q_{3}} − I_{F′_{−3}/Q_{3}(\sqrt{−3})}, so γ^{2} ∈ ⟨γ_{4}^{2}, γ_{3}⟩. As γγ_{3}γ^{−1} = γ_{3}^{2}, we may alter our choice of γ so that γ^{2} ∈ ⟨γ_{4}^{2}⟩. As γ\sqrt{−3} = −\sqrt{−3} we see that γβ = ±\sqrt{−1}β, so γ^{2} = γ_{4}^{2}. We will rename γ as γ_{4} and suppose it chosen so that γ_{4}β = \sqrt{−1}β. Thus, Gal(F′_{3}/Q_{3}) is generated by elements γ_{2}, γ_{3} and γ_{4} satisfying

• γ_{2}^{2} = γ_{3}^{3} = γ_{4}^{4} = 1,

• γ_{2}γ_{3} = γ_{3}γ_{2},

• γ_{4}γ_{2} = γ_{2}γ_{4},

• and γ_{4}γ_{3} = γ_{3}^{2}γ_{4}. The element γ_{4}^{2} is the unique element of I_{F′_{−3}/Q_{3}(\sqrt{−3})} of order 2 and hence coincides with our previous definition. The element γ_{2} is a Frobenius lift of order 2 and so we may take F_{−3} to be its fixed field, i.e. F_{−3} = Q_{3}(π), where π = α/β is a uniformiser for F′_{−3}. (We are not asserting that γ_{2} equals the element denoted γ̃_{2} in section 4.) One can check that

γ_{3}(π)/π ≡ 1 + π^{2} mod π^{4}.

Note also that ⟨γ_{2}, γ_{4}⟩ lifts tame inertia.

We conclude

• G(0) = −1,

• c_{π} ≡ 1 mod (3, u),

• H_{γ_{2}}(u) = 1,

• H_{γ_{4}}(u) = −\sqrt{−1},

• H_{γ_{3}}(u) ≡ 1 + u^{2} mod (3, u^{4}),

• t_{g} = f_{g,g′} = 0 for g, g′ ∈ ⟨γ_{2}, γ_{4}⟩.

6.5. The case of F′_{i}. Here i ∈ Z/3Z and we will let ĩdenote the unique lifting of i to Z with 0 ≤ ĩ < 3. Recall that τ′_{i} is the unique extended 3-types whose restrictions to G_{Q_{3}(\sqrt{−3})} correspond to the homomorphism

Q_{3}(\sqrt{−3})^{×} → GL_{2}(\overline{Q}_{3})

\begin{matrix}\sqrt{−3} & ⟼ & ζ − ζ^{−1} \\ −1 & ⟼ & −1 \\ 4 & ⟼ & 1 \\ 1 + 3\sqrt{−3} & ⟼ & ζ \\ 1 + \sqrt{−3} & ⟼ & ζ^{i}, \end{matrix}

where det ζ = 1 and ζ^{3} = 1 but ζ ≠ 1. Recall also that F′_{i} is the degree 12 abelian extension of Q_{3}(\sqrt{−3}) with norms the subgroup of Q_{3}(\sqrt{−3})^{×} topologically generated by −3, 4, 1 + 9\sqrt{−3} and 1 + (1 − 3ĩ)\sqrt{−3}. We let γ_{2}, γ_{3} and γ_{4}^{2} denote the elements of Gal(F′_{i}/Q_{3}) which correspond respectively to \sqrt{−3}, 1 − 3\sqrt{−3} and −1.

We claim that F′_{i} = Q_{3}(\sqrt{−3})(\sqrt{−1}, α, β) where α is a root of X^{3} − 3(1 + 3ĩ) and β a root of X^{2} + \sqrt{−3}. To verify this, set F′′ = Q_{3}(\sqrt{−3})(\sqrt{−1}, α, β), so F′′/Q_{3}(\sqrt{−3}) is abelian and we must check that N_{F′′/Q_{3}(\sqrt{−3})}(F′′)^{×} contains −3, 4, 1 + 9\sqrt{−3}, and 1 + (1 − 3ĩ)\sqrt{−3}. But note that N_{F′′/Q_{3}(\sqrt{−3})}(α/β) = −3(1 + 3ĩ)^{4}, N_{F′′/Q_{3}(\sqrt{−3})}(1 + α) = (4 + 9ĩ)^{4} and

N_{F′′/Q_{3}(\sqrt{−3})}(β(\sqrt{−3} − α)/α) = (1 + \sqrt{−3} + 3ĩ)/(1 + 3ĩ)^{4} ≡ 1 + (1 − 3ĩ)\sqrt{−3} mod 9.

Note that γ_{4}^{2} is an element of I_{F′_{i}/Q_{3}(\sqrt{−3})} of order 2, γ_{2} ≠ γ_{4}^{2} but also has order 2, and γ_{3} is an element of I_{F′_{i}/Q_{3}(\sqrt{−3})} of order 3. Thus,

• γ_{4}^{2}\sqrt{−1} = \sqrt{−1}, γ_{4}^{2}β = −β and γ_{4}^{2}α = α;

• γ_{2}\sqrt{−1} = −\sqrt{−1} and γ_{2}α = α;

• γ_{3}\sqrt{−1} = \sqrt{−1} and γ_{3}β = β. Moreover \sqrt{−3} is a norm from Q_{3}(\sqrt{−3})(α, β), because α/β has norm \sqrt{−3}(1 + 3ĩ)^{2}, so

• γ_{2}(β) = β.

The determination of γ_{3}(α) is more delicate. Let δ be a root of X^{3} − (1 + 3\sqrt{−3}), so δ = 1 + \sqrt{−3}μ where μ is a root of Y^{3} − \sqrt{−3}Y^{2} − Y + 1. Thus Q_{3}(\sqrt{−3})(δ)/Q_{3}(\sqrt{−3}) is unramified and

Frob_{3}(δ)/δ ≡ (1 + \sqrt{−3}μ^{3})/(1 + \sqrt{−3}μ) ≡ (−1 + \sqrt{−3})/2 mod 3.

The norms from Q_{3}(\sqrt{−3})(δ)^{×} to Q_{3}(\sqrt{−3})^{×} are generated by Z_{3}[\sqrt{−3}]^{×} and 3\sqrt{−3}. The norms from Q_{3}(\sqrt{−3})(α)^{×} to Q_{3}(\sqrt{−3})^{×} are generated by 1 + 9Z_{3}[\sqrt{−3}], 1 + (1 − 3ĩ)\sqrt{−3}, 4, −1 and \sqrt{−3}. The norms from Q_{3}(\sqrt{−3})(α, δ)^{×} to Q_{3}(\sqrt{−3})^{×} are generated by 1 + 9Z_{3}[\sqrt{−3}], 1 + (1 − 3ĩ)\sqrt{−3}, 4, −1 and 3\sqrt{−3}. Thus

(γ_{3}, Frob_{3}) ∈ Gal(Q_{3}(\sqrt{−3})(α)/Q_{3}(\sqrt{−3})) × Gal(Q_{3}(\sqrt{−3})(δ)/Q_{3}(\sqrt{−3}))
≅ Gal(Q_{3}(\sqrt{−3})(α, δ)/Q_{3}(\sqrt{−3}))

corresponds to \sqrt{−3}(1−3\sqrt{−3}) ∈ Q_{3}(\sqrt{−3})^{×}. As δα has norm to Q_{3}(\sqrt{−3}) the product of (\sqrt{−3}(1−3\sqrt{−3}))^{2} and −(1+3ĩ)(1+3\sqrt{−3})/(1−3\sqrt{−3})^{2}, we conclude that (γ_{3}, Frob_{3}) fixes δα. Thus γ_{3}(α)/α = δ/ Frob_{3}(δ) = (−1 − \sqrt{−3})/2. In other words

• γ_{3}(α) = (−1 − \sqrt{−3})α/2.

Choose an element γ ∈ I_{F′_{i}/Q_{3}} − I_{F′_{i}/Q_{3}(\sqrt{−3})}. Then γ^{2} ∈ ⟨γ_{4}^{2}, γ_{3}⟩. As γγ_{3}γ^{−1} = γ_{3}^{2} we may alter our choice of γ so that γ^{2} ∈ ⟨γ_{4}^{2}⟩. As γ\sqrt{−3} = −\sqrt{−3} we see that γβ = ±\sqrt{−1}β and so γ^{2} = γ_{4}^{2}. We will rename γ as γ_{4} and suppose it chosen so that γ_{4}β = \sqrt{−1}β. Thus, Gal(F′_{i}/Q_{3}) is generated by elements γ_{2}, γ_{3} and γ_{4} satisfying

• γ_{2}^{2} = γ_{3}^{3} = γ_{4}^{4} = 1,

• γ_{2}γ_{3} = γ_{3}γ_{2},

• γ_{4}γ_{2} = γ_{2}γ_{4},

• and γ_{4}γ_{3} = γ_{3}^{2}γ_{4}.

The element γ_{2} is a Frobenius lift and it has fixed field F_{i} = Q_{3}(π), where π = α/β is a uniformiser for F′_{i}. One can check that

γ_{3}^{±1}(π)/π = −(1 ∓ (1 + 3ĩ)^{−2}π^{6})/2.

We conclude

• G(u) = −(1 + 3ĩ)^{4},

• c_{π} ≡ 1 mod (3, u^{36}),

• H_{γ_{2}}(u) = 1,

• H_{γ_{4}}(u) = −\sqrt{−1},

• H_{γ_{3}^{±1}}(u) ≡ 1 ∓ u^{6} mod 3,

• t_{γ_{3}^{±1}}(u) ≡ −1 ∓ u^{6} mod (3, u^{12}),

• t_{g} = f_{g,g′} = 0 for g, g′ ∈ ⟨γ_{2}, γ_{4}⟩,

• f_{γ_{3}^{±1},γ_{3}^{±1}}(u), f_{γ_{3}^{±1},γ_{3}^{∓1}}(u) ≡ 0 mod (3, u^{12}).

7. Proof of Theorem 4.4.1.

In this section we will keep the notation of §4.4 and either §6.1 or §6.2 (depending if we are working with 𝒮_{1} or 𝒮_{−1}). We will set δ = ±1 in the case of 𝒮_{±1}. We will write F for F_{±1} and F′ for F′_{±1}. If 𝒢 (resp. ℳ) is a finite flat 𝒪_{F} -group scheme (resp. Breuil module over 𝒪_{F}) we will write 𝒢′ (resp. ℳ′) for the unramified base change to 𝒪_{F′}.

7.1. Rank one calculations. We recall from Lemma 5.2.1 that the only 𝒪_{F} -models for (Z/3Z)_{/F} are 𝒢(3, δ) ≅ (Z/3Z)_{/𝒪_{F}} and 𝒢(1, δ), and the only 𝒪_{F} -models for (μ_{3})_{/F} are 𝒢(0, 1) ≅ (μ_{3})_{/𝒪_{F}} and 𝒢(2, 1). In each case, by Lemma 5.7.1, the base change to 𝒪_{F′} admits unique descent data over Q_{3} compatible with the canonical descent data on the generic fibre of Z/3Z (resp. μ_{3}) over Q_{3}. We will refer to this descent data as the standard descent data on these finite flat group schemes.

7.2. Rank two calculations.

Lemma 7.2.1. The group of extensions of ℳ(2, 1) by ℳ(1, δ) over 𝒪_{F} is parametrised by c ∈ F_{3}. The Breuil module ℳ(1, δ; 2, 1; c) corresponding to c is free of rank two over F_{3}[u]/u^{9} with a basis {e_{1}, e_{ω}} such that

• ℳ_{1} = ⟨ue_{1}, u^{2}e_{ω} + ce_{1}⟩,

• ϕ_{1}(ue_{1}) = δe_{1}, ϕ_{1}(u^{2}e_{ω} + ce_{1}) = e_{ω},

• N(e_{1}) = 0, N(e_{ω}) = cu^{6}e_{1}. The standard descent data on ℳ(2, 1)′ and ℳ(1, δ)′ extends uniquely to descent data on ℳ(1, δ; 2, 1; c)′. The corresponding representations G_{3} → GL_{2}(F_{3}) are of the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

and are peu ramifié. Any such peu-ramifié extension arises for a suitable choice of c.

Proof. The classification of extensions of Breuil modules follows from Lemma 5.2.2. Next, we compute N on ℳ = ℳ(1, δ; 2, 1; c). By the last part of Lemma 5.2.1, N(e_{1}) = 0 and N(e_{ω}) = ge_{1} for some g ∈ F_{3}[u]/u^{eℓ} divisible by u. In F_{3}[u]/u^{9} we compute

c_{π} = −ϕ(G_{π}(u)) = −(u^{2} − δ)^{3} = −u^{6} + δ,

so

\frac{δu^{6}}{c_{π}} = u^{6}.

Using the defining properties of N, we compute in F_{3}[u]/u^{9}

N(e_{ω}) = N ∘ ϕ_{1}(u^{2}e_{ω} + ce_{1})
= ϕ ∘ N(u^{2}e_{ω} + ce_{1})
= ϕ(−u^{2}e_{ω} + u^{2}N(e_{ω}))
= \frac{ϕ_{1}}{c_{π}} (−u^{5}e_{ω} + u^{5}N(e_{ω}))
= \frac{ϕ_{1}}{c_{π}} (−u^{3}(u^{2}e_{ω} + ce_{1}) + cu^{3}e_{1} + u^{5}Ne_{ω}))
= \frac{ϕ_{1}}{c_{π}} (cu^{3}e_{1})

since u^{5}N(e_{ω}) ∈ u^{6} ℳ = u^{3} · u^{3}ℳ ⊆ u^{3}ℳ_{1} and the Frobenius-semilinear ϕ_{1} must kill u^{3} ℳ_{1}. Thus,

N(e_{ω}) = \frac{ϕ_{1}}{c_{π}} (cu^{2} · ue_{1}) = \frac{cu^{6}}{c_{π}} ϕ_{1}(ue_{1}) = \frac{cδu^{6}}{c_{π}} e_{1} = cu^{6}e_{1}.

To see existence and uniqueness of the descent data on ℳ(1, δ; 2, 1; c)′ compatible with the standard descent data on ℳ(1, δ)′ and ℳ(2, 1)′ we will work on the side of finite flat group schemes. Because 𝒢(1, δ; 2, 1; c)′ is the unique extension of 𝒢(1, δ)′ by 𝒢(2, 1)′ with generic fibre 𝒢(1, δ; 2, 1; c)′×F′ (by Lemma 4.1.2), uniqueness reduces to the corresponding questions on the generic fibre, which follows from the injectivity of

H^{1}(G_{3}, ω) ⟶ H^{1}(G_{F′}, ω).

For existence it suffices to exhibit a continuous representation G_{3} → GL_{2}(F_{3}) of the form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} )

which is peu ramifié but not split, with restriction to G_{F} corresponding to a local-local finite flat 𝒪_{F} -group scheme 𝒢. By Theorem 5.3.2 of [M] we can find an elliptic curve E_{/Q_{3}} such that E[3] furnishes the desired example. This also proves the final two assertions of the lemma.

Lemma 7.2.2. Suppose that F̃_{1} is a totally ramified abelian cubic extension of Q_{3} and suppose that 𝒢 is a local-local finite flat 𝒪F̃_{1}-group scheme killed by 3 such that 𝒢 × F̃_{1} is an extension of Z/3Z by μ_{3}. Then 𝒢 ×_{𝒪F̃_{1}} F̃_{1} ≅ G ×_{Q_{3}} F̃_{1} for some finite flat Q_{3}-group scheme G.

Proof. As in the proof of the last lemma we see that ℳ_{π}(𝒢) ≅ ℳ(1, 1; 2, 1; c) for some c ∈ F_{3}. As the only action of Gal(F̃_{1}/Q_{3}) on a one dimensional F_{3}-vector space is trivial, we see that each such c gives a class in H^{1}(GF̃_{1}, ω) which is invariant by Gal(F̃_{1}/Q_{3}). But

H^{1}(G_{3}, ω) ⟶^{∼} H^{1}(GF̃_{1}, ω)^{Gal(F̃_{1}/Q_{3})},

and so the lemma follows.

Lemma 7.2.3. The group of extensions of ℳ(1, δ) by ℳ(2, 1) over 𝒪_{F} is isomorphic to the group of linear polynomials c + c′u in F_{3}[u]. The Breuil module ℳ(2, 1; 1, δ; c + c′u) corresponding to c + c′u is free of rank two over F_{3}[u]/u^{9} with a basis {e_{ω}, e_{1}} such that

• ℳ(2, 1; 1, δ; c + c′u)_{1} = ⟨u^{2}e_{ω}, ue_{1} + (c + c′u)e_{ω}⟩,

• ϕ_{1}(u^{2}e_{ω}) = e_{ω}, ϕ_{1}(ue_{1} + (c + c′t)e_{ω}) = δe_{1}, Each ℳ(2, 1; 1, δ; c + c′u)′ admits unique descent data compatible with the standard descent data on ℳ(1, δ)′ and ℳ(2, 1)′. As c, c′ vary over F_{3} the corresponding descent to Q_{3} of the generic fibre of 𝒢_{π}(ℳ(2, 1; 1, δ; c+ c′u)′) runs over all 9 extensions of μ_{3} by Z/3Z. The corresponding representation of G_{3} is peu ramifié if and only if c = 0.

Proof. The classification of extensions of Breuil modules follows from Lemma 5.2.2. The uniqueness of the descent data on ℳ(2, 1; 1, δ; c + c′u)′ follows from Lemma 4.1.2 and the injectivity of H^{1}(G_{3}, ω^{−1}) ⟶ H^{1}(G_{F′}, ω^{−1}) as in the proof of Lemma 7.2.1. Note that Frobenius vanishes on the Dieudonné module of 𝒢(2, 1; 1, δ; c + c′u) if and only if c = 0. Thus the lemma will follow if for each 3-torsion extension G of μ_{3} by Z/3Z over Q_{3} which is très ramifié, we can find a finite flat 𝒪_{F} -group scheme 𝒢 such that

• the generic fibre of 𝒢 is isomorphic to G × F,

• the closed fibre of 𝒢 is local-local

• and Frobenius is not identically zero on D(𝒢).

The splitting field of G contains a cube root of 3v for some v ≡ 1 mod 3, where the three choices of v mod 9 correspond to the three choices of très ramifié \overline{ρ}. The calculations in §5.3 of [M] give explicit additive reduction elliptic curves E and E′ over Q_{3} with E[3] ≃ E′[3] ≃ G, where E acquires good supersingular reduction over the non-Galois cubic ramified extension

Q_{3}[X]/(X^{3} − 3X + 2b),

with b^{2} = 1 + 3v, and E′ acquires good supersingular reduction over the abelian cubic ramified extension of Q_{3} with norm group generated by 3v mod (Q_{3}^{×})^{3}. The appropriate 𝒢 are provided by the 3-torsion on the Néron models of E or E′ over 𝒪_{F}.

Corollary 7.2.4. Suppose that 𝒢 is a finite flat 𝒪_{F} -group scheme and that {[g]} is descent data on 𝒢′ = 𝒢 × 𝒪_{F′} such that (𝒢′, \{[g]\})_{Q_{3}}(\overline{Q}_{3}) corresponds to \overline{ρ}. Then

ℳ_{π}(𝒢) ≅ ℳ(2, 1; 1, δ; c + c′u)

for some c, c′ ∈ F_{3} with c ≠ 0.

Proof. From the connected-étale exact sequence and its dual we see that 𝒢 × F_{3} must be local-local. The corollary now follows from Lemma 7.2.3 and the discussion of §7.1.

Lemma 7.2.5. The group of extensions of ℳ(1, δ) by ℳ(1, δ) over 𝒪_{F} is isomorphic to the group of linear polynomials b + b′u in F_{3}[u]. The Breuil module ℳ(1, δ; 1, δ; b + b′u) corresponding to b + b′u is free rank two over F_{3}[u]/u^{9} with a basis {e, e′} such that

• ℳ(1, δ; 1, δ; b + b′u)_{1} = ⟨ue, ue′ + (b + b′u)e⟩,

• ϕ_{1}(ue) = δe, ϕ_{1}(ue′ + (b + b′u)e) = δe′, This extension splits over an unramified extension if and only if b = 0. If F′/Q_{3} is non-abelian, then any descent data on ℳ(1, −1; 1, −1; b + b′u)′ compatible with the standard descent data on ℳ(1, −1)′ satisfies

γ̂_{3}^{±1}(e) = H_{γ_{3}^{±1}}(u)^{3}e, γ̂_{3}^{±1}(e′) = H_{γ_{3}^{±1}}(u)^{3}e′ + h_{γ_{3}^{±1}}(u)e

where

h_{γ_{3}^{±1}}(0) = −bH′_{γ_{3}^{±1}}(0).

Proof. The classification of extensions of Breuil modules follows from Lemma 5.2.2. The computation of which of these split over an unramified extension follows from Lemma 5.2.2 and Corollary 5.4.2.

Now suppose that F′/Q_{3} is non-abelian. By Lemma 5.7.1, the only issue is to compute h_{γ_{3}}(0). Since H_{γ_{3}}(0) ≡ 1 mod 3, by evaluating the congruence

γ̂_{3} ∘ ϕ′_{1} ≡ ϕ′_{1} ∘ γ̂_{3} mod u ℳ(1, −1; 1, −1; b + b′u)′

on ue′ + (b + b′u)e and comparing constant terms of the coefficients of e on both sides we get

h_{γ_{3}}(0) = h_{γ_{3}}(0)^{3} + b (\frac{1 − H_{γ_{3}}(u)}{u})^{3} |_{u=0} = h_{γ_{3}}(0)^{3} − bH′_{γ_{3}}(0)^{3} = h_{γ_{3}}(0)^{3} + bH′_{γ_{3}}(0)

in F_{9}, where we have used the equality H′_{γ_{3}}(0)^{2} ≡ −1 mod 3 (see §6.2).

In other words h_{γ_{3}}(0) is a root of T^{3} − T + bH′_{γ_{3}}(0) = 0. Since H′_{γ_{3}}(0)^{2} = −1, we must have h_{γ_{3}}(0) = −bH′_{γ_{3}}(0) + a for some a ∈ F_{3}. Since γ_{2}(H_{γ_{3}}(u)) = H_{γ_{3}^{−1}}(u) and γ_{2}(h_{γ_{3}}(u)) = h_{γ_{3}^{−1}}(u) are forced by the identity γ_{2}(π) = π we see that h_{γ_{3}^{−1}}(0) = −bH′_{γ_{3}^{−1}}(0) + a for the same a ∈ F_{3}. The identity

γ̂_{3} ∘ γ̂_{3}^{−1} ∘ ϕ′_{1} ≡ ϕ′_{1} mod u ℳ(1, −1; 1, −1; b + b′u)′

then implies h_{γ_{3}}(0) + h_{γ_{3}^{−1}}(0) = 0, so a = 0.

Lemma 7.2.6. The group of extensions of ℳ(2, 1) by ℳ(2, 1) over 𝒪_{F} is isomorphic to the group of quadratic polynomials vanishing at 0, (b + b′u)u, in F_{3}[u]. The Breuil module ℳ(2, 1; 2, 1; (b + b′u)u) corresponding to (b + b′u)u is free rank two over F_{3}[u]/u^{9} with a basis {e, e′} such that

• ℳ(2, 1; 2, 1; (b + b′u)u)_{1} = ⟨u^{2}e, u^{2}e′ + (b + b′u)ue⟩,

• ϕ_{1}(u^{2}e) = e, ϕ_{1}(u^{2}e′ + (b + b′u)ue) = e′, This extension splits over an unramified extension if and only if b = 0. If F′/Q_{3} is non-abelian, then any descent data on ℳ(2, 1; 2, 1; (b + b′u)u)′ compatible with the standard descent data on ℳ(2, 1)′ satisfies

γ̂_{3}(e) = H_{γ_{3}}(u)^{6}e, γ̂_{3}(e′) = H_{γ_{3}}(u)^{6}e′ + h_{γ_{3}}(u)e

where

h_{γ_{3}}(0) = −bH′_{γ_{3}}(0).

The sign in h_{γ_{3}}(0) = −bH′_{γ_{3}}(0) will be very important in §7.4. The proof of this lemma is essentially the same as that of Lemma 7.2.5, but we repeat it anyway.

Proof. The classification of extensions of Breuil modules follows from Lemma 5.2.2. The computation of which of these split over an unramified extension follows from Lemma 5.2.2 and Corollary 5.4.2.

Now suppose that F′/Q_{3} is non-abelian. By Lemma 5.7.1, the only issue is to compute h_{γ_{3}}(0). Since H_{γ_{3}}(0) ≡ 1 mod 3, by evaluating the congruence

γ̂_{3} ∘ ϕ′_{1} ≡ ϕ′_{1} ∘ γ̂_{3} mod u ℳ(2, 1; 2, 1; (b + b′u)u)′

on ue′ + (b + b′u)e and comparing constant terms of the coefficients of e on both sides we get

h_{γ_{3}}(0) = h_{γ_{3}}(0)^{3} + b (\frac{1 − H_{γ_{3}}(u)}{u})^{3} |_{u=0} = h_{γ_{3}}(0)^{3} − bH′_{γ_{3}}(0)^{3} = h_{γ_{3}}(0)^{3} + bH′_{γ_{3}}(0)

in F_{9}, where we have used the equality H′_{γ_{3}}(0)^{2} = −1 (see §6.2).

In other words h_{γ}(0) is a root of T^{3} − T + bH′_{γ_{3}}(0) = 0. Since H′_{γ_{3}}(0)^{2} = −1, we must have h_{γ_{3}}(0) = −bH′_{γ_{3}}(0) + a for some a ∈ F_{3}. Since γ_{2}(H_{γ_{3}}(u)) = H_{γ_{3}^{−1}}(u) and τ(h_{γ_{3}}(u)) = h_{γ_{3}^{−1}}(u) are forced by the identity τ(π) = π we see that h_{γ_{3}^{−1}}(0) = −bH′_{γ_{3}}(0) + a for the same a ∈ F_{3}. The identity

γ̂_{3} ∘ γ̂_{3}^{−1} ∘ ϕ′_{1} ≡ ϕ′_{1} mod u ℳ(2, 1; 2, 1; b + b′u)′

then implies h_{γ}(0) + h_{γ^{−1}}(0) = 0, so a = 0.

7.3. Rank three calculations.

Lemma 7.3.1. Suppose that 𝒢 is a finite flat group scheme over 𝒪_{F} which is killed by 3. Suppose that there is filtration by closed finite flat subgroupschemes 𝒢_{1} ⊂ 𝒢_{2} ⊂ 𝒢 such that 𝒢_{1} ≅ 𝒢(1, δ), 𝒢_{2}/𝒢_{1} ≅ 𝒢(2, 1) and 𝒢/𝒢_{2} ≅ 𝒢(1, δ). Suppose finally that 𝒢_{2} ×_{𝒪_{F}} F′ descends to Q_{3} in such a way that it is a très ramifié extension of μ_{3} by Z/3Z. Then

𝒢/𝒢_{1} ≅ 𝒢(2, 1) ⊕ 𝒢(1, δ)

compatibly with the extension class structure.

Proof. Let ℳ = ℳ_{π}(𝒢) and 𝒩 = ℳ_{π}(𝒢/𝒢_{1}). Using Lemmas 7.2.1 and 7.2.3 we see that we can write • ℳ = (F_{3}[u]/u^{9})e_{1} ⊕ (F_{3}[u]/u^{9})e_{ω} ⊕ (F_{3}[u]/u^{9})e′_{1},

• ℳ_{1} = ⟨ue_{1}, u^{2}e_{ω} + be_{1}, ue′_{1} + (c + c′u)e_{ω} + fe_{1}⟩ for b, c, c′ ∈ F_{3} with c ≠ 0 and with f ∈ F_{3}[u]/u^{9}. It suffices to show b = 0. Since we must have u^{3} ℳ ⊆ ℳ_{1}, we see that

(c + c′u)(u^{2}e_{ω} + be_{1}) − u^{2}(ue′_{1} + (c + c′u)e_{ω} + fe_{1}) + u^{3}e′_{1} = (bc + bc′u − u^{2}f)e_{1} ∈ ℳ_{1} .

The Breuil module 𝒩 is spanned as a F_{3}[u]/u^{9}-module by e_{1} and e_{ω}, so by Lemma 7.2.1 u must divide bc + bc′u − u^{2}f. As c ≠ 0 we must have b = 0, as desired.

Combining this with Lemma 7.2.1 and the injectivity of H^{1}(G_{3}, ω) → H^{1}(G_{F′}, ω) we get the following corollary, which is also the first part of Theorem 4.7.3.

Corollary 7.3.2. The natural map

θ_{0} : Ext_{𝒮_{±1}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω)

is zero.

7.4. Conclusion of proof of Theorem 4.4.1. Consider first the case of F_{1}. We still have to explain why

\overline{θ}_{ω} : H_{𝒮_{1}}^{1}(G_{3}, ad^{0} \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3})

is zero. Suppose x ∈ H_{𝒮_{1}}^{1}(G_{3}, ad^{0} \overline{ρ}) does not map to zero in H^{1}(I_{3}, F_{3}).

By our hypothesis on x we may choose a totally ramified abelian cubic extension F̃_{1}/Q_{3} such that x restricts to zero under the natural map H^{1}(G_{3}, ad^{0} \overline{ρ}) → H^{1}(GF̃_{1}, F_{3}). Then the image of x under the natural map H^{1}(G_{3}, ad^{0} \overline{ρ}) → H^{1}(GF̃_{1}, \overline{ρ} ⊗ ω) is the image of some x̃ ∈ H^{1}(GF̃_{1}, ω) under the natural map H^{1}(GF̃_{1}, ω) → H^{1}(GF̃_{1}, \overline{ρ} ⊗ ω). The element x̃ parametrises a finite flat F̃_{1}-group scheme H which is an extension of μ_{3} by Z/3Z and which is a subquotient of the restriction to GF̃_{1} of the extension of \overline{ρ} by itself paramitrised by x. It follows that H has a finite flat model ℋ_{/𝒪F̃_{1}} (see Lemma 4.1.1) and the special fibre of

ℋ must be local-local (if x̃ = 0 then the extension of \overline{ρ} by itself parametrised by x splits over F̃_{1} and this is clear, while if x̃ ≠ 0 we would otherwise get a contradiction from the connected-étale sequence). By Lemma 7.2.2, we may therefore lift x̃ to H^{1}(G_{3}, ω). Using the commutative diagram

H^{1}(G_{3}, ω) ⟶ H^{1}(G_{3}, \overline{ρ} ⊗ ω)
res ↓  ↓ res
H^{1}(GF̃_{1}, ω) ⟶ H^{1}(GF̃_{1}, \overline{ρ} ⊗ ω)

and noting that the right hand vertical map is injective we conclude that

x ∈ H_{𝒮_{1}}^{1}(G_{3}, ad^{0} \overline{ρ}) ⊂ H^{1}(G_{3}, \overline{ρ} ⊗ ω)

is in the image of H^{1}(G_{3}, ω) → H^{1}(G_{3}, \overline{ρ} ⊗ ω), a contradiction with the hypothesis that even the image of x in H^{1}(I_{3}, F_{3}) is non-zero.

Now consider the case F′ = F′_{−1} which is nonabelian over Q_{3}. We must show that

\overline{θ}_{ω} : Ext_{𝒮_{−1}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3})

is zero.

An element x ∈ Ext_{𝒮_{−1}}^{1}(\overline{ρ}, \overline{ρ}) gives rise to a finite flat 𝒪_{F_{−1}}-group scheme 𝒢 killed by 3 and descent data \{[g]\} for F′_{−1}/Q_{3} on 𝒢′ = 𝒢 ×_{𝒪_{F{′ \atop −1}}} F′_{−1}, such that (𝒢′, \{[g]\})_{Q_{3}} corresponds to the extension of \overline{ρ} by itself

classified by x. Let 𝒩 denote the Breuil module for 𝒢 and let 𝒩′ = 𝒩 ⊗ F_{9}. According to Lemmas 7.2.1, 7.2.3, 7.2.5, 7.2.6 and 7.3.1 we see that we can write

𝒩 = (F_{3}[u]/u^{9})e_{ω} ⊕ (F_{3}[u]/u^{9})e_{1} ⊕ (F_{3}[u]/u^{9})e′_{ω} ⊕ (F_{3}[u]/u^{9})e′_{1}

with

(7.4.1) 𝒩_{1} = ⟨u^{2}e_{ω}, ue_{1} + (c + c′u)e_{ω}, u^{2}e′_{ω} + (au + a′u^{2})e_{ω}, ue′_{1} + (c + c′u)e′_{ω} + (b + b′u)e_{1} + he_{ω}⟩

where h ∈ F_{3}[u]/u^{9} is some polynomial and where a, a′, b, b′, c, c′ ∈ F_{3} with c ≠ 0 (as \overline{ρ} is très ramifié). By Lemma 7.2.6 what we must show is that a = 0.

Note that H′_{γ_{3}}(0) ≠ 0 in F_{9} by §6.2. By Lemmas 5.7.1 and 7.2.1, the action γ̂_{3} is determined by

γ̂_{3}(e_{ω}) = H_{γ_{3}}(u)^{6}e_{ω}, γ̂_{3}(e_{1}) = H_{γ_{3}}(u)^{3}e_{1} + g_{γ_{3}}(u)e′_{ω}, γ̂_{3}(e′_{ω}) = H_{γ_{3}}(u)^{6}e′_{ω} + h_{γ_{3},ω}(u)e_{ω},
γ̂_{3}(e′_{ω}) = H_{γ_{3}}(u)^{3}e′_{1} + g_{γ_{3}}(u)e′_{ω} + h_{γ_{3},1}e_{1} + G_{γ_{3}}(u)e_{ω},

where g_{γ_{3}}(u), G_{γ_{3}}(u) ∈ F_{9}[u]/u^{9} and h_{γ_{3},ω} and h_{γ_{3},1} are as in Lemmas 7.2.6 and 7.2.5 respectively. Due to the requirement γ̂_{3}(𝒩′_{1}) ⊆ 𝒩′_{1}, we must have

γ̂_{3}(ue′_{1} + (c + c′u)e′_{ω} + (b + b′u)e_{1} + h(u)e_{ω}) ∈ 𝒩′_{1},

and this element is obviously equal to

(uH_{γ_{3}})(H_{γ_{3}}^{3}e′_{1} + g_{γ_{3}}e′_{ω} + h_{γ_{3},1}e_{1} + G_{γ_{3}}e_{ω}) + (c + c′uH_{γ_{3}})(H_{γ_{3}}^{6}e′_{ω} + h_{γ_{3},ω}e_{ω})+
+(b + b′uH_{γ_{3}})(H_{γ_{3}}^{3}e_{1} + g_{γ_{3}}e_{ω}) + h(uH_{γ_{3}})H_{γ_{3}}^{6}e_{ω}.

We now try to express this as a linear combination of the generators of 𝒩′_{1} listed in (7.4.1), while working modulo ⟨u^{3} 𝒩′, u^{2}e_{ω}⟩ ⊆ 𝒩′_{1}. Using that H_{γ_{3}}(0) = 1 in F_{9} and h(uH_{γ_{3}}) ≡ h(u) mod u^{2}, we arrive at the expression

H_{γ_{3}}(ue′_{1} + (c + c′u)e′_{ω} + (b + b′u)e_{1} + he_{ω}) + (\frac{c((1 − H_{γ_{3}})/u) + g_{γ_{3}}}{u} ) (u^{2}e′_{ω} + (au + a′u^{2})e_{ω})

+ (H_{γ_{3}}h_{γ_{3},1} + b (\frac{1 − H_{γ_{3}}}{u} )) (ue_{1} + (c + c′u)e_{ω}) + F_{γ_{3}}(u)e_{ω},

where

F_{γ_{3}}(u) = uH_{γ_{3}}G_{γ_{3}} + (c + c′uH_{γ_{3}})h_{γ_{3},ω} + (b + b′uH_{γ_{3}})g_{γ_{3}} + h(u)(1 − H_{γ_{3}})
−(a + a′u)(c(1 − H_{γ_{3}})/u + g_{γ_{3}}) − (c + c′u)(H_{γ_{3}}h_{γ_{3},1} + b((1 − H_{γ_{3}})/u))

in F_{9}[u]/u^{9}. In particular, c(1 − H_{γ_{3}}(u))/u + g_{γ_{3}} ≡ 0 mod u and F_{γ_{3}}(u) ≡ 0 mod u^{2}. The condition c((1 − H_{γ_{3}})/u) + g_{γ_{3}} ≡ 0 mod u can be reformulated as

g_{γ_{3}}(0) = cH′_{γ_{3}}(0).

Since F_{γ_{3}}(u) ≡ 0 mod u^{2}, we have to have F_{γ_{3}}(0) = 0. But a direct calculation using g_{γ_{3}}(0) = cH′_{γ_{3}}(0) and the definition of F_{γ_{3}} gives

F_{γ_{3}}(0) = 0 + ch_{γ_{3},ω}(0) + bg_{γ_{3}}(0) + 0 − 0 − c(h_{γ_{3},1}(0) − bH′_{γ_{3}}(0)) = c(h_{γ_{3},ω}(0) − h_{γ_{3},1}(0) − bH′_{γ_{3}}(0)),

so the non-vanishing of c forces

h_{γ_{3},ω}(0) − h_{γ_{3},1}(0) = bH′_{γ_{3}}(0).

Lemmas 7.2.6 and 7.2.5 give us the values

h_{γ_{3},ω}(0) = −aH′_{γ_{3}}(0), h_{γ_{3},1}(0) = −bH′_{γ_{3}}(0).

Thus (−a + b)H′_{γ_{3}}(0) = bH′_{γ_{3}}(0), and so a = 0. This completes the proof of Theorem 4.7.3 and hence of Theorem 4.4.1.

8. Proof of Theorem 4.5.1.

In this section we will keep the notation of §4.5 and either §6.3 or §6.4 (depending if we are working with 𝒮_{3} or 𝒮_{−3}). We will set δ = ±1 in the case of 𝒮_{∓3} (so that c_{π} ≡ δ mod (3, u). Note the signs. We will write F for F_{±3}, F′ for F′_{±3} and ℐ for ℐ_{±3}. If 𝒢 (resp. ℳ) is a finite flat 𝒪_{F} -group scheme (resp. Breuil module over 𝒪_{F}) we will write 𝒢′ (resp. ℳ′) for the base change to 𝒪_{F′}.

8.1. Rank one calculations. We remark that with our choice of polynomials H_{g}(u) in §6.3 and §6.4, any object ℳ in \underline{ϕ_{1}DD}_{F′/Q_{3}} has an action of ⟨γ_{2}, γ_{4}⟩ (via γ̂_{2} and γ̂_{4}, the action of γ_{2} being Frob_{3}-semilinear).

Also, since γ_{3} and γ_{2} commute, H_{γ_{2}} = 1 and H_{γ_{3}^{±1}}(u) ∈ Z_{3}[u], we see that γ̂_{2} must commute with γ̂_{3}^{±1} by Corollary 5.6.2.

We recall from Lemma 5.2.1 that the only models for (Z/3Z)_{/F} over 𝒪_{F} are 𝒢(r, δ) for r = 0, 2, 4, 6, 8, 10, 12 with 𝒢(12, δ) ≅ (Z/3Z)_{/𝒪_{F}} , and the only models for (μ_{3})_{/F} over 𝒪_{F} are 𝒢(r, 1) for r = 0, 2, 4, 6, 8, 10, 12 with 𝒢(0, 1) ≅ (μ_{3})_{/𝒪_{F}} . In each case, the base change to 𝒪_{F′} admits unique descent data over Q_{3} such that descent of the generic fibre to Q_{3} is Z/3Z (resp. μ_{3}). (See Lemma 5.7.1.) We will write 𝒢′_{r,1} (resp. 𝒢′_{r,ω}) for the corresponding pair (𝒢(r, δ) × 𝒪_{F′}, {[g]}) (resp. (𝒢(r, 1) × 𝒪_{F′}, {[g]})). We will also let ℳ′_{r,1} (resp. ℳ′_{r,ω}) denote the corresponding object of \underline{ϕ_{1}DD}_{F′/Q_{3}}. In particular the underlying F_{9}[u]/u^{36}-module has the form (F_{9}[u]/u^{36})e_{χ} with e_{χ} the standard generator, though we write e rather than e_{χ} if χ is understood.

We have the following useful lemma.

Lemma 8.1.1. Let 0 ≤ r ≤ e = 12 be an even integer. The descent data on ℳ′_{r,1} is determined by

γ̂_{2}(e) = e, γ̂_{4}(e) = −(−\sqrt{−1})^{r/2}e, γ̂_{3}^{±1}(e) = H_{γ_{3}^{±1}}(u)^{−3r/2}e,

and the descent data on ℳ′_{r,ω} is determined by

γ̂_{2}(e) = e, γ̂_{4}(e) = (−\sqrt{−1})^{r/2}e, γ̂_{3}^{±1}(e) = H_{γ_{3}^{±1}}(u)^{−3r/2}e.

In particular, γ_{4}^{2} = −1 on D(𝒢′_{r,1}) if and only if γ_{4}^{2} = −1 on D(𝒢′_{r,ω}) if and only if r = 2, 6 or 10.

Proof. Certainly γ̂_{2}(e) = e. We have already seen in Lemma 5.7.1 that descent data must exist in each case, so our task is to compute the unique units ξ_{γ_{4}}, ξ_{γ_{3}}^{±1} ∈ (F_{9}[u]/u^{36})^{×} so that

γ̂_{4}(e) = ξ_{γ_{4}}e, \begin{matrix} \\ γ_{3} & ±1 \end{matrix}(e) = ξ_{γ_{3}}^{±1}e

corresponds to generic fiber descent data for the mod 3 cyclotomic or trivial character on D_{3}. The case of γ_{3}^{±1} follows from Lemma 5.7.1.

From the condition

γ̂_{4} ∘ ϕ′_{1}(u^{r}e) = ϕ′_{1} ∘ γ̂_{4}(u^{r}e)

we get ξ_{γ_{4}}^{2}(u) = (−\sqrt{−1})^{r}, so

ξ_{γ_{4}}(u) = ±(−\sqrt{−1})^{r/2}.

The non-zero maps ℳ_{r,1} → ℳ_{12,1} are given by e ↦ ±u^{3(12−r)/2}e and the non-zero maps ℳ_{0,ω} → ℳ_{r,ω} are given by e ↦ u^{3r/2}e. Thus, it suffices to check that γ̂_{4}e = e on ℳ′_{12,1} and γ̂_{4}e = e on ℳ′_{0,ω}. In both cases we have shown that γ̂_{4}e = ±e and so we only need to check that γ_{4} = 1 on D(𝒢′_{12,1}) and D(𝒢′_{0,ω}). That is, we have to show that the 𝒪_{F′}-group scheme maps Z/3Z → ^{γ_{4}}(Z/3Z) and μ_{3} → ^{γ_{4}}μ_{3} arising from the canonical generic fibre descent data induce the identity on the special fibres. This is easy.

Lemma 8.1.2. Let ℳ be an object of \underline{ϕ_{1}−mod}_{F} corresponding to a finite flat group scheme 𝒢 and let \{[g]\} be descent data on 𝒢′ = 𝒢 × 𝒪_{F′} relative to Q_{3}. Assume that (𝒢′, {ĝ})_{Q_{3}} can be filtered so that each graded piece is isomorphic to Z/3Z or μ_{3} and so that the corresponding filtration of (ℳ′, \{ĝ\}) in \underline{ϕ_{1}DD}_{F′/Q_{3}} has

successive quotients of the form ℳ′_{r_{j},χ_{j}} with r_{j} ∈ {2, 6, 10} and χ_{j} ∈ {1, ω}. Then γ_{4}^{2} = −1 on ℳ′ /u ℳ′ and there exists a basis {e_{j}} of ℳ over F_{3}[u]/u^{36} so that for all j

• e_{j} ∈ ϕ_{1}(ℳ_{1}),

• e_{j} is an eigenvector of the F_{9}-linear map γ̂_{4} on ℳ′,

• e_{j} lies in the part of the filtration of ℳ′ which surjects onto ℳ′_{r_{j},χ_{j}} and this surjection sends e_{j} onto the standard basis vector e of ℳ′_{r_{j},χ_{j}} over F_{9}[u]/u^{36}.

Proof. Since γ_{4}^{2} acts linearly on ℳ′ /u ℳ′ and (γ_{4}^{2})^{2} = 1, the action of γ_{4}^{2} must be semisimple. The eigenvalues of γ_{4}^{2} are all equal to −1, so necessarily γ_{4}^{2} = −1 on ℳ′ /u ℳ′.

We now induct on the number of Jordan-Hölder factors in the generic fiber, the case of length 1 being clear. Thus, we can assume we have a short exact sequence in \underline{ϕ_{1}DD}_{F′/Q_{3}}

0 → 𝒩′ → ℳ′ → ℳ′_{r,χ} → 0,

so the lemma is known for 𝒩′. We just have to find e_{0} ∈ ϕ_{1}(ℳ_{1}) mapping onto the standard basis vector e in ℳ′_{r,χ} such that e_{0} is an eigenvector of γ̂_{4}. Since ϕ′_{1}(ℳ′_{1}) → ϕ′_{1}((ℳ′_{r,χ})_{1}) is a surjective map of F_{9}- vector spaces which is compatible with the semisimple F_{9}-linear endomorphism γ̂_{4} on each side, we can find e′_{0} ∈ ϕ′_{1}(ℳ′_{1}) mapping onto e with e′_{0} an eigenvector of γ̂_{4}, say γ̂_{4}(e′_{0}) = (\sqrt{−1})^{±1}e′_{0}. Since

γ̂_{4} ∘ γ̂_{2}(e′_{0}) = γ̂_{2} ∘ γ_{4}^{3}(e′_{0}) = γ̂_{2}(\sqrt{−1}^{∓1}e′_{0}) = \sqrt{−1}^{±1}γ̂_{2}(e′_{0}),

the element e_{0} = (1/2)(e′_{0} + γ̂_{2}(e′_{0})) maps to e and is an an eigenvector for γ̂_{4}. Also, e_{0} ∈ ϕ′_{1}(ℳ′_{1}) is γ̂_{2}-invariant and γ̂_{2} commutes with ϕ′_{1}, so e_{0} ∈ ϕ_{1}(ℳ_{1}).

8.2. Models for \overline{ρ}.

Proposition 8.2.1. There exists a unique object (𝒢′, {[g]}) of ℱ𝒟_{F′/Q_{3},ℐ} such that (𝒢′, {[g]})_{Q_{3}} corresponds to \overline{ρ}. If we set (ℳ(\overline{ρ})′, \{ĝ\}) = ℳ_{π}(𝒢′, \{[g]\}) then (ℳ(\overline{ρ})′, \{ĝ\}) is an extension of ℳ′_{2,ω} by ℳ′_{10,1} in \underline{ϕ_{1}DD}_{F′/Q_{3}}. Moreover Frobenius is not identically zero on D(𝒢′).

Proof. Let (𝒢′, \{[g]\}) be an object of ℱ𝒟_{F′/Q_{3},ℐ} such that (𝒢′, \{[g]\})_{Q_{3}} corresponds to \overline{ρ}, and set (ℳ′, \{ĝ\}) = ℳ_{π}(𝒢′, {[g]}). As in the discussion following Theorem 5.6.1, we have canonically ℳ′ ≃ F_{9} ⊗_{F_{3}} ℳ for a Breuil module ℳ over 𝒪_{F}, with γ̂_{2} acting as γ_{2} ⊗ 1. By Lemma 8.1.1, there is a short exact sequence of Breuil modules over 𝒪_{F}

0 → ℳ(s, δ) → ℳ → ℳ(r, 1) → 0

with r, s ∈ {2, 6, 10} and this is compatible with descent data after base change to 𝒪_{F′} in the sense that we obtain an exact sequence

0 → ℳ′_{s,1} → ℳ′ → ℳ′_{r,ω} → 0

compatible with descent data. Because \overline{ρ} is très ramifié, it follows that \overline{ρ}|_{F′} is non-split, so the sequence

0 → ℳ(s, δ) → ℳ → ℳ(r, 1) → 0

is non-split.

We first show that we must have (r, s) = (2, 10). Since \overline{ρ} is self-dual, in order to prove (r, s) = (2, 10) we may use Cartier duality (and Lemma 5.2.1) in order to reduce to the case where r + s ≤ e = 12. We will first rule out cases with r ≥ s and then the case (r, s) = (2, 6).

By Lemmas 8.1.1 and 8.1.2, we can write

ℳ = (F_{3}[u]/u^{36})e_{1} ⊕ (F_{3}[u]/u^{36})e′_{ω}, ℳ_{1} = ⟨u^{s}e_{1}, u^{r}e′_{ω} + he_{1}⟩

for some h ∈ F_{3}[u]/u^{36} so that

ϕ_{1}(u^{s}e_{1}) = δe_{1}, ϕ_{1}(u^{r}e′_{ω} + he_{1}) = e′_{ω}

and

γ̂_{4}(e_{1}) = −(−\sqrt{−1})^{s/2}e_{1}, γ̂_{4}(e′_{ω}) = (−\sqrt{−1})^{r/2}e′_{ω}.

Recall from Lemma 5.2.2 that the ‘parameter’ h gives an isomorphism of abstract groups

(F_{3}[u]/u^{36})/\{u^{s}t − δu^{r}t^{3}|t ∈ F_{3}[u]/u^{36}\} ≃ Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ℳ(r, 1), ℳ(s, δ)).

It is easy to see that

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1}

if and only if γ̂_{4}(u^{r}e′_{ω} + he_{1}) ∈ ℳ′_{1} and γ̂_{4}(e′_{ω}) = ϕ′_{1} ∘ γ̂_{4}(u^{r}e′_{ω} + he_{1}), or equivalently

(\sqrt{−1})^{r/2}h(u) ≡ −(−\sqrt{−1})^{s/2}h(−\sqrt{−1}u) mod u^{12+s}.

This says exactly that

j ≡ 2 − (r + s)/2 mod 4

for any j < 12 + s with a non-zero u^{j} term appearing in h.

If (r, s) = (6, 2) this would force h ≡ 0 mod u^{2}, yet {u^{2}t − δu^{6}t^{3}|t ∈ F_{3}[u]/u^{36}} contains all multiples of u^{2}, so

0 → ℳ(2, δ) → ℳ → ℳ(6, 1) → 0

is split, a contradiction.

When (r, s) = (10, 2) or (r, s) = (2, 2) we see that h ≡ h(0) mod u^{4}, yet

u^{4}(F_{3}[u]/u^{36}) ⊆ {u^{s}t − δu^{r}t^{3}|t ∈ F_{3}[u]/u^{36}},

so the choice of e′_{ω} may be changed in order to arrange that

h ∈ F_{3}

(though making this change of basis of ℳ may destroy the ‘diagonal’ form of γ̂_{4}). Since

0 → ℳ(s, δ) → ℳ → ℳ(r, 1) → 0

is non-split, necessarily h ≠ 0, so by rescaling e′_{ω} it can be assumed that h = 1. Then V_{ℳ} (e′_{ω}) ≡ e_{1} mod u ℳ (by Theorem 5.1.3) and

ϕ(e′_{ω}) ≡ −(δ/c_{π})u^{3(12−r−s)}e_{1} ≡ −u^{3(12−r−s)}e_{1} mod u ℳ .

This forces r + s = 12. In particular, (r, s) = (2, 2) is ruled out.

For (r, s) = (10, 2), a splitting of the generic fiber \overline{ρ}|_{F} is induced by the Breuil module map

ℳ(0, 1) → ℳ

defined by

e ↦ u^{15}e′_{ω} + u^{3}fe_{1} = u^{5}(u^{10}e′_{ω} + e_{1}) + (uf − u^{3})u^{2}e_{1},

where f ∈ F_{3}[u]/u^{36} satisfies f^{3} − δf = u^{6} (i.e., f = −δu^{6} − u^{18}, and a constant c ∈ F_{3} can even be added to this if δ = 1). But \overline{ρ}|_{G_{F′}} must be non-split, so this rules out (r, s) = (10, 2).

The remaining case with r ≥ s is (r, s) = (6, 6). In this case {u^{s}t − δu^{r}t^{3}|t ∈ F_{3}[u]/u^{36}} contains all multiples of u^{8}. But we have j ≡ 0 mod 4 for all j < 12 + s = 18 such that a non-zero u^{j} term appears in h, so again (at the expense of possibly making the γ̂_{4}-action non-diagonal) we may assume

h = c + c′u^{4}

for some c, c′ ∈ F_{3}. Writing γ̂_{4}(e′_{ω}) = (\sqrt{−1})e′_{ω} + h_{γ_{4}}(u)e_{1}, the commutativity of γ̂_{4} and ϕ′_{1} amounts to h_{γ_{4}} = −δh_{γ_{4}}^{3}, so h_{γ_{4}}(u) = b\sqrt{−δ} for some b ∈ F_{3}. The condition γ_{4}^{4}(e′_{ω}) = e′_{ω} forces b = 0, so γ̂_{4} still has diagonal action. This analysis shows that the map of F_{3}-vector spaces

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ℳ(6, 1), ℳ(6, −1))

has at most a 2-dimensional image. If c′ + δc = 0, then the Breuil module map

F_{9} ⊗ ℳ(0, 1) → ℳ′

defined by

e ↦ fu^{6}e_{1} + u^{3}(u^{6}e′_{ω} + (c + c′u^{4})e_{1}), (8.2.1)

with f^{3} = δ(fu^{6} + (c + c′u^{4})u^{3}) (i.e., f = cδu + a\sqrt{δ}u^{3} with a ∈ F_{3}) gives a splitting of \overline{ρ}|_{F′}, which is non-split. Thus, c′ + δc ≠ 0.

At this point, we treat the cases δ = ±1 separately. Consider first the case δ = 1. We must have

γ̂_{3}(e_{1}) = H_{γ_{3}}(u)^{−9}e_{1}, γ̂_{3}(e′_{ω}) = H_{γ_{3}}(u)^{−9}e′_{ω} + h_{γ_{3}}(u)e_{1},

where h_{γ_{3}}(u) ∈ F_{9}[u]/u^{36} lies in F_{3}[u]/u^{36} because γ̂_{3} commutes with γ̂_{2}. Evaluating γ̂_{3} ∘ ϕ′_{1} ≡ ϕ′_{1} ∘ γ̂_{3} mod u ℳ′ on u^{6}e′_{ω} + (c + c′u^{4})e_{1} ∈ ℳ′_{1} and using our knowledge of H_{γ_{3}}(u) mod 3, we arrive at

h_{γ_{3}}(0) = δ(h_{γ_{3}}(0)^{3} + (c + c′)),

which is impossible for h_{γ_{3}}(0) ∈ F_{3} with δ = 1 because c + c′ = c + δc′ ∈ F_{3}^{×}.

Now let us turn to the case δ = −1, still in the case (r, s) = (6, 6). This case is more difficult. We will first show that

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ℳ(6, 1), ℳ(6, −1))

is injective. Consider a typical element (ℳ′, \{ĝ\}) in Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}), so

ℳ = (F_{3}[u]/u^{36})e_{1} ⊕ (F_{3}[u]/u^{36})e′_{ω}, ℳ_{1} = ⟨u^{6}e_{1}, u^{6}e′_{ω} + (c + c′u^{4})e_{1}⟩

with

ϕ_{1}(u^{6}e_{1}) = −e_{1}, ϕ_{1}(u^{6}e′_{ω} + (c + c′u^{4})e_{1}) = e′_{ω}, γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω}.

The wild descent data must have the form

γ̂_{3}(e_{1}) = H_{γ_{3}}(u)^{−9}e_{1}, γ̂_{3}(e′_{ω}) = H_{γ_{3}}(u)^{−9}e′_{ω} + h_{γ_{3}}(u)e_{1}

for some h_{γ_{3}} ∈ F_{3}[u]/u^{36}. What we have to show is that h_{γ_{3}} is uniquely determined by the pair (c, c′). As h_{γ_{3}}(0) = −(h_{γ_{3}}(0)^{3} + c + c′) we see that h_{γ_{3}}(0) ∈ F_{3} is given by

h_{γ_{3}}(0) = c + c′.

On the other hand, evaluation of

γ̂_{3} ∘ ϕ′_{1} = (1 + t_{γ_{3}}N) ∘ ϕ′_{1} ∘ γ̂_{3}

on u^{6}e′_{ω} + (c + c′u^{4})e_{1} gives

h_{γ_{3}} = −h_{γ_{3}}^{3}H_{γ_{3}}^{18} + f_{c,c′,γ_{3}}(u)

where f_{c,c′,γ_{3}}(u) ∈ F_{3}[u]/u^{36} depends on H_{γ_{3}}, t_{γ_{3}}, and c, c′ (the monodromy operator N depends only on ℳ and hence only on c, c′). But any solution h_{γ_{3}}(u) of this equation is determined by its constant term. Thus h_{γ_{3}}(u) is uniquely determined by c and c′, as desired.

We show next that in this case ((r, s) = (6, 6) and δ = −1) the map

(8.2.2) Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[D_{3}]}^{1}(1, ω)

is not surjective. Suppose it were surjective. Then the injection

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) ↪ Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ℳ(6, 1), ℳ(6, −1)) ≅ \{(c, c′) ∈ F_{3}^{2}\}

would be an isomorphism of two-dimensional F_{3}-vector spaces. In the case c−c′ = 0, we have seen via (8.2.1) that the corresponding element of Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ℳ(6, 1), ℳ(6, −1)) has vanishing image in Ext_{F_{3}[G_{F′}]}^{1}(1, ω). As we are in the case δ = −1, the restriction map H^{1}(G_{3}, ω) → H^{1}(G_{F′}, ω) is injective. Thus the line c = c′ is in the kernel of

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[D_{3}]}^{1}(1, ω)

and so the image of this map is at most one dimensional, a contradiction.

To finally exclude the case (r, s) = (6, 6) and δ = −1, it suffices to show that the image of

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[D_{3}]}^{1}(1, ω)

contains the peu ramifié line (as \overline{ρ} is trés ramifié and (8.2.2) is not surjective). By Proposition 5.2.1 of [M], there is an elliptic curve E′_{/Q_{3}} which has supersingular reduction over Q_{3}(\sqrt{−1}, β), with \overline{ρ}_{E′,3} a non-split, peu ramifié extension of 1 by ω. The representation \overline{ρ}_{E′,3}|_{F′} is non-split (again because H^{1}(G_{3}, ω) → H^{1}(G_{F′}, ω) is injective in the δ = −1 case). Let 𝒩′ be the Breuil module corresponding to the 3-torsion on the Néron model of E′×_{Q_{3}} F′, so 𝒩′ admits descent data {g′} via the Néronian property of Néron models. The filtration of \overline{ρ} induces a short exact sequence in \underline{ϕ_{1}DD}_{F′/Q_{3}}

0 → ℳ′_{a,1} → (𝒩′, {g′}) → ℳ′_{b,ω} → 0

for some even a, b with 2 ≤ a, b ≤ 10. The Néron model of E′ ×_{Q_{3}} Q_{3}(\sqrt{−1}, β) has local-local 3-torsion, and the induced local-local integral models 𝒢_{ω} and 𝒢_{1} of the diagonal characters ω|_{Q_{3}(\sqrt{−1},β)} and 1|_{Q_{3}(\sqrt{−1},β)} must be the unique local-local models (uniqueness follows from Corollary 1.5.1 of [Ra]). Moreover, Corollary 1.5.1 of [Ra] makes it clear that base change to 𝒪_{F′} takes the order 3 group schemes 𝒢_{ω} and 𝒢_{1} to the integral models that lie in the middle of the well-ordered sets of integral models of ω|_{F′} and 1|_{F′}. It follows that a = b = 6, so the map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[D_{3}]}^{1}(1, ω)

indeed hits the peu ramifié line.

We next exclude the case (r, s) = (2, 6). As a first step, we check that there is at most one possibility for the underlying Breuil module ℳ (ignoring the extension class structure) if (r, s) = (2, 6). We can write

ℳ = (F_{3}[u]/u^{36})e_{1} ⊕ (F_{3}[u]/u^{36})e′_{ω}, ℳ_{1} = ⟨u^{6}e_{1}, u^{2}e′_{ω} + he_{1}⟩

for some necessarily non-zero h ∈ F_{3}[u]/u^{36} with

ϕ_{1}(u^{6}e_{1}) = δe_{1}, ϕ_{1}(u^{2}e′_{ω} + he_{1}) = e′_{ω}

and

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω}.

The combined conditions γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1} and ϕ′_{1} ∘ γ̂_{4} = γ̂_{4} ∘ ϕ′_{1} on ℳ′_{1} are equivalent to

h(u) ≡ −h(−\sqrt{−1}u) mod u^{18}.

Since {u^{6}t−δu^{2}t^{3}|t ∈ F_{3}[u]/u^{36}} contains u^{6} −δu^{2} and all multiples of u^{9}, we may change e′_{ω} (at the expense of possibly losing the diagonal form for γ̂_{4}) so that h = cu^{2} for some c ∈ F_{3}. Since h is necessarily non-zero, we may rescale to get h = u^{2}, so there is indeed at most one possibility for the underlying Breuil module ℳ when (r, s) = (2, 6).

Again we treat the cases δ = ±1 separately. Consider first the case δ = −1. We have seen above that there is an extension ℰ_{6,6} = (𝒩′, \{g′\}) of ℳ′_{6,ω} by ℳ′_{6,1} in \underline{ϕ_{1}DD}_{F′/Q_{3}} corresponding to a non-split, peu ramifié extension of 1 by ω. Pulling back ℰ_{6,6} by a non-zero map

ℳ′_{2,ω} → ℳ′_{6,ω}

in \underline{ϕ_{1}DD}_{F′/Q_{3}} given by e ↦ ±u^{6}e, we get an extension ℰ_{2,6} of ℳ′_{2,ω} by ℳ′_{6,1} in \underline{ϕ_{1}DD}_{F′/Q_{3}} corresponding to a non-split, peu ramifié extension of 1 by ω. The underlying Breuil module of ℰ_{2,6} must be isomorphic to F_{9} ⊗_{F_{3}} ℳ for our uniquely determined ℳ (with h = u^{2}). By the injectivity of H^{1}(G_{3}, ω) → H^{1}(G_{F′}, ω) in the δ = −1 case, we conclude that F_{9} ⊗_{F_{3}} ℳ cannot admit descent data giving rise to a très ramifié element in Ext_{F_{3}[G_{3}]}^{1}(1, ω). This rules out the case (r, s) = (2, 6) and δ = −1.

Now turn to the case (r, s) = (2, 6) and δ = 1. We will show that with the Breuil module ℳ constructed above (with h = u^{2}), the Breuil module ℳ′ = F_{9} ⊗_{F_{3}} ℳ does not admit descent data relative to F′/Q_{3} (with γ̂_{2} = γ_{2} ⊗ 1, without loss of generality). One checks that N(e_{1}) = N(e′_{ω}) = 0, so

N ∘ ϕ_{1} = 0.

We must have

γ̂_{3}(e_{1}) = H_{γ_{3}}(u)^{−9}e_{1}, γ̂_{3}(e′_{ω}) = H_{γ_{3}}(u)^{−3}e′_{ω} + h_{γ_{3}}(u)e_{1}

for some h_{γ_{3}} ∈ F_{9}[u]/u^{36}. As usual, since γ̂_{3} and γ̂_{2} must commute, we have h_{γ_{3}} ∈ F_{3}[u]/u^{36}. The condition γ̂_{3}(ℳ′_{1}) ⊆ ℳ′_{1} is equivalent to

γ̂_{3}(u^{2}e′_{ω} + u^{2}e_{1}) ∈ ℳ′_{1},

which amounts to

h_{γ_{3}}(u) ≡ H_{γ_{3}}^{−3} − H_{γ_{3}}^{−9} ≡ 0 mod u^{4},

so

γ̂_{3}(u^{2}e′_{ω} + u^{2}e_{1}) = H_{γ_{3}}(u)^{−1}(u^{2}e′_{ω} + u^{2}e_{1}) + (\frac{h_{γ_{3}} − H_{γ_{3}}^{−3} + H_{γ_{3}}^{−9}}{u^{4}} ) H_{γ_{3}}^{2}u^{6}e_{1}.

As N ∘ ϕ_{1} = 0, we have

γ̂_{3} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{3}

on ℳ′_{1}. Evaluating this identity on u^{2}e′_{ω} + u^{2}e_{1} ∈ ℳ_{1} gives

h_{γ_{3}} = H_{γ_{3}}^{6} · (\frac{h_{γ_{3}} − H_{γ_{3}}^{−3} + H_{γ_{3}}^{−9}}{u^{4}} )^{3} ,

so h_{γ_{3}} is a cube. Thus, h_{γ_{3}} = u^{6}g_{γ_{3}} for some g_{γ_{3}} ∈ F_{3}[u]/u^{36}.

Since H_{γ_{3}}^{3} ≡ 1 + u^{6} mod u^{12}, we compute

H_{γ_{3}}^{−9} − H_{γ_{3}}^{−3} ≡ u^{6} mod u^{12},

so

h_{γ_{3}} ≡ H_{γ_{3}}^{6} · ((\frac{h_{γ_{3}}}{u^{4}} )^{3} + u^{6}) mod u^{7}

and

g_{γ_{3}}(0) = g_{γ_{3}}(0)^{3} + 1

in F_{3}. This is absurd. This rules out all possibilities for (r, s) aside from (r, s) = (2, 10). Uniqueness now follows from Corollary 4.1.5.

From Theorem 5.4.2 of [M] and Proposition B.4.2 of [CDT] we see that there is an elliptic curve E_{/Q_{3}} such that E[3](\overline{Q}_{3}) ≅ \overline{ρ} and ρ_{E,3} has type τ_{±3}. Let ℰ denote the Néron model of E ×_{Q_{3}} F′ over 𝒪_{F′}. By the Néron property of ℰ_{/𝒪_{F′}} we see that ℰ[3^{∞}] has descent data over Q_{3}. As in §4.5 we see that ℐ acts trivially on the Dieudonné module of ℰ[3^{∞}] × F_{9}. Thus ℳ(\overline{ρ})′ ≅ ℳ_{π}(ℰ[3]) in \underline{ϕ_{1}DD}_{F′/Q_{3}} and it follows that Frobenius is non-zero on D(𝒢′).

8.3. Completion of proof of Theorem 4.5.1.

Lemma 8.3.1. Let (𝒢′, \{[g]\}) be the unique object of ℱ𝒟_{F′/Q_{3},ℐ} such that (𝒢′, \{[g]\})_{Q_{3}} corresponds to \overline{ρ}. Set (ℳ(\overline{ρ})′, \{ĝ\}) = ℳ_{π}(𝒢′, \{[g]\}). The natural map of groups

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} ((ℳ(\overline{ρ})′, \{ĝ\}), (ℳ(\overline{ρ})′, \{ĝ\})) → Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{10,1}, ℳ′_{2,ω}),

using pushout by (ℳ(\overline{ρ})′, \{ĝ\}) → ℳ′_{2,ω} and pullback by ℳ′_{10,1} → (ℳ(\overline{ρ})′, \{ĝ\}), is zero.

Proof. Let (ℳ̃′, \{ĝ\}) represent a class in Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} ((ℳ(\overline{ρ})′, \{ĝ\}), (ℳ(\overline{ρ})′, \{ĝ\})) and let (ℳ′, \{ĝ\}) be its

image in Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{10,1}, ℳ′_{2,ω}). By Lemma 5.2.2, ℳ′ = F_{9} ⊗ ℳ with γ̂_{2} = γ_{2} ⊗ 1 and

ℳ = (F_{3}[u]/u^{36})e_{ω} ⊕ (F_{3}[u]/u^{36})e′_{1}, ℳ_{1} = ⟨u^{2}e_{ω}, u^{10}e′_{1} + (c + c′u)e_{ω}⟩,

with c, c′ ∈ F_{3}. Also,

ϕ_{1}(u^{2}e_{ω}) = e_{ω}, ϕ_{1}(u^{10}e′_{1} + (c + c′u)e_{ω}) = δe′_{1},

and

γ̂_{4}(e_{ω}) = −\sqrt{−1}e_{ω}, γ̂_{4}(e′_{1}) = \sqrt{−1}e′_{1} + h_{γ_{4}}(u)e_{ω}

for some h_{γ_{4}} ∈ F_{9}[u]/u^{36}.

The properties γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1} and γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1} amount to

c′ = 0, h_{γ_{4}} = −δh_{γ_{4}}^{3}u^{24},

so h_{γ_{4}} = 0. If c = 0 then N ∘ ϕ_{1} = 0, so γ̂_{3} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{3} on ℳ′_{1}. From this we readily see that (ℳ′, {ĝ}) is split in \underline{ϕ_{1}DD}_{F′/Q_{3}}, as desired.

Now assume c ≠ 0; we will deduce a contradiction. Consider

(𝒩, {ĝ}) = (ℳ̃′, {ĝ})/(ℳ′_{10,1}, {ĝ}),

where ℳ′_{10,1} ↪ ℳ(\overline{ρ})′ ↪ ℳ̃′. Then 𝒩 has an ordered basis \{e_{ω}, e′_{1}, e′_{ω}\} with respect to which

𝒩_{1} = ⟨u^{2}e_{ω}, u^{10}e′_{1} + e_{ω}, u^{2}e′_{ω} + (a + a′u)e′_{1} + (b + b′u)e_{ω}⟩

for some a, a′, b, b′ ∈ k. Since our base field F′ has absolute ramification degree 12, 𝒩_{1} contains

u^{12}e′_{ω} = u^{10}(u^{2}e′_{ω} + (a + a′u)e′_{1} + (b + b′u)e_{ω}) − (a + a′u)(u^{10}e′_{1} + e_{ω}) + ((a + a′u) − u^{10}(b + b′u))e_{ω}.

From the list of generators of 𝒩_{1}, it is not difficult to check that in the above expression for u^{12}e′_{ω} ∈ 𝒩_{1}, u^{2} must divide the coefficient of e_{ω}. Thus a = a′ = 0.

We must have 𝒩 /⟨e_{ω}⟩ ≅ ℳ(\overline{ρ})′. Since a = a′ = 0, ℳ(\overline{ρ}) has basis \{e′_{1}, e′_{ω}\} and

ℳ(\overline{ρ})_{1} = ⟨u^{10}e′_{1}, u^{2}e′_{ω}⟩.

It follows immediately from this that ϕ ≡ 0 mod uℳ(\overline{ρ}), which (using Theorem 5.1.3) contradicts Proposition 8.2.1.

Corollary 8.3.2. The natural map

θ_{0} : Ext_{𝒮_{±3}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω)

is zero.

Theorem 4.7.4, and hence Theorem 4.4.1, now follow from the first case of the following lemma. We include the second case to simplify the proof.

Lemma 8.3.3. The maps of groups

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{10,1}, ℳ′_{10,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, 1), Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{2,ω}, ℳ′_{2,ω}) → Ext_{F_{3}[G_{3}]}^{1}(ω, ω)

have images inside the line of extension classes that split over an unramified extension of Q_{3}.

Proof. Since

H^{1}(G_{3}, Z/3) ⟶ H^{1}(G_{F′}, Z/3)

is injective and induces an isomorphism between the subgroups of unramified classes, it suffices to check that

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{10,1}, ℳ′_{10,1}) → Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (1, 1), Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{2,ω}, ℳ′_{2,ω}) → Ext_{\underline{ϕ_{1}−mod}_{F}}^{1} (ω, ω)

have images consisting of elements split over an unramified extension of F. By Cartier duality it suffices to consider only the second map.

Consider a representative (ℳ′, \{ĝ\}) of an element in Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{2,ω}, ℳ′_{2,ω}). Lemma 5.2.2 ensures that we can write

ℳ = (F_{3}[u]/u^{36})e_{ω} ⊕ (F_{3}[u]/u^{36})e′_{ω}, ℳ_{1} = ⟨u^{2}e_{ω}, u^{2}e′_{ω} + he_{ω}⟩

for some h = c + c′u + c′′u^{2} with c, c′, c′′ ∈ F_{3} and

ϕ_{1}(u^{2}e_{ω}) = e_{ω}, ϕ_{1}(u^{2}e′_{ω} + he_{ω}) = e′_{ω}.

We have

γ̂_{4}(e_{ω}) = −\sqrt{−1}e_{ω}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{ω}

for some h_{γ_{4}} ∈ F_{9}[u]/u^{36}, and the condition γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1} is equivalent to

h(u) ≡ −h(−\sqrt{−1}u) mod u^{2},

so c = c′ = 0. The Breuil module extension class ℳ over 𝒪_{F} (ignoring descent data) therefore only depends on the parameter c′′ ∈ F_{3}. We then have a splitting \overline{F}_{3} ⊗_{F_{3}} ℳ(2, 1) → \overline{F}_{3} ⊗_{F_{3}} ℳ determined by

e ↦ aue_{ω} + ue′_{ω}

where a ∈ \overline{F}_{3} satisfies a^{3} = a + c′′.

9. Proof of Theorems 4.6.1, 4.6.2 and 4.6.3.

In this section we will keep the notation of §4.6 and §6.5. We will write F for F_{i}, F′ for F′_{i} and ℐ for ℐ_{i}. If 𝒢 (resp. ℳ) is a finite flat 𝒪_{F} -group scheme (resp. Breuil module over 𝒪_{F}) we will write 𝒢′ (resp. ℳ′) for the base change to 𝒪_{F′}.

9.1. Rank one calculations. We remark that with our choice of polynomials H_{g}(u) in §6.5, any object ℳ in \underline{ϕ_{1}DD}_{F′/Q_{3}} has an action of ⟨γ_{2}, γ_{4}⟩ via γ̂_{2} and γ̂_{4}. (The action of γ_{2} is Frob_{3}-semilinear). Since γ_{3} and

γ_{2} commute and H_{γ_{3}^{±1}}(u) ∈ Z_{3}[u] we see that γ̂_{2} must commute with γ̂_{3}^{±1} (see Corollary 5.6.2).

By Lemma 5.2.1, the only models for (Z/3Z)_{/F} over 𝒪_{F} are 𝒢(r, 1) for r = 0, 2, 4, 6, 8, 10, 12 with 𝒢(12, 1) ≅ (Z/3Z)_{/𝒪_{F}} , and the only models for (μ_{3})_{/F} over 𝒪_{F} are 𝒢(r, 1) for r = 0, 2, 4, 6, 8, 10, 12 with 𝒢(0, 1) ≅ (μ_{3})_{/𝒪_{F}} . Lemma 5.7.1 ensures that the base changes to 𝒪_{F′} admit unique descent data over Q_{3} such that descent of the generic fibre to Q_{3} is Z/3Z (resp. μ_{3}). We will write 𝒢′_{r,1} (resp. 𝒢′_{r,ω}) for the corresponding pair (𝒢(r, 1) ×_{𝒪_{F}} 𝒪_{F′}, {[g]}) (resp. (𝒢(r, 1) ×_{𝒪_{F}} 𝒪_{F′}, {[g]})). We will also let ℳ′_{r,1} (resp. ℳ′_{r,ω}) denote the corresponding object of \underline{ϕ_{1}DD}_{F′/Q_{3}}.

We have the following useful lemmas, for which the proofs are identical to the proofs of Lemmas 8.1.1 and 8.1.2.

Lemma 9.1.1. Let 0 ≤ r ≤ e = 12 be an even integer. The descent data on ℳ_{r,1} is determined by

γ̂_{2}(e) = e, γ̂_{4}(e) = −(−\sqrt{−1})^{r/2}e, γ̂_{3}^{±1}(e) = H_{γ_{3}^{±1}}(u)^{−3r/2}e,

and the descent data on ℳ_{r,ω} is determined by

γ̂_{2}(e) = e, γ̂_{4}(e) = (−\sqrt{−1})^{r/2}e, γ̂_{3}^{±1}(e) = H_{γ_{3}^{±1}}(u)^{−3r/2}e.

In particular, γ_{4}^{2} = −1 on D(𝒢_{r,1}) if and only if γ_{4}^{2} = −1 on D(𝒢_{r,ω}) if and only if r = 2, 6 or 10.

Lemma 9.1.2. Let ℳ be an object of \underline{ϕ_{1}−mod}_{F} corresponding to a finite flat group scheme 𝒢 and let \{[g]\} be descent data on 𝒢′ = 𝒢 ×_{𝒪_{F}} 𝒪_{F′} over Q_{3}. Assume that (𝒢′, {ĝ})_{Q_{3}} can be filtered so that each graded piece is isomorphic to Z/3Z or μ_{3} and so that the corresponding filtration of (ℳ′, \{ĝ\}) in \underline{ϕ_{1}DD}_{F′/Q_{3}} has

successive quotients of the form ℳ′_{r_{j},χ_{j}} with r_{j} ∈ {2, 6, 10} and χ_{j} ∈ {1, ω}. Then γ_{4}^{2} = −1 on ℳ′ /u ℳ′ and there exists a basis {e_{j}} of ℳ over F_{3}[u]/u^{36} so that for all j

• e_{j} ∈ ϕ_{1}(ℳ_{1}),

• e_{j} is an eigenvector of the F_{9}-linear map γ̂_{4} on ℳ′,

• e_{j} lies in the part of the filtration of ℳ′ which surjects onto ℳ′_{r_{j},χ_{j}} and this surjection sends e_{j} onto the standard basis vector e of ℳ′_{r_{j},χ_{j}} over F_{9}[u]/u^{36}.

9.2. Models for \overline{ρ}. Recall that we are assuming that \overline{ρ} has the très ramifié form

( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ,

and is not split over F′. We will let \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ} denote the full subcategory of \underline{ϕ_{1}DD}_{F′/Q_{3}} consisting of

objects ℳ′ for which the ideal ℐ acts trivially on (ℳ′/uℳ′) ⊗_{F_{9},Frob_{3}} F_{9}.

Proposition 9.2.1. Suppose that (ℳ′, \{ĝ\}) is an object of \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ} such that (ℳ′, \{ĝ\})_{Q_{3}} is an extension of Z/3Z by μ_{3}. Then we have an exact sequence

(0) ⟶ ℳ′_{s,1} ⟶ ℳ′ ⟶ ℳ′_{r,ω} ⟶ (0)

with (r, s) = (2, 6), (6, 10), (2, 10) or (6, 6). Moreover we can write ℳ′ = ℳ ⊗_{F_{3}} F_{9} with γ̂_{2} = 1 ⊗ Frob_{3}, where ℳ has an F_{3}[u]/(u^{36})-basis {e_{1}, e′_{ω}} with e_{1} the standard basis element of ℳ(s, 1) and e′_{ω} mapping to the standard basis element of ℳ(r, 1). More precisely we have the following exhaustive list of extension class possibilities, all of which are well-defined.

1. \underline{(r, s) = (2, 6)}: The natural map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{2,ω}, ℳ′_{6,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, ω)

is an isomorphism, with elements parameterized by pairs (c, c_{1}) ∈ F_{3}^{2} corresponding to

ℳ_{1} = ⟨u^{6}e_{1}, u^{2}e′_{ω} + cu^{2}e_{1}⟩, ϕ_{1}(u^{6}e_{1}) = e_{1}, ϕ_{1}(u^{2}e′_{ω} + cu^{2}e_{1}) = e′_{ω}

(so N ∘ ϕ_{1} = 0) with

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})(e′_{ω} ± c_{1}u^{6}e_{1}).

The pairs with c = 0 are the ones which generically split over F′. In all cases ϕ ≡ 0 mod uℳ.

2. \underline{(r, s) = (6, 10)}: The natural map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{10,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, ω)

is an isomorphism, with elements parameterized by pairs (c, c_{1}) ∈ F_{3}^{2} corresponding to

ℳ_{1} = ⟨u^{10}e_{1}, u^{6}e′_{ω} + cu^{6}e_{1}⟩, ϕ_{1}(u^{10}e_{1}) = e_{1}, ϕ_{1}(u^{6}e′_{ω} + cu^{6}e_{1}) = e′_{ω}

(so N_{ℳ} ∘ ϕ_{1} = 0) with

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = (1 ∓ u^{18})e_{1}, γ̂_{3}^{±1}(e′_{ω}) = e′_{ω} ± c_{1}u^{6}e_{1}.

The pairs with c = 0 are the ones which generically split over F′. In all cases ϕ ≡ 0 mod uℳ. These cases are Cartier dual to the (2, 6) cases above.

3. \underline{(r, s) = (2, 10)}: The natural map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{2,ω}, ℳ′_{10,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, ω)

is an isomorphism, with elements parameterized by pairs (c, c_{1}) ∈ F_{3}^{2} corresponding to

ℳ_{1} = ⟨u^{10}e_{1}, u^{2}e′_{ω} + cu^{8}e_{1}⟩, ϕ_{1}(u^{10}e_{1}) = e_{1}, ϕ_{1}(u^{2}e′_{ω} + cu^{8}e_{1}) = e′_{ω}

(so N ∘ ϕ_{1} = 0) with

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = (1 ∓ u^{18})e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})(e′_{ω} ± c_{1}u^{12}e_{1}).

The pairs with c = 0 are the ones which generically split over F′. In all cases ϕ ≡ 0 mod uℳ.

4. \underline{(r, s) = (6, 6)}: The natural map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, ω)

is an isomorphism, with elements parameterized by pairs (c, c′) ∈ F_{3}^{2} corresponding to

ℳ_{1} = ⟨u^{6}e_{1}, u^{6}e′_{ω} + (c + c′u^{4})e_{1}⟩, ϕ_{1}(u^{6}e_{1}) = e_{1}, ϕ_{1}(u^{6}e′_{ω} + (c + c′u^{4})e_{1}) = e′_{ω}

(it is easily checked that N(e_{1}) = 0 and N(e′_{ω}) = c′u^{30}e_{1}) and

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂(e′_{ω}) = \sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{ω}) = e′_{ω} + (±c ∓ c′u^{12} − c′u^{30})e_{1}.

In particular, ϕ ≡ 0 mod uℳ if and only if c = 0. In the first three cases, the peu ramifié condition on a class in Ext_{F_{3}[G_{3}]}^{1}(1, ω) is equivalent to the vanishing

of c_{1}. In the fourth case it is equivalent to the vanishing of c.

Proof. By Lemma 9.1.1 we have an exact sequence

(0) ⟶ ℳ′_{s,1} ⟶ ℳ′ ⟶ ℳ′_{r,ω} ⟶ (0)

with r, s ∈ {2, 6, 10}.

In the cases (r, s) = (2, 2) and (6, 2) as in the proof of Proposition 8.2.1 we see that h = 0. We will show that in the case (r, s) = (10, 2) we also have h = 0. Following the proof of Proposition 8.2.1 we may suppose that h ∈ F_{3}. Without loss of generality we can take h = 1 and look for a contradiction. Again following the proof of Proposition 8.2.1 and using

ℳ_{1} = ⟨u^{2}e_{1}, u^{10}e′_{ω} + e_{1}⟩

we find that ϕe′_{ω} ≡ −e_{1} mod uℳ. Also

\begin{aligned}γ̂_{3}^{±1}e_{1} & = & (1 ± u^{18})e_{1} \\ γ̂_{3}^{±1}e′_{ω} & = & (1 ∓ u^{18})e′_{ω} + h_{±1}(u)e_{1} \end{aligned}

for some h_{±1}(u) ∈ F_{9}[u]/(u^{36}), which, as usual, must actually lie in F_{3}[u]/(u^{3}6). Thus

−e_{1} ≡ ϕe′_{ω}
≡ (γ̂_{3}γ̂_{2} − γ̂_{3}^{−1}γ̂_{2})(e′_{ω})
≡ (γ̂_{3} − γ̂_{3}^{−1})(e′_{ω})
≡ (h_{1}(0) − h_{−1}(0))e_{1} mod uℳ′.

The inverse linear maps γ̂_{3}^{±1} on ℳ′/uℳ′ have matrices

( \begin{matrix}1 & h_{±1}(0) \\ 0 & 1 \end{matrix} )

with respect to the basis {e_{1}, e′_{ω}}, so that h_{−1}(0) = −h_{1}(0). Thus h_{1}(0) = 1. On the other hand evaluating γ̂_{3}ϕ′_{1} ≡ ϕ′_{1}γ̂_{3} mod uℳ′ on u^{10}e′_{ω} + e_{1} and comparing coefficuients of e_{1} gives h_{1}(0) = 0, a contradiction.

Thus if any case (r, 2) arises the underlying Breuil module must be a split extension

ℳ = (F_{3}[u]/(u^{36}))e_{1} ⊕ (F_{3}[u]/(u^{36}))e′_{ω}, ℳ_{1} = ⟨u^{2}e_{1}, u^{r}e′_{ω}⟩
ϕ_{1}(u^{2}e_{1}) = e_{1},  ϕ_{1}(u^{r}e′_{ω}) = e′_{ω}

(so N ∘ ϕ_{1} = 0), with

γ̂_{2}e_{1} = e_{1},  γ̂_{2}e′_{ω} = e′_{ω}
γ̂_{4}e_{1} = \sqrt{−1}e_{1}, γ̂_{4}e′_{ω} = (−\sqrt{−1})^{r/2}e′_{ω}.

We also have

γ̂_{3}^{±1}e_{1} = H_{γ_{3}^{±1}}(u)^{−3}e_{1} γ̂_{3}^{±1}e′_{ω} = H_{γ_{3}^{±1}}(u)^{−3r/2}e′_{ω} + h_{±1}(u)e_{1}

for some h_{±1} ∈ F_{3}[u]/(u^{36}). Since N ∘ ϕ_{1} = 0 we have γ̂_{3}^{±1}ϕ′_{1} = ϕ′_{1}γ̂_{3}^{±1} on ℳ′_{1}. Evaluating this on u^{r}e′_{ω} and comparing coefficients of e_{1} gives h_{±1}(u) = u^{3(r−2)}h_{±1}(u)^{3}H_{γ_{3}^{±1}}(u)^{3}. This forces h_{±1}(u) = 0 if r ≠ 2. If r = 2 it forces h_{±1}(u) = c_{±1}(1 ∓ u^{18}) for some c_{±1} ∈ F_{3}. Wer will show c_{−1} = c_{1} = 0. Indeed, evaluating the congruence

ϕ ≅ (γ̂_{3}γ̂_{2} − γ̂_{3}^{−1}γ̂_{2}) mod uℳ′

on e′_{ω} gives

0 = ϕ(e′_{ω}) ≡ (c_{1} − c_{−1})e_{1} mod uℳ′

so that c_{−1} = c_{1}. On the other hand the congruence

γ̂_{3}γ̂_{4} ≡ γ̂_{4}γ̂_{3}^{−1} mod uℳ′

gives

( \begin{matrix}1 & c_{1} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\sqrt{−1} & 0 \\ 0 & −\sqrt{−1} \end{matrix} ) = ( \begin{matrix}\sqrt{−1} & 0 \\ 0 & −\sqrt{−1} \end{matrix} ) ( \begin{matrix}1 & c_{−1} \\ 0 & 1 \end{matrix} )

in M_{2}(F_{3}), so c_{−1} = −c_{1}. Thus c_{−1} = c_{1} = 0 and h_{±1} = 0 for r = 2 as well. Thus for r = 2, 6 and 10 the Breuil module with descent data ℳ′ is split so that \overline{ρ} is split, a contradiction.

This rules out the possibilities (2, 2), (6, 2) and (10, 2). Using Cartier duality we can also rule out (10, 10) and (10, 6). We are left with the four possible pairs (r, s) as asserted in the proposition and must determine which possibilities arise in each case.

Next consider the case (r, s) = (2, 6). Using the same analysis as in the (r, s) = (2, 6) case in Proposition 8.2.1, we find that the possibilities for the Breuil module ℳ are the ones in the statement of the proposition (and N ∘ ϕ_{1} = 0 is easy to check), though we only know that

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{1}

for some h_{γ_{4}}(u) ∈ F_{9}[u]/u^{36}. The conditions

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1}

are equivalent to

h_{γ_{4}} ≡ 0 mod u^{4}, h_{γ_{4}} = − (\frac{h_{γ_{4}}}{u^{4}} )^{3} .

The solutions to this are h_{γ_{4}} = a\sqrt{−1}u^{6} for a ∈ F_{3}. Replacing e′_{ω} by e′_{ω} + au^{6}e_{1} preserves our standardized form but makes h_{γ_{4}} = 0:

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω}.

The wild descent data must have the form

γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})e′_{ω} + h_{±1}(u)e_{1}

for some h_{±1} ∈ F_{9}[u]/u^{36}. The conditions

γ̂_{3}^{±1}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{3}^{±1} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{3}^{±1} on ℳ′_{1}

(recall N ∘ ϕ_{1} = 0) are equivalent to

h_{±1} ≡ 0 mod u^{4}, h_{±1} = (1 ± u^{18}) (\frac{h_{±1}}{u^{4}} )^{3} ,

whose solutions are

h_{±1} = c_{±1}u^{6}(1 ± u^{18})

for some c_{±1} ∈ F_{3}. Since N ∘ ϕ_{1} = 0, we have

γ̂_{3}^{±1} ∘ γ̂_{3}^{∓1} ∘ ϕ′_{1} = ϕ′_{1},

so

c_{−1} = −c_{1}.

Using Lemma 5.2.2 and Corollary 5.6.2, we see that all of these possibilities are well-defined. We also see that ℐ acts trivially on ℳ/uℳ ⊗ F_{9}. It is straightforward to check that generic splitting over F′ (which is equivalent to generic splitting over F) is equivalent to c = 0, and that such splitting is compatible with descent data (i.e., descends to Q_{3}) if and only if c = c_{1} = 0. For dimension reasons, the map on Ext^{1}’s is therefore an isomorphism.

Now consider the case (r, s) = (2, 10). Here we have

ℳ_{1} = ⟨u^{10}e_{1}, u^{2}e′_{ω} + he_{1}⟩

for some h ∈ F_{3}[u]/u^{36}, with

ϕ_{1}(u^{10}e_{1}) = e_{1}, ϕ_{1}(u^{2}e′_{ω} + he_{1}) = e′_{ω}

and

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω}.

In order that

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1},

it is necessary and sufficient that

h ≡ h(−\sqrt{−1}u) mod u^{22}.

But {u^{10}t − u^{2}t^{3}|t ∈ F_{3}[u]/u^{36}} is spanned by u^{13} − u^{11}, u^{12} − u^{8}, u^{11} − u^{5}, u^{10} − u^{2}, and all multiples of u^{15}, so we may suppose

h = c′′ + c′u^{4} + cu^{8}, (9.2.1)

for some c′′, c′, c ∈ F_{3}, at the expense of possibly losing the diagonal form of γ̂_{4}.

The monodromy operator satisfies

N(e_{1}) = 0, N(e′_{ω}) = (c′′u^{6} − c′u^{18} + c′′u^{30})e_{1}.

Since the wild descent data must take the form

γ̂_{3}^{±1}(e_{1}) = (1 ∓ u^{18})e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})e′_{ω} + h_{±1}e_{1}

for some h_{±1} ∈ F_{9}[u]/u^{36}, we compute

γ̂_{3}^{±1}(u^{2}e′_{ω} + he_{1}) = H_{γ_{3}^{±1}}^{2}(u)(1 ± u^{18})(u^{2}e′_{ω} + he_{1}) + f_{±1}(u)e_{1},

where

f_{±1}(u) = −hH_{γ_{3}^{±1}}^{2}(u)(1 ± u^{18}) + u^{2}H_{γ_{3}^{±1}}^{2}h_{±1} + (1 ∓ u^{18})h(uH_{γ_{3}^{±1}}). (9.2.2)

Thus, in order that γ̂_{3}^{±1}(ℳ′_{1}) ⊆ ℳ′_{1}, it is necessary and sufficient that f_{±1} satisfies

f_{±1} ≡ 0 mod u^{10}.

Using (9.2.1) and H_{γ_{3}^{±1}} ≡ 1 ∓ u^{6} mod 3, this amounts to

h_{±1} ≡ ±c′′u^{4} mod u^{8}. (9.2.3)

However, N ∘ ϕ_{1}(ℳ_{1}) ⊆ u^{6} ℳ, so

γ̂_{3}^{±1} ∘ ϕ′_{1} ≡ ϕ′_{1} ∘ γ̂_{3}^{±1} mod u^{6} ℳ′

when evaluated on ℳ′_{1}. This gives

h_{±1} ≡ (\frac{f_{±1}}{u^{10}} )^{3} mod u^{6}.

Since h_{±1} is a cube modulo u^{6}, by (9.2.3) we must have c′′ = 0, and so N ∘ ϕ_{1} ≡ 0 mod u^{18} ℳ. Thus, γ̂_{3}^{±1} and ϕ′_{1} commute modulo u^{18} ℳ′ when evaluated on ℳ′_{1}, so we get

h_{±1} ≡ (\frac{f_{±1}}{u^{10}} )^{3} mod u^{18}, (9.2.4)

and h_{±1} is a cube modulo u^{18}.

On the other hand, with c′′ = 0, we see from (9.2.1) and (9.2.2) that

f_{±1} ≡ u^{2}H_{γ_{3}^{±1}}^{2}h_{±1} mod u^{10},

so the condition f_{±1} ≡ 0 mod u^{10} gives us

h_{±1} ≡ 0 mod u^{8}.

Because h_{±1} is a cube modulo u^{18}, so we get the slight improvement

h_{±1} ≡ 0 mod u^{9}.

Combining this with the vanishing of c′′, we deduce from (9.2.2) that f_{±1} ≡ ±c′u^{10} mod u^{11}, so by (9.2.4)

h_{±1} ≡ ±c′ mod u.

This forces c′ = 0, so N ∘ ϕ_{1} = 0. Thus, γ̂_{3}^{±1} and ϕ′_{1} commute on ℳ′_{1}, so

h_{±1} = (\frac{f_{±1}}{u^{10}} )^{3} .

in F_{9}[u]/u^{36}. Using h = cu^{8} this becomes (via (9.2.2))

h_{±1} = (1 ± u^{18}) (\frac{h_{±1}}{u^{8}} )^{3} ,

so

h_{±1} = c_{±1}u^{12}(1 ± u^{18})

for some c_{±1} ∈ F_{3}. As before, we get c_{−1} = −c_{1}.

Now we “diagonalise” γ̂_{4}. Since we have

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{1}

for some h_{γ_{4}} ∈ F_{9}[u]/u^{36}, the conditions

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1}

are equivalent to

h_{γ_{4}} ≡ 0 mod u^{8}, h_{γ_{4}} = − (\frac{h_{γ_{4}}}{u^{8}} )^{3} ,

which is to say

h_{γ_{4}} = a\sqrt{−1}u^{12}

for some a ∈ F_{3}. Replacing e′_{ω} by e′_{ω} + au^{12}e_{1} then puts us in a setting with a = 0. Thus all extensions have the form asserted in the proposition. It is easy to check that in each case ℐ acts trivially on (ℳ/uℳ) ⊗ F_{9}.

Pushout by the non-zero map ℳ′_{6,1} → ℳ′_{10,1} in \underline{ϕ_{1}DD}_{F′/Q_{3}} induced by e ↦ u^{6}e takes our (2,6) examples to our (2,10) examples (compatibly with the labelling of parameters c, c_{1} as in the statement of the proposition). Thus all 9 possibilities for (c, c_{1}) do occur and we get an isomorphism of Ext^{1}’s as asserted. Moreover, generic splitting over F′ (which is equivalent to generic splitting over F) is equivalent to c = 0, and such splitting is compatible with descent data (i.e., descends to Q_{3}) if and only if c = c_{1} = 0.

Using Cartier duality and the case (r, s) = (2, 6), we see that in the case (r, s) = (6, 10) the map of Ext^{1}’s is an isomorphism. It is easy to check that the objects in our asserted list of 9 possibilities for (r, s) = (6, 10) are well-defined and that pullback by the non-zero map ℳ′_{2,ω} → ℳ′_{6,ω} induced by e ↦ u^{6}e takes these to our (2, 10) examples (compatibly with the labelling of parameters c, c_{1}).

Finally, we turn to the case (r, s) = (6, 6). Choosing a basis with respect to which γ̂_{4} has a diagonal action, the conditions

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1} (9.2.5)

are equivalent to

h(u) ≡ h(−\sqrt{−1}u) mod u^{18}.

Since {u^{6}t − u^{6}t^{3}|t ∈ F_{3}[u]/u^{36}} consists of multiples of u^{7}, we may change e′_{ω} to get

h = c + c′u^{4}

for some c, c′ ∈ F_{3}, with

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{1}

for some h_{γ_{4}} ∈ F_{9}[u]/u^{36}. Feeding this into (9.2.5) we get h_{γ_{4}} = −h_{γ_{4}}^{3}, so h_{γ_{4}} = \sqrt{−1}a for some a ∈ F_{3}. Replacing e′_{ω} by e′_{ω}−ae_{1} returns us to the setting with ‘diagonal’ γ̂_{4}-action and preserves the standardizations we have made so far.

It is easy to compute N(e′_{ω}) = c′u^{30}e_{1} (and we know N(e_{1}) = 0). The ‘wild’ descent data is

γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{ω}) = e′_{ω} + h_{±1}e_{1}

for some h_{±1} ∈ F_{9}[u]/u^{36}. Using the congruence for t_{γ_{3}^{±1}} in §6.5, the identity

γ̂_{3}^{±1} ∘ ϕ′_{1} = (1 + t_{γ_{3}^{±1}} · N) ∘ ϕ′_{1} ∘ γ̂_{3}^{±1}

on ℳ′_{1} amounts to the condition

h_{±1} = h_{±1}^{3} ∓ c′u^{12} − c′u^{30},

whose solutions are

h_{±1} = c_{±1} ∓ c′u^{12} − c′u^{30}

for some c_{±1} ∈ F_{3}. The identity

γ̂_{3}^{±1} ∘ γ̂_{3}^{∓1} ≡ 1 mod u ℳ′

implies c_{−1} = −c_{1}. Thus

−V_{ℳ} ′(e′_{ω}) ≡ ϕ(e′_{ω}) ≡ −ce_{1} mod u ℳ′, (γ̂_{3} ∘ γ̂_{2} − γ̂_{3}^{−1}γ̂_{2})(e′_{ω}) ≡ −c_{1}e_{1} mod u ℳ′ .

Thus ℐ acts trivially on (ℳ/uℳ) ⊗_{F_{3}} F_{9} if and only if c_{1} = c.

By Lemma 5.2.2 and Corollary 5.6.2, it is easy to see that all of these objects are well-defined. The kernel of

(9.2.6) Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[G_{F}]}^{1}(1, ω)

consists of pairs (c, −c), where generic splittings are induced by any of the (non-zero) Breuil module maps

ℳ(0, 1) → ℳ

defined by

e ↦ u^{9}e′_{ω} + (c̃u^{9} + cu^{3})e_{1} = u(c + u^{2}c̃)u^{6}e_{1} + u^{3}(u^{6}e′_{ω} + (c − cu^{4})e_{1})

with c̃ ∈ F_{3}. Thus, the pairs (c, c′) corresponding to the \overline{ρ} which are split over F (or equivalently, split over F′) are exactly those for which c + c′ = 0. The map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, ω)

is therefore injective, because the splitting given above respects descent data if and only if c̃ = c_{1} = 0.

It remains to establish which of the given extensions of Breuil modules correspond to peu ramifié extensions of Z/3Z by μ_{3} over Q_{3}. We noted above that the maps among the Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} ’s in the (2,6), (6,10), (2,10)

cases induced by pushout/pullback along e ↦ u^{6}e are compatible with the parameterization by pairs (c, c_{1}). With a little more care, one checks that the maps

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{2,ω}, ℳ′_{6,1}) ← Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) → Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{10,1})

induced by e ↦ u^{6}e send the pair (c, c′) in the middle to the pair (c + c′, c) on either end (to construct the necessary commutative diagrams of short exact sequences in the two cases, use the maps

(e′_{ω}, e_{1}) ↦ (u^{6}e′_{ω} − c′e_{1}, e_{1}), (e′_{ω}, e_{1}) ↦ (e′_{ω} + c′e_{1}, u^{6}e_{1})

respectively). This reduces us to checking the (6, 6) case.

By Corollary 2.3.2, the two très ramifié extensions, \overline{ρ}_{1} and \overline{ρ}_{2}, of 1 by ω which are non-split over F arise from elliptic curves, E_{1} and E_{2}, over Q_{3} for which ρ_{E_{i},3} is potentially Barsotti-Tate with extended type τ′_{i}. Let 𝒢_{i} denote the 3-torsion in the Néron model of E_{i} over 𝒪_{F}. From the universal property of Néron models we see that 𝒢′_{i} = 𝒢_{i} ×_{𝒪_{F}} 𝒪_{F′} inherits descent data {[g]} over Q_{3}. By the same argument used at the end of §4.6 we see that (𝒢′_{i}, \{[g]\}) is an object of \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}. Moreover we see that F ≠ 0 on D(𝒢_{i}). Since all non-(6, 6) cases above have ϕ ≡ 0 mod uℳ, by the parts of Proposition 9.2.1 which we have already proved we see that ℳ_{π}(𝒢′_{i}, {[g]}) is an extension of ℳ′_{6,ω} by ℳ′_{6,1} and correspond to a pair (c, c′) with c ≠ 0 (since F ≠ 0) and c + c′ ≠ 0 (by our analysis of (9.2.6), since \overline{ρ}_{i} is not-split over F). Hence ℳ_{π}(𝒢′_{1}, \{[g]\}) and ℳ_{π}(𝒢′_{2}, {[g]}) must correspond in some order to the lines ⟨(1, 0)⟩ and ⟨(1, 1)⟩ in F_{3}^{2}.

As a non-split peu ramifié extension of 1 by ω remains non-split over F′, we see that the peu ramifié line in

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,ω}, ℳ′_{6,1}) ≅ Ext_{F_{3}[G_{3}]}^{1}(1, ω)

cannot correspond to c + c′ = 0. By the above analysis it cannot correspond to c′ = 0 or c − c′ = 0. Thus it must correspond to the remaining line c = 0.

The properties of ϕ in the cases listed in Proposition 9.2.1 make it clear that the (6, 6) case there is “different”. We will see further manifestations of this difference later.

9.3. Further rank two calculations.

Lemma 9.3.1. For (r, s) = (2, 6), (6, 10) and (2, 10) we have

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{s,1}, ℳ′_{r,ω}) = (0).

Proof. The (6,10) case follows from the (2,6) case by Cartier duality. Thus, we assume r = 2, s ∈ {6, 10}. Let (ℳ′, {ĝ}) be such an extension. By Lemma 8.1.2, (ℳ′, {ĝ}) arises from a Breuil module over 𝒪_{F} of the form

ℳ = (F_{3}[u]/u^{36})e_{ω} ⊕ (F_{3}[u]/u^{36})e′_{1}, ℳ_{1} = ⟨u^{2}e_{ω}, u^{s}e′_{1} + he_{ω}⟩

with

ϕ_{1}(u^{2}e_{ω}) = e_{ω}, ϕ_{1}(u^{s}e′_{1} + he_{ω}) = e′_{1}

and

γ̂_{4}(e_{ω}) = −\sqrt{−1}e_{ω}, γ̂_{4}(e′_{1}) = −(−\sqrt{−1})^{s/2}e′_{1},

where h ∈ F_{3}[u]/u^{36}.

The combined conditions

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1}

are equivalent to

(\sqrt{−1})^{s/2}h(u) ≡ (\sqrt{−1})h(−\sqrt{−1}u) mod u^{14}.

Treating the cases s = 6 and s = 10 separately, we conclude from Lemma 5.2.2 that we may change e′_{1} so that h = 0 when s = 6 and h ∈ F_{3} when s = 10. As a result of this change, we only have

γ̂_{4}(e_{ω}) = −\sqrt{−1}e_{ω}, γ̂_{4}(e′_{1}) = −(−\sqrt{−1})^{s/2}e′_{1} + h_{γ_{4}}(u)e_{ω}.

However, with h ∈ F_{3} when s = 10 and h = 0 when s = 6, the conditions

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1}

force h_{γ_{4}} = −u^{3(s−2)}h_{γ_{4}}^{3}, so that in fact h_{γ_{4}} = 0 after all.

When h = 0, so ℳ is split in \underline{ϕ_{1}−mod}_{F} (compatibly with γ̂_{4} on ℳ′), and it is easy to check that the ‘wild’

descent data γ̂_{3}^{±1} must also be diagonal, so we have the desired splitting in \underline{ϕ_{1}DD}_{F′/Q_{3}}.

It remains to consider the case (r, s) = (2, 10) with h = c ∈ F_{3}. It is easy to compute

N(e_{ω}) = 0, N(e′_{1}) = −cu^{30}e_{ω}.

The wild descent data must have the form

γ̂_{3}^{±1}(e_{ω}) = (1 ± u^{18})e_{ω}, γ̂_{3}^{±1}(e′_{1}) = (1 ∓ u^{18})e′_{1} + h_{±1}e_{ω}

with h_{±1} ∈ F_{9}[u]/u^{36}.

It is straightforward to check that γ̂_{3}^{±1}(ℳ′_{1}) ⊆ ℳ′_{1}, and then the condition

γ̂_{3}^{±1} ∘ ϕ′_{1} = (1 + t_{γ_{3}^{±1}} · N) ∘ ϕ′_{1} ∘ γ̂_{3}^{±1}

on ℳ′_{1} gives

h_{±1} = ±cu^{12} + h_{±1}^{3}u^{24}(1 ∓ u^{18}) + cu^{30}.

The unique solution to this is

h_{±1} = c(±u^{12} + u^{30}).

Thus γ̂_{3}γ̂_{2} − γ̂_{3}^{−1}γ̂_{2} ≡ 0 mod u, while ϕ(e′_{1}) ≡ −ce_{ω} mod u. This forces c = 0. With c = 0 we obviously have only the split extension class.

Lemma 9.3.2. The natural map

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}}^{1} (ℳ′_{6,1}, ℳ′_{6,ω}) → Ext_{F_{3}[D_{3}]}^{1}(ω, 1)

is an isomorphism, with elements parametrised by pairs (c, c′) ∈ F_{3}^{2} corresponding to

ℳ = (F_{3}[u]/u^{36})e_{ω} ⊕ (F_{3}[u]/u^{36})e′_{1}, ℳ_{1} = ⟨u^{6}e_{ω}, u^{6}e′_{1} + (c + c′u^{4})e_{ω}⟩,

where

ϕ_{1}(u^{6}e_{ω}) = e_{ω}, ϕ_{1}(u^{6}e′_{1} + (c + c′u^{4})e_{ω}) = e′_{1}, N(e_{ω}) = 0, N(e′_{1}) = c′u^{30}e_{ω}

and the descent data is

γ̂_{4}(e_{ω}) = \sqrt{−1}e_{ω}, γ̂_{4}(e′_{1}) = −\sqrt{−1}e′_{1},
γ̂_{3}^{±1}(e_{ω}) = e_{ω}, γ̂_{3}^{±1}(e′_{1}) = e′_{1} + (±c ∓ c′u^{12} − c′u^{30})e_{ω}.

Proof. The proof is identical to the proof of the case (r, s) = (6, 6) in Proposition 9.2.1, except \sqrt{−1} is everywhere replaced by −\sqrt{−1} and when we study splitting we give ℳ(0, 1) the descent data for the trivial mod 3 character (which amounts to using γ̂_{4}(e) = −e rather than γ̂_{4}(e) = e).

Lemma 9.3.3. For r ∈ {2, 10}, the maps

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{r,ω}, ℳ′_{r,ω}) → Ext_{F_{3}[G_{3}]}^{1}(ω, ω)

and

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{r,1}, ℳ′_{r,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, 1)

are injective and have image consisting of the 1-dimensional space of classes which split over an unramified extension of Q_{3}.

Proof. The cases r = 10 follow from the cases r = 2 using Cartier duality. Thus we suppose r = 2. We treat only the case of ℳ′_{2,ω}, the case ℳ_{2,1} being exactly the same except that −\sqrt{−1} replaces \sqrt{−1} everywhere.

Let (ℳ′, \{ĝ\}) represents an element in Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{2,ω}, ℳ′_{2,ω}). Lemma 8.1.2 ensures the existence of

an ordered F_{3}[u]/u^{36}-basis e_{ω}, e′_{ω} of ℳ such that

ℳ_{1} = ⟨u^{2}e_{ω}, u^{2}e′_{ω} + he_{ω}⟩, ϕ_{1}(u^{2}e_{ω}) = e_{ω}, ϕ_{1}(u^{2}e′_{ω} + he_{ω}) = e′_{ω}

with

γ̂_{4}(e_{ω}) = −\sqrt{−1}e_{ω}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω}.

Carrying out the usual calculation,

γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1}, ϕ′_{1} ∘ γ̂_{4} = γ̂_{4} ∘ ϕ′_{1} on ℳ′_{1} (9.3.1)

if and only if

h ≡ −h(−\sqrt{−1}u) mod u^{14}.

Combining this with Lemma 5.2.2, we may change e′_{ω} so that h = cu^{2}, with c ∈ F_{3}, at the expense of possibly losing the diagonal form of γ̂_{4}. But with h = cu^{2} and γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{ω}, the conditions (9.3.1) imply h_{γ_{4}} = −h_{γ_{4}}^{3}, and so h_{γ_{4}}(u) = (\sqrt{−1})a for some a ∈ F_{3}. Then γ_{4}^{4} = 1 forces a = 0, so γ̂_{4} still has diagonal form.

It is easy to check that N(e′_{ω}) = 0, so N ∘ ϕ_{1} = 0. Thus, we must have

γ̂_{3}^{±1} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{3}^{±1} (9.3.2)

on ℳ′_{1}. Since the wild descent data has to be of the form

γ̂_{3}^{±1}(e_{ω}) = (1 ± u^{18})e_{ω}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})e′_{ω} + h_{±1}e_{ω}

for some h_{±1} ∈ F_{3}[u]/u^{36}, evaluation of (9.3.2) on u^{2}e′_{ω} + cu^{2}e_{ω} ∈ ℳ_{1} gives h_{±1} = (1 ± u^{18})h_{±1}^{3}, so

h_{±1} = c_{±1}(1 ± u^{18}) for some c_{±1} ∈ F_{3}. The relation γ̂_{3}^{±1} ∘ γ̂_{4} ∘ γ̂_{3}^{±1} ∘ γ̂_{4}^{3}(e′_{ω}) = e′_{ω} forces c_{±1} = 0.

We now have described all possibilities in terms of the single parameter c ∈ F_{3}, and it is straightforward to use Corollary 5.6.2 to check that all of these examples are in fact well-defined. Generic splittings over unramified extension of Q_{3} correspond to the maps

\overline{F}_{3} ⊗_{F_{3}} ℳ(0, 1) → \overline{F}_{3} ⊗_{F_{3}} ℳ

given by

e ↦ au^{3}e_{ω} + u(u^{2}e′_{ω} + cu^{2}e_{ω}),

where a ∈ \overline{F}_{3} satisfies a^{3} = a + c. Such generic splittings can be defined over Q_{3} (i.e. without extending the residue field) if and only if c = 0.

Lemma 9.3.4. 1. The map of groups

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,1}, ℳ′_{6,1}) → Ext_{F_{3}[G_{3}]}^{1}(1, 1)

is an isomorphism.

Explicitly, the group Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,1}, ℳ′_{6,1}) is parameterized by pairs (c, c′) ∈ F_{3}^{2} corresponding to

ℳ = (F_{3}[u]/u^{36})e_{1} ⊕ (F_{3}[u]/u^{36})e′_{1}, ℳ_{1} = ⟨u^{6}e_{1}, u^{6}e′_{1} + (cu^{2} + c′u^{6})e_{1}⟩,

with

ϕ_{1}(u^{6}e_{1}) = e_{1}, ϕ_{1}(u^{6}e′_{1} + (cu^{2} + c′u^{6})e_{1}) = e′_{1}, N(e_{1}) = 0, N(e′_{1}) = −cu^{24}e_{1}

and descent data

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{1}) = −\sqrt{−1}e′_{1},
γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{1}) = e′_{1} + c(±u^{6} ± u^{18} − u^{24} ± u^{30})e_{1}.

The classes in Ext_{F_{3}[G_{3}]}^{1}(1, 1) which split over an unramified extension of Q_{3} correspond to the pairs with c = 0.

2. The map of groups

Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,ω}) → Ext_{F_{3}[G_{3}]}^{1}(ω, ω)

is an isomorphism.

Explicitly, the group Ext_{\underline{ϕ_{1}DD}_{F′/Q_{3}}}^{1} (ℳ′_{6,ω}, ℳ′_{6,ω}) is parameterized by pairs (c, c′) ∈ F_{3}^{2} corresponding to

ℳ = (F_{3}[u]/u^{36})e_{ω} ⊕ (F_{3}[u]/u^{36})e′_{ω}, ℳ_{1} = ⟨u^{6}e_{ω}, u^{6}e′_{ω} + (cu^{2} + c′u^{6})e_{ω}⟩,

with

ϕ_{1}(u^{6}e_{ω}) = e_{ω}, ϕ_{1}(u^{6}e′_{ω} + (cu^{2} + c′u^{6})e_{ω}) = e′_{ω}, N(e_{ω}) = 0, N(e′_{ω}) = −cu^{24}e_{ω}

and descent data

γ̂_{4}(e_{ω}) = \sqrt{−1}e_{ω}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{ω}) = e_{ω}, γ̂_{3}^{±1}(e′_{ω}) = e′_{ω} + c(±u^{6} ± u^{18} − u^{24} ± u^{30})e_{ω}.

The classes in Ext_{F_{3}[G_{3}]}^{1}(ω, ω) which split over an unramified extension of Q_{3} correspond to the pairs with c = 0.

Proof. We treat the first part of the lemma; replacing \sqrt{−1} with −\sqrt{−1} throughout gives the proof of the second part.

As usual, we can find an ordered F_{3}[u]/u^{36}-basis e_{1}, e′_{1} of ℳ so that

ℳ_{1} = ⟨u^{6}e_{1}, u^{6}e′_{1} + he_{1}⟩, ϕ_{1}(u^{6}e_{1}) = e_{1}, ϕ_{1}(u^{6}e′_{1} + he_{1}) = e′_{1},

and γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{1}) = −\sqrt{−1}e′_{1}. The conditions γ̂_{4}(ℳ′_{1}) ⊆ ℳ′_{1} and γ̂_{4} ∘ ϕ′_{1} = ϕ′_{1} ∘ γ̂_{4} on ℳ′_{1} amount to

h(u) ≡ −h(−\sqrt{−1}u) mod u^{18}.

Since {u^{6}t − u^{6}t^{3}|t ∈ F_{3}[u]/u^{36}} consists of multiples of u^{7}, we can change the choice of e′_{1} so that

h = cu^{2} + c′u^{6}

for some c, c′ ∈ F_{3}, where we may a priori lose the diagonal form of γ̂_{4}. But the same kind of calculation as in Lemma 9.3.3 shows γ̂_{4}(e′_{1}) = −\sqrt{−1}e′_{1} + a\sqrt{−1}e_{1} for some a ∈ F_{3}, so the condition γ_{4}^{4} = 1 forces a = 0 (i.e. γ̂_{4} still has diagonal action).

It is straightforward to compute the asserted formula for N, and then the wild descent data can be computed exactly as in our previous computations of wild descent data; this yields the formulas

γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{1}) = e′_{1} + (c_{±1} + c(±u^{6} ± u^{18} − u^{24} ± u^{30}))e_{1},

where c_{±1} ∈ F_{3}. Modulo u, the linear action of γ_{3}^{±1}γ_{4}γ_{3}^{±1}γ_{4}^{3} sends e′_{1} to e′_{1} − c_{±1}e_{1}, but γ_{3}^{±1}γ_{4}γ_{3}^{±1}γ_{4}^{3} = 1, so c_{ϵ} = 0 for ϵ = ±1. Thus, we obtain the asserted list of possibilities. The well-definedness of these examples follows from Lemma 5.2.2 and Corollary 5.6.2.

It is easy to see that there is a non-zero map \overline{F}_{3} ⊗_{F_{3}} ℳ(0, 1) → \overline{F}_{3} ⊗_{F_{3}} ℳ if and only if c = 0, in which case such non-zero maps are precisely those induced by

e ↦ au^{9}e_{1} + u^{3}(u^{6}e′_{1} + c′u^{6}e_{1}),

where a ∈ \overline{F}_{3} satisfies a^{3} = a + c′. The verification that c = c′ = 0 corresponds to being in the kernel of our map of Ext^{1}’s is now clear, since X^{3} = X + c′ has a solution in F_{3} if and only if c′ = 0.

9.4. Completion of the proof of Theorem 4.6.1. Everything in Theorem 4.6.1 is now clear except for the third assertion, to the proof of which we now turn. Let (𝒢′, {[g]}) be as in the third part of that theorem. We may suppose that 𝒢′ = 𝒢 ×_{𝒪_{F}} 𝒪_{F′} for some 𝒢_{/𝒪_{F}} . The filtration on \overline{ρ} ⊗_{F_{3}} k gives a filtration

(0) ⟶ 𝒢_{ω} ⟶ 𝒢 ⟶ 𝒢_{1} ⟶ (0),

which is compatible with the descent data over Q_{3}. According to Lemma 5.2.3 we have ℳ_{π}(𝒢_{ω}) ≅ ℳ(k; r_{ω}, f_{ω}) and ℳ_{π}(𝒢_{1}) ≅ ℳ(k; r_{1}, f_{1}) for some 0 ≤ r_{1}, r_{ω} ≤ 12 and some f_{1}, f_{ω} ∈ k[u]/u^{36}. We will let χ denote either 1 or ω. In particular ℳ_{π}(𝒢_{χ})_{1} = u^{r_{χ}}ℳ_{π}(𝒢_{χ}) for χ = 1, ω. From this one can conclude that if ℋ is an subquotient of 𝒢_{χ} then ℳ_{π}(ℋ)_{1} = u^{r_{χ}}ℳ_{π}(ℋ). Quite generally, for any Breuil module ℳ over 𝒪_{F} with ℳ_{1} = u^{r} ℳ and any short exact sequence of Breuil modules

0 → ℳ′ → ℳ → ℳ′′ → 0,

we must also have

ℳ′_{1} = u^{r} ℳ′, ℳ′′_{1} = u^{r} ℳ′′ .

Indeed, ℳ → ℳ′′ is a surjection taking ℳ_{1} onto ℳ′′_{1}, so the assertion for ℳ′′ is clear. Since ℳ′ is a F_{3}[u]/u^{36}-module direct summand of ℳ and

ℳ′_{1} = ℳ′ ∩ ℳ_{1} = ℳ′ ∩u^{r} ℳ,

the assertion for ℳ′ is likewise clear. We conclude that (ℳ_{π}(𝒢_{χ})′, {ĝ}) admits a filtration with successive quotients ℳ′_{r_{χ},χ}. Thus r_{χ} ∈ {2, 6, 10}.

Consider a fixed surjection of F_{3}[D_{3}]-modules

\overline{ρ} ⊗ k ↠ \overline{ρ}.

This gives rise to a finite flat 𝒪_{F} -group scheme ℋ with descent data on ℋ′ = ℋ×𝒪_{F′} over Q_{3} corresponding to \overline{ρ} and an epimorphism

𝒢 → → ℋ

compatible with descent data. Consider the commutative diagram

0 → ℳ_{π}(ℋ_{1}) → ℳ_{π}(ℋ) → ℳ_{π}(ℋ_{ω}) → 0
↓ ↓ ↓
0 → ℳ_{π}(𝒢_{1}) → ℳ_{π}(𝒢) → ℳ_{π}(𝒢_{ω}) → 0

where the top row corresponds to the non-split filtration of \overline{ρ}. The middle vertical map is an isomorphism of the source onto a F_{3}[u]/u^{36}-module direct summand of the target, so the left vertical map is as well, because an injection of F_{3}[u]/u^{36} into a free F_{3}[u]/u^{36}-module must be an identification with such a direct summand (consider torsion). This forces ℳ_{π}(ℋ_{1})′ = ℳ′_{r_{1},1} and so, by Proposition 9.2.1, we see that r_{1} ≠ 2. Repeating the analogous argument applied to a submodule \overline{ρ} ⊆ \overline{ρ} ⊗ k one sees that r_{ω} ≠ 10.

Thus (𝒢′, {[g]}) is weakly filtered by {𝒢_{s,1}, 𝒢_{r,ω}} for (r, s) = (2, 6), (6, 10), (2, 10) or (6, 6), as desired.

9.5. Completion of the proof of Theorem 4.6.3. Write A_{N} for F_{3}[[T]]/(T^{N}). For (r, s) = (2, 6), (6, 10) and (2, 10), we will define a Breuil module ℳ_{N,(r,s)} over 𝒪_{F} and descent data {ĝ} for Gal(F′/Q_{3}) on ℳ′_{N,(r,s)} = ℳ_{N,(r,s)} ⊗_{F_{3}} F_{9} such that ℳ_{N,(r,s)} and (ℳ′_{N,(r,s)}, {ĝ}) have compatible actions of A_{N} (and γ̂_{2} = 1 ⊗ Frob_{3}). More specifically set t = 2, 6 or 8 according as (r, s) = (2, 6), (6, 10) or (2, 10). Viewing \overline{ρ} as an extension class, it corresponds to a particular pair (c, c_{1}) ∈ F_{3}^{2} in Proposition 9.2.1. Fix these values. Motivated by the idea of deforming the formulae in Proposition 9.2.1, we are led to define

ℳ_{N,(r,s)} = (A_{N}[u]/u^{36})e_{1} ⊕ (A_{N}[u]/u^{36})e′_{ω}, (ℳ_{N,(r,s)})_{1} = ⟨u^{s}e_{1}, u^{r}e′_{ω} + (c + T)u^{t}e_{1}⟩

with

ϕ_{1}(u^{s}e_{1}) = e_{1}, ϕ_{1}(u^{r}e′_{ω} + (c + T)u^{t}e_{1}) = e′_{ω}.

It is straightforward to check that N ∘ϕ_{1} = 0 on ℳ_{N,(r,s)}. We may define A_{N}-linear descent data on ℳ′_{N,(r,s)} by setting γ̂_{2} = 1 ⊗ Frob_{3} and using the following formulae.

1. When (r, s) = (2, 6), set

γ̂_{4}(e_{1}) = −\sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})(e′_{ω} ± c_{1}u^{6}e_{1}).

2. When (r, s) = (6, 10), set

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = (1 ∓ u^{18})e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (e′_{ω} ± c_{1}u^{6}e_{1}).

3. When (r, s) = (2, 10), set

γ̂_{4}(e_{1}) = \sqrt{−1}e_{1}, γ̂_{4}(e′_{ω}) = −\sqrt{−1}e′_{ω},
γ̂_{3}^{±1}(e_{1}) = (1 ∓ u^{18})e_{1}, γ̂_{3}^{±1}(e′_{ω}) = (1 ± u^{18})(e′_{ω} ± c_{1}u^{12}e_{1}).

It is readily checked that this defines an object of \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ} with an action of A_{N}. Let 𝒢_{N,(r,s)} and (𝒢′_{N,(r,s)}, {[g]}) be the corresponding finite flat 𝒪_{F} -group scheme and finite flat 𝒪_{F′}-group scheme with descent data.

If 1 ≤ M < N then we have a short exact sequence in \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ}

(0) ⟶ ℳ′_{M,(r,s)} ⟶ ℳ′_{N,(r,s)} ⟶ ℳ′_{N−M,(r,s)} ⟶ (0),

where the first map is induced by multiplication by T^{N−M}. The case M = 1 shows that

(𝒢_{N,(r,s)}, {[g]})_{Q_{3}}/T (𝒢_{N,(r,s)}, {ĝ})_{Q_{3}}

corresponds to \overline{ρ}. Thus we get a surjection of A_{N}[G_{3}]-modules A_{N}^{2} → → 𝒢_{N,(r,s)}(\overline{Q}_{3}), which must in fact be an isomorphism (count orders). Thus (𝒢_{N,(r,s)}, \{[g]\})_{Q_{3}} defines a deformation ρ_{N,(r,s)} of \overline{ρ} to A_{N}^{2}. For N ≥ 2 we have ρ_{N,(r,s)} mod T^{2} ≅ ρ_{2,(r,s)}.

We also have an exact sequence

(0) ⟶ ℳ′_{s,1} ⊗_{F_{3}} A_{N} ⟶ (ℳ′_{N,(r,s)}, {ĝ}) ⟶ ℳ′_{r,ω} ⊗_{F_{3}} A_{N} ⟶ (0)

in \underline{ϕ_{1}DD}_{F′/Q_{3}}, from which we obtain an exact sequence of A_{N}[G_{3}]-modules

(0) ⟶ X_{ω} ⟶ ρ_{N} ⟶ X_{1} ⟶ (0).

Note that X_{1} ≅ F_{3}^{N} and X_{ω} ≅ F_{3}(ω)^{N} as F_{3}[G_{3}]-modules. Moreover, this sequence must split as a sequence of A_{N}-modules. (Use, for instance, the kernel of ρ_{N}(σ) − 1 for any σ ∈ G_{3} − G_{Q_{3}(\sqrt{−3})}.) Thus X_{1} ≅ A_{N} and X_{ω} ≅ A_{N}(ω) as A_{N}[G_{3}]-modules, so det ρ_{N} = ω.

Finally, we must check that the exact sequence

(0) ⟶ \overline{ρ} ⟶ ρ_{2} ⟶ \overline{ρ} ⟶ (0)

is not split. We have maps of Breuil modules

\begin{matrix}f_{1} : ℳ_{s,1} & ⟶ & ℳ_{2,(r,s)} \\ e & ⟼ & e_{1} \end{matrix}

and

\begin{matrix}f_{2} : ℳ_{2,(r,s)} & ⟶ & ℳ_{r,ω} \\ e_{1} & ⟼ & 0 \\ T e_{1} & ⟼ & 0 \\ e′_{ω} & ⟼ & 0 \\ T e′_{ω} & ⟼ & e \end{matrix}

compatible with descent data. These give rise to maps

f_{1}^{∗} : ρ_{2} ⟶ 1

and

f_{2}^{∗} : ω ⟶ ρ_{2},

such that the composites

\overline{ρ} ↪ ρ_{2} ⟶^{f_{1}^{∗}} 1

and

ω ⟶^{f_{2}^{∗}} ρ_{2} → → \overline{ρ}

are non-zero.

To check that

(0) ⟶ \overline{ρ} ⟶ ρ_{2} ⟶ \overline{ρ} ⟶ (0)

is non-split, it suffices to check that

(0) ⟶ ω ⟶ ker f_{1}^{∗}/ Im f_{2}^{∗} ⟶ 1 ⟶ (0) (9.5.1)

is non-split. However ker f_{1}^{∗}/ Im f_{2}^{∗} corresponds to an object (𝒩′, \{ĝ\}) of \underline{ϕ_{1}DD}_{F′/Q_{3},ℐ} satisfying

𝒩′ = (F_{9}[u]/u^{36})(T e_{1}) ⊕ (F_{9}[u]/u^{36})e′_{ω}, 𝒩′_{1} = ⟨u^{s}(T e_{1}), u^{r}e′_{ω} + u^{t}(T e_{1})⟩

with

ϕ_{1}(u^{s}(T e_{1})) = (T e_{1}), ϕ_{1}(u^{r}e′_{ω} + u^{t}(T e_{1})) = e′_{ω}.

By Lemma 5.2.2, the sequence of Breuil modules with descent data

(0) ⟶ ℳ′_{s,1} ⟶ 𝒩′ ⟶ ℳ′_{r,ω} ⟶ (0)

is not split. This sequence recovers (9.5.1) under generic fibre descent, so by Proposition 9.2.1

(0) ⟶ ω ⟶ ker f_{1}^{∗}/ Im f_{2}^{∗} ⟶ 1 ⟶ (0)

is not split.

9.6. Completion of the proof of Theorem 4.6.2. Suppose first that (r, s) = (2, 6), (6, 10) or (2, 10). By Lemma 9.3.1

θ_{0} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω)

is the zero map. Lemma 9.3.3 then tells us that if r ≠ 6 then

\overline{θ}_{1} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3})

is the zero map; while if s ≠ 6 then

\overline{θ}_{ω} : Ext_{𝒮_{i,(r,s)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3})

is the zero map. Thus Theorem 4.7.5, and hence Theorem 4.6.2, follows in these cases.

Now consider the case (r, s) = (6, 6). Choose x ∈ H_{𝒮_{i,(6,6)}}^{1}(G_{3}, ad^{0} \overline{ρ}). Let 𝒢 denote the corresponding rank 81 finite flat 𝒪_{F} -group scheme with descent data {[g]} on 𝒢′ = 𝒢 ×_{𝒪_{F}} 𝒪_{F′}. Set ℳ = ℳ_{π}(𝒢). Let ℋ ⊂ 𝒢 denote the closed subgroup scheme (with descent data) corresponding to the kernel of the map (𝒢′, \{[g]\})_{Q_{3}} → → \overline{ρ} → → F_{3} and let 𝒩 = ℳ_{π}(ℋ). Then 𝒩 has F_{3}[u]/u^{36}-basis e_{ω}, e′_{1}, e′_{ω} with respect to which

𝒩_{1} = ⟨u^{6}e_{ω}, u^{6}e′_{1} + (b + b′u^{4})e_{ω}, u^{6}e′_{ω} + (c + c′u^{4})e′_{1} + fe_{ω}⟩,

where b, b′, c, c′ ∈ F_{3}, f ∈ F_{3}[u]/u^{36}, and ϕ_{1} sends the indicated generators of 𝒩_{1} to e_{ω}, e′_{1}, e′_{ω} respectively. Also, the descent data has the form

γ̂_{4}(e_{ω}) = \sqrt{−1}e_{ω}, γ̂_{4}(e′_{1}) = −\sqrt{−1}e′_{1}, γ̂_{4}(e′_{ω}) = \sqrt{−1}e′_{ω} + h_{γ_{4}}(u)e_{ω}

for some h_{γ_{4}} ∈ F_{9}[u]/u^{36}, and

γ̂_{3}^{±1}(e_{ω}) = e_{ω}, γ̂_{3}^{±1}(e′_{1}) = e′_{1} + (±b − b′(±u^{12} + u^{30}))e_{ω}, γ̂_{3}^{±1}(e′_{ω}) = e′_{ω} + (±c − c′(±u^{12} + u^{30}))e′_{1} + h_{±1}e_{ω},

where h_{±1} ∈ F_{9}[u]/u^{36}. Also, as \overline{ρ} is très ramifié, we see that c ≠ 0 by Proposition 9.2.1 and Lemma 9.3.4. The requirement that u^{12} 𝒩 ⊆ 𝒩_{1} forces 𝒩_{1} to contain

u^{12}e′_{ω} = u^{6}(u^{6}e′_{ω} + (c + c′u^{4})e′_{1} + fe_{ω}) − (c + c′u^{4})(u^{6}e′_{1} + (b + b′u^{4})e_{ω}) + ((b + b′u^{4})(c + c′u^{4}) − fu^{6})e_{ω},

so 𝒩_{1} must contain (b + b′u^{4})(c + c′u^{4})e_{ω}. As c ≠ 0 we get (b + b′u^{4})e_{ω} ∈ 𝒩_{1}, and since e_{ω}, u^{4}e_{ω} ∉ 𝒩_{1}, we must have b = b′ = 0. We conclude that the natural map

θ_{0} : Ext_{𝒮_{i,(6,6)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(G_{3}, ω)

is the zero map.

Let us further analyse 𝒩. Replacing e′_{ω} by e′_{ω} + t^{3}e_{ω} for t ∈ F_{3}[u]/u^{36} causes f to be replaced by f −u^{6}t^{3} +u^{6}t and otherwise leaves our standardized form unchanged (except that h_{γ_{4}} and h_{±1} may change). Using a suitable choice of such t, we may assume f has degree at most 6. On the other hand,

γ̂_{4}(u^{6}e′_{ω} +(c+c′u^{4})e′_{1} +fe_{ω}) = −\sqrt{−1}(u^{6}e′_{ω} +(c+c′u^{4})e′_{1} +fe_{ω})+(\sqrt{−1}(f(u)+f(−\sqrt{−1}u))−u^{6}h_{γ_{4}}(u))e_{ω},

so γ̂_{4}(𝒩′_{1}) ⊆ 𝒩′_{1} if and only if

f(u) + f(−\sqrt{−1}u) ≡ 0 mod u^{6},

which forces

f = a_{2}u^{2} + a_{6}u^{6}

for some a_{2}, a_{6} ∈ F_{3}. From the wild descent data formulae derived in the proof of Lemma 9.3.4 we also see that h_{±1} ≡ 0 mod u^{6}.

Now ℳ has an ordered basis e_{1}, e_{ω}, e′_{1}, e′_{ω} with respect to which

(9.6.1) ℳ_{1} = ⟨u^{6}e_{1}, u^{6}e_{ω} + (c + c′u^{4})e_{1}, u^{6}e′_{1} + he_{1}, u^{6}e′_{ω} + (c + c′u^{4})e′_{1} + (a_{2}u^{2} + a_{6}u^{6})e_{ω} + ge_{1}⟩,

where g, h ∈ F_{3}[u]/u^{36} and ϕ_{1} sends the indicated generators of ℳ_{1} to e_{1}, e_{ω}, e′_{1}, e′_{ω}. If we try to expand out u^{12}e′_{ω} as a linear combination of the indicated generators of ℳ_{1}, we find that

u^{12}e′_{ω} ≡ ((c + c′u^{4})h + ca_{2}u^{2})e_{1} mod ℳ_{1} .

It follows that u^{12}e′_{ω} ∈ ℳ_{1} if and only if

(c + c′u^{4})h + ca_{2}u^{2} ≡ 0 mod u^{6}.

Since \overline{ρ} is très ramifié, the last part of Proposition 9.2.1 tells us that c ≠ 0. Thus, u^{12}e′_{ω} ∈ ℳ_{1} if and only if h ≡ −a_{2}u^{2} mod u^{6}. We can now use Lemma 9.3.4 to see that the wild descent data action is determined by

γ̂_{3}^{±1}(e_{1}) = e_{1}, γ̂_{3}^{±1}(e_{ω}) = e_{ω} + (±c − c′(±u^{12} + u^{30}))e_{1}, γ̂_{3}^{±1}(e′_{1}) = e′_{1} + f_{±1}e_{1}

(with f_{±1} ≡ 0 mod u^{6}), and

γ̂^{±1}(e′_{ω}) = e′_{ω} + (±c − c′(±u^{12} + u^{30}))e′_{1} + h_{±1}e_{ω} + g_{±1}e_{1}

where g_{±1} ∈ F_{9}[u]/u^{36} and h_{±1} ≡ 0 mod u^{6}.

We must have

γ̂_{3}^{±1}(u^{6}e′_{ω} + (c + c′u^{4})e′_{1} + (a_{2}u^{2} + a_{6}u^{6})e_{ω} + g(u)e_{1}) ∈ ℳ′_{1}, (9.6.2)

and this expression is easily computed to equal

u^{6}H_{γ_{3}^{±1}}^{6} · (e′_{ω} + (±c − c′(±u^{12} + u^{30}))e′_{1} + h_{±1}e_{ω} + g_{±1}e_{1}) + (c + c′u^{4}H_{γ_{3}^{±1}}^{4})(e′_{1} + f_{±1}e_{1})
+(a_{2}u^{2}H_{γ_{3}^{±1}}^{2} + a_{6}u^{6}H_{γ_{3}^{±1}}^{6})(e_{ω} + (±c − c′(±u^{12} + u^{30}))e_{1}) + g(uH_{γ_{3}^{±1}})e_{1}.

Remembering that ⟨u^{6}e_{1}, u^{12} ℳ′⟩ ⊆ ℳ′_{1}, (9.6.2) becomes

u^{6}(e′_{ω} ± ce′_{1}) + (c + c′u^{4}H_{γ_{3}^{±1}}^{4})e′_{1} + (a_{2}u^{2}H_{γ_{3}^{±1}}^{2} + a_{6}u^{6})e_{ω} ± a_{2}cu^{2}e_{1} + g(u)e_{1} ∈ ℳ′_{1}.

Using the explicit generators of ℳ_{1} given in (9.6.1) and recalling that h ≡ −a_{2}u^{2} mod u^{6}, this simplifies to

±a_{2}cu^{2}e_{1} ∈ ℳ′_{1} .

Thus a_{2}cu^{2} is divisible by u^{6}, so a_{2} = 0.

The image of the class x in Ext_{F_{3}[G_{3}]}^{1}(ω, ω) under θ_{ω} corresponds to a finite flat 𝒪_{F} -group scheme with Breuil module ℳ_{x} free of rank two over F_{3}[u]/u^{36} with basis e_{ω}, e′_{ω}, and with

(ℳ_{x})_{1} = ⟨u^{6}e_{ω}, u^{6}e′_{ω} + a_{6}u^{6}e_{ω}⟩,

where ϕ_{1} sends the indicated generators of (ℳ_{x})_{1} to e_{ω} and e′_{ω} respectively. According to the proof of Lemma 9.3.4 this implies that the image of the class x in Ext_{F_{3}[G_{3}]}^{1}(ω, ω) is split over an unramified extension of Q_{3}. Thus,

\overline{θ}_{ω} : Ext_{𝒮_{i,(6,6)}}^{1}(\overline{ρ}, \overline{ρ}) ⟶ H^{1}(I_{3}, F_{3})

is the zero map. This completes the proof of Theorem 4.7.5, and hence of Theorem 4.6.2.

10. Corrigenda for [CDT].

We would like to take this opportunity to record a few corrections to [CDT].

• Page 532, line -6: “The semisimplicity of σ_{n} follows from that of σ_{1}” is false and should be deleted. This assertion was not used anywhere in the rest of the paper.

• Page 538, line -10: replace “of type (S, τ)” by “such that ρ|_{G_{ℓ}} is of type τ and ρ is of type (S, τ)”.

• Page 539, lines 18-20: replace each ω_{1} by η_{1} and each ω_{2} by η_{2}.

• Page 544, line -6: “the discrete topology on V_{p}” should read “the ℓ-adic topology on M_{p}”.

• Page 545, part 4 of Lemma 6.1.2: V′ should be assumed to be a normal subgroup of V.

• Page 546, line 1: We should have noted that the key component of this argument is very similar to the main idea of [Kh].

• §6.2: There are two significant errors in this section. The assertion “Γ = SL_{2}(Z) ∩ U_{S} satisfies the hypotheses of Theorem 6.1.1” is false and Hom(L_{n}, k) should be L_{n} ⊗ k. The argument of this section can be repaired by making the following changes.

– Page 546, lines 5 and 6: Replace “Setting S = T (\overline{ρ}) ∪ \{r\}, we find that the group Γ = SL_{2}(Z) ∩ U_{S} satisfies the hypotheses of Theorem 6.1.1.” by “Set S = T (\overline{ρ}) ∪ \{r\}; U′_{S} = ∏_{p} U′_{S,p} where U′_{S,p} = U_{1}(p^{c_{p}}) if p ∈ T (\overline{ρ}) and U′_{S,p} = U_{S,p} otherwise; V′_{S} = ∏_{p} V′_{S,p} where V′_{S,p} = U_{1}(p^{c_{p}}) if p ∈ T (\overline{ρ}) and V′_{S,p} = V_{S,p} otherwise; and L′_{S} = Hom_{𝒪[U′_{S}/V′_{S}]} (M_{ℓ}, H^{1}(X_{V′_{S}}, 𝒪))[I′_{S}]. Then Γ = SL_{2}(Z) ∩ U′_{S} satisfies the hypotheses of Theorem 6.1.1.”

– Page 546, lines 7-13: Replace Y_{S} by Y_{U′_{S}}, Hom(L_{n}, k) by L_{n} ⊗ k, M_{S} by M_{ℓ}, ℱ_{S} by ℱ_{Hom_{𝒪} (M_{ℓ},𝒪)} and L_{S} by L′_{S}.

– Page 546, line 13: Replace “ and 𝒩_{S} is non-empty.” by “. Using the fact that lemma 5.1.1 holds with U′_{S} replacing U_{S} and σ_{ℓ} replacing σ_{S} and the discussion on page 541 we conclude that 𝒩_{S} is non-empty.”

• Page 549, line -15: Replace U_{{r,r′},p} by U_{{r},p}.

• Page 549, line -11: Replace U′_{S}/U′_{0,S} by V_{0}/V_{1}.

• Page 552, line 4: The assertion is false in the case ℓ ≥ 5. It can be corrected by adding “and j(E) ≢ 1728 mod ℓ” after “if ℓ ≥ 5”.

• Page 553, line -5: After “e(F) = 3.” add the following explanation. “(If E has potentially supersingular reduction at 5 and 5 ≡ 1 mod 4 we must have j(E) ≢ 1728 mod 5 and so E acquires good reduction over an extension of Q_{5} with ramification index dividing 6.)”

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Département de Mathématiques, CNRS, Université Paris-Sud, 91405 Orsay cedex, FRANCE.

E-mail address: Christophe.BREUIL@math.u-psud.fr

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.

E-mail address: bconrad@math.harvard.edu

Department of Mathematics, Brandeis University, Waltham, MA 02454, USA.

E-mail address: fdiamond@euclid.math.brandeis.edu

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.

E-mail address: rtaylor@math.harvard.edu