Potentially Crystalline Lifts

of Certain Prescribed Types

Toby Gee, Florian Herzig, Tong Liu, David Savitt^{1}

Received: December 15, 2015 Revised: January 11, 2017

Communicated by Don Blasius

Abstract. We prove several results concerning the existence of potentially crystalline lifts of prescribed Hodge–Tate weights and inertial types of a given representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}), where K/ℚ_{p} is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.

1. Introduction

Let p be a prime, let K/ℚ_{p} be a finite extension, and let \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) be a continuous representation. For many reasons, it is a natural and important question to study the lifts of \overline{r} to de Rham representations r : G_{K} → GL_{n}(\overline{ℤ}_{p}); for example, the de Rham lifts of fixed Hodge and inertial types are parameterised by a universal (framed) deformation ring thanks to [Kis08], and the study of these deformation rings is an important step in proving automorphy lifting theorems, going back to Wiles’ proof of Fermat’s Last Theorem, which made use of Ramakrishna’s work on flat deformations [Ram02]. It is therefore slightly vexing that (as far as we are aware) it is currently an open problem to prove that for a general choice of \overline{r}, a single such lift r exists (equivalently, to show for each \overline{r} that at least one of Kisin’s deformation rings is nonzero). Some results in this direction can be found in the Ph.D. thesis of Alain Muller [Mul13]. This note sheds little further light on this question, but rather investigates the question of congruences between de Rham representations of different Hodge and inertial types; that is, in many of our results we suppose the existence of a single lift, and see what other lifts (of differing Hodge and inertial types) we can produce from this. The existence of congruences between representations of differing such types is conjecturally governed by the (generalised) Breuil–Mézard conjecture (at least for regular Hodge types; see [EG14]). This conjecture is almost completely open beyond the case of GL_{2} /ℚ_{p}, so it is of interest to prove unconditional results. We prove several such results in this paper, by a variety of different methods. Some of our results make use of the notion of a potentially diagonalisable Galois representation, which was introduced in [BLGGT14], and is very important in automorphy lifting theorems. It is expected ([EG14, Conj. A.3]) that every \overline{r} admits a potentially diagonalisable lift of regular weight^{2}, but this is at present known only if n ≤ 3 or \overline{r} is semisimple; see for example [CEG^{+}16, Lem. 2.2], and the proof of [Mul13, Prop. 2.5.7] for the case n = 3. It seems plausible that these arguments could be extended to cover other small dimensions, but the case of general n seems to be surprisingly difficult. We recall that an n-dimensional de Rham representation of G_{K} is said to have Hodge type 0 if for any continuous embedding K ↪ \overline{ℚ}_{p} the corresponding Hodge–Tate weights are 0, 1, …, n−1; while if K/ℚ_{p} is unramified, a crystalline representation of G_{K} is said to be Fontaine–Laffaille if for each continuous embedding K ↪ \overline{ℚ}_{p} the corresponding Hodge–Tate weights are all contained in an interval of the form [i, i + p − 2]. We remark that we will normalise Hodge–Tate weights so that the cyclotomic character ε has Hodge–Tate weight −1. Our first result is the following theorem, which will be used in forthcoming work of Arias de Reyna and Dieulefait.

1The first author was partially supported by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326. The second author was partially supported by a Sloan Fellowship and an NSERC grant The third author was partially supported by NSF grants DMS-0901360 and DMS-1406926 The fourth author was partially supported by NSF grant DMS-0901049 and NSF CAREER grant DMS-1054032

Theorem A. (Cor. 2.3.4) Suppose that K/ℚ_{p} is unramified, and fix an integer n ≥ 1. Then there is a finite extension K′/K, depending only on n and K, with the following property: if \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) has a Fontaine–Laffaille lift, then it also has a potentially diagonalisable lift r : G_{K} → GL_{n}(\overline{ℤ}_{p}) of Hodge type 0 with the property that r|_{G_{K′}} is crystalline.

In fact this is a special case of a result (Cor. 2.1.11) that holds for a more general class of representations \overline{r} that we call peu ramifiée, and with no assumption that the finite extension K/ℚ_{p} is unramified. We expect that this result should even be true without the assumption that \overline{r} is peu ramifiée, but we do not know how to prove this; indeed, as mentioned above, we do not know how to produce a single de Rham lift in general! To explain why this result is reasonable, and to give some indication of the proof, we focus on the case that K = ℚ_{p} and n = 2. Assume for simplicity in the following discussion that p > 2. One way to see that we should expect the result to be true (at least if we remove “potentially diagonalisable” from the statement) is that it is then the local Galois analogue of the well-known statement that every modular eigenform of level prime to p is congruent to one of weight 2 and bounded level at p. Indeed, via the mechanism of modularity lifting theorems and potential modularity, it is possible to turn this analogy into a proof. (See Theorem C below. Since all potentially Barsotti–Tate representations are known to be potentially diagonalisable, this literally proves Theorem A in this case, but this deduction cannot be made if n > 2.) Since these global methods are (at least at present) unable to handle the case n > 2, a local approach is needed, which we again motivate via the case K = ℚ_{p} and n = 2. The possible \overline{r} : G_{ℚ_{p}} → GL_{2}(\overline{𝔽}_{p}) are well-understood; they are either irreducible representations, in which case they are induced from characters of the unramified quadratic extension ℚ_{p^{2}} of ℚ_{p}, or they are reducible, and are extensions of unramified twists of powers of the mod p cyclotomic character ω. In the first case, the representations are induced from characters of G_{ℚ_{p^{2}}} which become unramified after restriction to any totally ramified extension of degree p^{2} − 1, and it is straightforward to produce the required lifts by considering inductions of potentially crystalline characters of G_{ℚ_{p^{2}}} which become crystalline over such an extension; see Lemma 2.1.12. Such representations are automatically potentially diagonalisable, as after restriction to some finite extension they are even a direct sum of crystalline characters. This leaves the case that \overline{r} is reducible. After twisting, we may assume that \overline{r} is an extension of an unramified twist of ω^{−i} by the trivial character, for some 0 ≤ i ≤ p − 2. Then the natural way to lift to characteristic zero and Hodge type 0 is to try to lift to an extension of an unramified twist of ε^{−1}ω^{1−i} by the trivial character, where ω̃ is the Teichmüller lift of ω; this is promising because any such extension is at least potentially semistable, and becomes semistable over ℚ_{p}(ζ_{p}) (which is in particular independent of the specific reducible \overline{r} under consideration), and if it is potentially crystalline, then it is also potentially diagonalisable (as it is known that any successive extension of characters which is potentially crystalline is also potentially diagonalisable). The problem of producing such lifts is one of Galois cohomology, and Tate’s duality theorems show that when i ≠ 1 there is no obstruction to lifting.^{3} It is also easy to check that in this case the lifts are automatically potentially crystalline. However, when i = 1 the situation is more complicated. Then one can check that très ramifiée extensions of ω^{−1} by the trivial character do not lift to extensions of a non-trivial unramified twist of ε^{−1} by the trivial character, but only lift to semistable non-crystalline extensions of ε^{−1} by the trivial character. However, this is the only obstruction to carrying out the strategy in this case; and in fact, since très ramifiée representations do not have Fontaine–Laffaille lifts, the result also follows in the case i = 1. We prove Theorem A by a generalisation of this strategy: we write \overline{r} as an extension of irreducible representations, lift the irreducible representations as inductions of crystalline characters, and then lift the extension classes. However, the issues that arose in the previous paragraph in the case i = 1 are more complicated in general. To address this, we make use of the following observation: in the case considered in the previous paragraphs (that is, K = ℚ_{p}, n = 2, and \overline{r} has a trivial subrepresentation), if \overline{r} is not très ramifiée then it admits “many” reducible crystalline lifts; indeed, it can be lifted as an extension by the trivial character of any unramified twist of ε^{−i} that lifts the corresponding character mod p. This freedom to twist by unramified characters is in marked contrast to the behaviour in the très ramifiée case, and can be exploited in the Galois cohomology calculations used to produce the potentially crystalline lifts of Hodge type 0. Motivated by these observations, we introduce a generalisation (Definition 2.1.3) of the classical notion of peu ramifiée representations, and we prove by direct Galois cohomology arguments that the peu ramifiée condition allows great flexibility in the production of lifts to varying reducible representations (see Theorem 2.1.8 and Corollary 2.1.11). Conversely, every representation that admits enough lifts of the sort promised by Theorem 2.1.8 must in fact be peu ramifiée (see Proposition 2.2.4 for a precise statement); such a representation is said to admit “highly twisted lifts.” We show that representations that admit Fontaine–Laffaille lifts also admit highly twisted lifts (Proposition 2.3.1), and so deduce that Corollary 2.1.11 applies whenever the residual representation is Fontaine–Laffaille. Theorem A follows. Using roughly the same purely local methods, we additionally prove the following.

2Recall that a de Rham representation of G_{K} is said to have regular weight if for any continuous embedding K ↪ \overline{ℚ}_{p}, the corresponding Hodge–Tate weights are all distinct.

3This is true even for the ramified self-extensions of the trivial character in the case i = 0, which are not Fontaine–Laffaille, although they are peu ramifiée in the sense of this paper (Definition 2.1.3).

Theorem B. (Cor. 2.1.13) Suppose that \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) is peu ramifiée. Then \overline{r} has a crystalline lift of some Serre weight (in the sense of Section 1.2.4).

In contrast to these relatively concrete local arguments, in Section 3 we use global methods, and in particular the potential automorphy machinery of [BLGGT14]. Our first result is the following, which takes as input a potentially crystalline lift that could have highly ramified inertial type, or highly spread out Hodge–Tate weights, and produces a crystalline lift of small Hodge– Tate weights.

Theorem C. (Thm. 3.1.2) Suppose that p ∤ 2n, and that \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) has a potentially diagonalisable lift of some regular weight. Then the following hold.

(1) There exists a finite extension K′/K (depending only on n and K, and not on \overline{r}) such that \overline{r} has a lift r : G_{K} → GL_{n}(\overline{ℤ}_{p}) of Hodge type 0 that becomes crystalline over K′.

(2) \overline{r} has a crystalline lift of some Serre weight.

The first part of this result should be contrasted with Theorem A above, while the second part should be contrasted with Theorem B. For instance, we remark that it follows from Theorem A (or more precisely, from its more general statement for peu ramifiée representations) that every peu ramifiée representation \overline{r} admits a potentially diagonalisable lift of some regular weight, whereas this latter condition on \overline{r} is an input to Theorem C. If K/ℚ_{p} is unramified and \overline{r} admits a lift of extended FL weight (see Section 1.2.4 for this terminology), we also show the following “weak Breuil– Mézard result”.

Theorem D. (Thm. 3.1.5) Suppose that p ≠ n, that K/ℚ_{p} is unramified, and that \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) has a crystalline lift of some extended FL weight F. If F is a Jordan–Hölder factor of \overline{σ}(λ, τ) for some λ, τ, then \overline{r} has a potentially crystalline lift of type (λ, τ).

Since there is no restriction on λ or τ, this result seems to be well beyond anything that can currently be proved directly using integral p-adic Hodge theory. If we knew that all potentially crystalline lifts were potentially diagonalisable, then the special case of Theorem A in which the given Fontaine–Laffaille lift is regular would be an easy consequence of part (1) of Theorem C (note that the existence of a regular Fontaine–Laffaille lift implies that p > n). However, we do not know how to prove that general potentially crystalline representations are potentially diagonalisable (and we do not have any strong evidence that it should be true).

1.1. Acknowledgements. We would like to thank Luis Dieulefait for asking a question which led to us writing this paper, as well as Alain Muller for valuable discussions.

1.2. Notation and conventions. Fix a prime p, and let K/ℚ_{p} be a finite extension with ring of integers O_{K}. Write G_{K} for the absolute Galois group of K, I_{K} for the inertia subgroup of G_{K}, and Frob_{K} ∈ G_{K} for a choice of geometric Frobenius. All representations of G_{K} are assumed without further comment to be continuous. Write v_{K} for the p-adic valuation on K taking the value 1 on a uniformiser of K, as well as for the unique extension of this valuation to any algebraic extension of K.

1.2.1. Inertial types. An inertial type is a representation τ : I_{K} → GL_{n}(\overline{ℚ}_{p}) with open kernel which extends to the Weil group W_{K}. We say that a de Rham representation r : G_{K} → GL_{n}(\overline{ℚ}_{p}) has inertial type τ if the restriction to I_{K} of the Weil–Deligne representation WD(r) associated to r is equivalent to τ. Given an inertial type τ, there is a (not necessarily unique) finite-dimensional smooth irreducible \overline{ℚ}_{p}-representation σ(τ) of GL_{n}(O_{K}) associated to τ by the “inertial local Langlands correspondence”, which we normalise as in [EG14, Conj. 4.1.3]. (Note that there is an unfortunate difference in conventions between this and that of [EG14, Thm. 4.1.5], but it is this normalisation that is used in the remainder of [EG14].) We can and do suppose that σ(τ) is defined over \overline{ℤ}_{p}.

1.2.2. Hodge–Tate weights and Hodge types. If W is a de Rham representation of G_{K} over \overline{ℚ}_{p}, and κ : K ↪ \overline{ℚ}_{p}, then we will write HT_{κ}(W) for the multiset of Hodge–Tate weights of W with respect to κ. By definition, the multiset HT_{κ}(W) contains i with multiplicity dim_{\overline{ℚ}_{p}}(W ⊗_{κ,K} ˆ \overline{K}(i))^{G_{K}}. Thus for example if ε denotes the p-adic cyclotomic character of G_{K}, then HT_{κ}(ε) = {−1} for all κ. We say that W has regular Hodge–Tate weights if for each κ, the elements of HT_{κ}(W) are pairwise distinct. Let ℤ_{+}^{n} denote the set of tuples (λ_{1}, …, λ_{n}) of integers with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}. Then if W has regular Hodge–Tate weights, there is a unique λ = (λ_{κ,i}) ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,\overline{ℚ}_{p})} such that for each κ : K ↪ \overline{ℚ}_{p},

HT_{κ}(W) = {λ_{κ,1} + n − 1, λ_{κ,2} + n − 2, …, λ_{κ,n}},

and we say that W is regular of Hodge type λ.

1.2.3. Representations of GL_{n} and Serre weights. For any λ ∈ ℤ_{+}^{n}, view λ as a dominant weight (with respect to the upper triangular Borel subgroup) of the algebraic group GL_{n} in the usual way, and let M′_{λ} be the algebraic O_{K}- representation of GL_{n} given by

M′_{λ} := Ind_{B_{n}}^{GL_{n}}(w_{0}λ)_{/O_{K}}

where B_{n} is the Borel subgroup of upper-triangular matrices of GL_{n}, and w_{0} is the longest element of the Weyl group (see [Jan03] for more details of these notions, and note that M′_{λ} has highest weight λ). Write M_{λ} for the O_{K}- representation of GL_{n}(O_{K}) obtained by evaluating M′_{λ} on O_{K}. For any λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,\overline{ℚ}_{p})} we write L_{λ} for the \overline{ℤ}_{p}-representation of GL_{n}(O_{K}) defined by

L_{λ} := ⊗_{κ:K↪\overline{ℚ}_{p}}M_{λ_{κ}} ⊗_{O_{K},κ} \overline{ℤ}_{p}.

Let k be the residue field of K. We call isomorphism classes of irreducible \overline{𝔽}_{p}- representations of GL_{n}(k) Serre weights; they can be parameterised as follows. We say that an element (a_{i}) of ℤ_{+}^{n} is p-restricted if p − 1 ≥ a_{i} − a_{i+1} for all 1 ≤ i ≤ n−1, and we write X_{1}^{(n)} for the set of p-restricted elements. Given any a ∈ X_{1}^{(n)}, we define the k-representation P_{a} of GL_{n}(k) to be the representation obtained by evaluating Ind_{B_{n}}^{GL_{n}}(w_{0}a)_{/k} on k, and let N_{a} be the irreducible sub-k-representation of P_{a} generated by the highest weight vector (that this is indeed irreducible follows from the analogous result for the algebraic group GL_{n}, cf. II.2.2–II.2.6 in [Jan03], and the appendix to [Her09]). If a = (a_{\overline{κ},i}) ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})}, write a\overline{κ} for the component of a indexed by \overline{κ} ∈ Hom(k, \overline{𝔽}_{p}). If a ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})} then we define an irreducible \overline{𝔽}_{p}- representation F_{a} of GL_{n}(k) by

F_{a} := ⊗_{\overline{κ}∈Hom(k,\overline{𝔽}_{p})}N_{a\overline{κ}} ⊗_{k,\overline{κ}} \overline{𝔽}_{p}.

The representations F_{a} are irreducible, and every Serre weight is (isomorphic to one) of the form F_{a} for some a. The choice of a is not unique: one has F_{a} ≅ F_{a′} if and only if there exist integers x\overline{κ} such that a_{\overline{κ},i} − a′_{\overline{κ},i} = x\overline{κ} for all \overline{κ}, i and, for any labeling \overline{κ}_{j} of the elements of Hom(k, \overline{𝔽}_{p}) such that \overline{κ}_{j}^{p} = \overline{κ}_{j+1} we have ∑_{j=0}^{f−1} p^{j}x_{\overline{κ}_{j}} ≡ 0 (mod p^{f} − 1), where f = [k : 𝔽_{p}]. In this case we write a ∼ a′. We remark that if K/ℚ_{p} is unramified and a ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})} satisfies a_{\overline{κ},1} − a_{\overline{κ},n} ≤ p − (n − 1) for each \overline{κ}, then L_{a} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ F_{a} as representations of GL_{n}(O_{K}). The reason is that P_{b} = N_{b} whenever b ∈ ℤ_{+}^{n} satisfies b_{1} − b_{n} ≤ p − (n − 1) (cf. [Jan03, II.5.6]).

1.2.4. Potentially crystalline representations. An element λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,\overline{ℚ}_{p})} is said to be a lift of an element a ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})} if for each \overline{κ} ∈ Hom(k, \overline{𝔽}_{p}) there exists κ\overline{κ} ∈ Hom_{ℚ_{p}}(K, \overline{ℚ}_{p}) lifting \overline{κ} such that λ_{κ\overline{κ}} = a\overline{κ}, and λ_{κ′} = 0 for all other κ′ ≠ κ\overline{κ} in Hom_{ℚ_{p}}(K, \overline{ℚ}_{p}) lifting \overline{κ}. If λ is a lift of a, then F_{a} is a Jordan–Hölder factor of L_{λ} ⊗ \overline{𝔽}_{p}. Given a pair (λ, τ), we say that a potentially crystalline representation W of G_{K} over \overline{ℚ}_{p} has type (λ, τ) if it is regular of Hodge type λ, and has inertial type τ. Write σ(λ, τ) for L_{λ} ⊗_{\overline{ℤ}_{p}} σ(τ), a \overline{ℤ}_{p}-representation of GL_{n}(O_{K}), and write \overline{σ}(λ, τ) for the semisimplification of σ(λ, τ) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}. Then the action of GL_{n}(O_{K}) on \overline{σ}(λ, τ) factors through GL_{n}(k), so that the Jordan–Hölder factors of \overline{σ}(λ, τ) are Serre weights. If \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) has a crystalline lift W of type (λ, 1) (that is, W is crystalline of Hodge type λ), and λ is a lift of some a ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})}, then we say that \overline{r} has a crystalline lift of Serre weight F_{a}. This terminology is sensible because the existence of a crystalline lift of Hodge type λ for some lift λ of a does not depend on the choice of the element a in its equivalence class under the equivalence relation ∼ (cf. [GHS15, Lem. 7.1.1]). If furthermore K/ℚ_{p} is unramified, and a_{\overline{κ},1} − a_{\overline{κ},n} ≤ p − 1 − n for all \overline{κ}, then we say that a (or F_{a}) is an FL weight, and that \overline{r} has a crystalline lift of FL weight F_{a}. If instead a_{\overline{κ},1} − a_{\overline{κ},n} ≤ p − n for all \overline{κ}, then we say that a (or F_{a}) is an extended FL weight, and that \overline{r} has a crystalline lift of extended FL weight F_{a}.

1.2.5. Potential diagonalisability. Following [BLGGT14], we say that a potentially crystalline representation r : G_{K} → GL_{n}(\overline{ℤ}_{p}) with distinct Hodge–Tate weights is potentially diagonalisable if for some finite extension K′/K, r|_{G_{K′}} is crystalline, and the corresponding \overline{ℚ}_{p} point of the corresponding crystalline deformation ring lies on the same irreducible component as some direct sum of crystalline characters. (For example, it follows from the main theorem of [GL14] that any crystalline representation of extended FL weight is potentially diagonalisable.)

2. Local existence of lifts in the residually Fontaine–Laffaille case 2.1. Peu ramifiée representations. Recall that for any discrete G_{K}- module X, the space H_{ur}^{1}(G_{K}, X) of unramified classes in H^{1}(G_{K}, X) is the kernel of the restriction map H^{1}(G_{K}, X) → H^{1}(I_{K}, X); by the inflationrestriction sequence, this is the same as the image of the inflation map H^{1}(G_{K}/I_{K}, X^{I_{K}}) ↪ H^{1}(G_{K}, X). We will make use of the following wellknown fact.

Lemma 2.1.1. Suppose that X is a discrete G_{K}-module that is moreover a finite-dimensional vector space over a field 𝔽. Then

dim_{𝔽} H_{ur}^{1}(G_{K}, X) = dim_{𝔽} H^{0}(G_{K}, X).

Proof. We have

dim_{𝔽} H^{1}(G_{K}/I_{K}, X^{I_{K}}) = dim_{𝔽} H^{0}(G_{K}/I_{K}, X^{I_{K}}) = dim_{𝔽} H^{0}(G_{K}, X),

the first equality coming from the fact that H^{i}(G_{K}/I_{K}, X^{I_{K}}) for i = 0, 1 are, respectively, the invariants and co-invariants of X^{I_{K}} under Frob_{K} − 1. □

Definition 2.1.2. Suppose that K/ℚ_{p} is a finite extension and 𝔽 is a field of characteristic p. Consider a representation \overline{r} : G_{K} → GL_{n}(𝔽), let \overline{V} be the underlying 𝔽[G_{K}]-module of \overline{r}, and suppose that 0 = \overline{U}_{0} ⊂ \overline{U}_{1} ⊂ ⋯ ⊂ \overline{U}_{ℓ} = \overline{V} is an increasing filtration on \overline{V} by 𝔽[G_{K}]-submodules. Write \overline{V}_{i} := \overline{U}_{i}/\overline{U}_{i−1}. We say that \overline{r} is peu ramifiée with respect to the filtration \{\overline{U}_{i}\} if for all 1 ≤ i ≤ ℓ the class in H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{i}, \overline{U}_{i−1})) defined by \overline{U}_{i} (regarded as an extension of \overline{V}_{i} by \overline{U}_{i−1}) is annihilated under Tate local duality by H_{ur}^{1}(G_{K}, Hom_{𝔽}(\overline{U}_{i−1}, \overline{V}_{i}(1))).

Since group cohomology is compatible with base change for field extensions, so is Definition 2.1.2: that is, if 𝔽′/𝔽 is any field extension, then \overline{r} is peu ramifiée with respect to some filtration \{\overline{U}_{i}\} if and only if \overline{r} ⊗_{𝔽} 𝔽′ is peu ramifiée with respect to the filtration \{\overline{U}_{i} ⊗_{𝔽} 𝔽′\}. Definition 2.1.2 is most interesting in the case where the filtration \{\overline{U}_{i}\} is saturated, i.e., where the graded pieces \overline{V}_{i} are irreducible. (For instance, any \overline{r} will trivially be peu ramifiée with respect to the one-step filtration 0 = \overline{U}_{0} ⊂ \overline{U}_{1} = \overline{V}.) This motivates the following further definition.

Definition 2.1.3. We say that \overline{r} is peu ramifiée if there exists a saturated filtration \{\overline{U}_{i}\} with respect to which \overline{r} is peu ramifiée as in Definition 2.1.2.

Examples 2.1.4.

(1) If n = 2 and \overline{r} ≅ (\begin{matrix}χω & ∗ \\ 0 & χ \end{matrix}) for some character χ, then Definition 2.1.3

coincides with the usual definition of peu ramifiée. (Recall that ω denotes the mod p cyclotomic character.) Indeed, the duality pairing H^{1}(G_{K}, 𝔽_{p}(1)) × H^{1}(G_{K}, 𝔽_{p}) → ℚ_{p}/ℤ_{p} can be identified (via the Kummer and Artin maps) with the evaluation map

K^{×}/(K^{×})^{p} × Hom(K^{×}, 𝔽_{p}) → 𝔽_{p} ↪ ℚ_{p}/ℤ_{p}, from which it is immediate that the classes in H^{1}(G_{K}, 𝔽_{p}(1)) that are annihilated by H_{ur}^{1}(G_{K}, 𝔽_{p}) are precisely those which are identified with O_{K}^{×}/(O_{K}^{×})^{p} by the Kummer map.

(2) If \overline{r} is semisimple then trivially \overline{r} is peu ramifiée.

(3) If there are no nontrivial G_{K}-maps \overline{U}_{i−1} → \overline{V}_{i}(1) for any i (e.g. if one has \overline{V}_{j} ≅ \overline{V̸}_{i}(1) for all j < i) then \overline{r} is necessarily peu ramifiée because by Lemma 2.1.1 we have H_{ur}^{1}(G_{K}, Hom_{𝔽}(\overline{U}_{i−1}, \overline{V}_{i}(1))) = 0.

(4) Suppose K/ℚ_{p} is unramified. We will prove in Section 2.3 that Fontaine–Laffaille representations are peu ramifiée, so that all of the main results in this section will apply to Fontaine–Laffaille representations.

Example 2.1.5. If \overline{r} is peu ramifiée, it is natural to ask whether \overline{r} is peu ramifiée with respect to every (saturated) filtration on \overline{r}. This is not the case. Suppose, for instance, that K does not contain the p-th roots of unity (so ω ≠ 1) and

\overline{r} ≅ (\begin{matrix}ω & ∗_{1} & ∗_{2} \\ & 1 & 0 \\ & & 1 \end{matrix})

where the class of the cocycle ∗_{1} is nontrivial and peu ramifiée, and the cocycle ∗_{2} is très ramifiée. For the filtration on \overline{r} in which \overline{U}_{i} is the span of the first i vectors giving rise to the above matrix representation (so that the action of G_{ℚ_{p}} on \overline{U}_{i} is given by the upper-left i × i block), the representation \overline{r} is peu ramifiée. This is clear at the first two steps in the filtration, and for the third step one notes (as in Example 2.1.4(3)) that there are no nontrivial maps \overline{U}_{2} → \overline{V}_{3}(1). On the other hand, if one defines a new filtration on \overline{r} by replacing \overline{U}_{2} with the span of the first and third basis vectors giving rise to the above matrix representation, then \overline{r} is not peu ramifiée with respect to the new filtration, because the new \overline{U}_{2} is très ramifiée.

Remark 2.1.6. One consequence of the preceding example is that the collection of peu ramifiée representations is not closed under taking arbitrary subquotients. On the other hand, if \overline{r} is peu ramifiée with respect to the filtration \{\overline{U}_{i}\}, then for any a ≤ b it is not difficult to check that \overline{U}_{b}/\overline{U}_{a} is peu ramifiée with respect to the induced filtration \{\overline{U}_{a+i}/\overline{U}_{a}\}_{0≤i≤b−a}. Using the preceding example one can similarly see that the collection of peu ramifiée representations is not closed under contragredients.

Remark 2.1.7. In some sense we are making an arbitrary choice by demanding that we first lift \overline{U}_{1}, then to \overline{U}_{2}, then to \overline{U}_{3}, and so forth. One could equally well lift in other orders, and as Example 2.1.5 shows, this can make a difference. However, since Definition 2.1.2 will suffice for our purposes, we do not elaborate further on this point.

We say that a \overline{ℤ}_{p}-lift of an \overline{𝔽}_{p}[G_{K}]-module \overline{V} is a \overline{ℤ}_{p}[G_{K}]-module V that is free as a \overline{ℤ}_{p}-module, together with a \overline{𝔽}_{p}[G_{K}]-isomorphism V ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ \overline{V}. We have introduced the notion of a peu ramifiée representation (Definition 2.1.2) in order to prove the following result, to the effect that peu ramifiée representations have many \overline{ℤ}_{p}-lifts.

Theorem 2.1.8. Suppose that K/ℚ_{p} is a finite extension. Consider a representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) that is peu ramifiée with respect to the increasing filtration \{\overline{U}_{i}\}, so that \overline{r} may be written as

\overline{r} = (\begin{matrix}\overline{V}_{1} & … & ∗ \\ & ... & ⋮ \\ \overline{V} & ℓ \end{matrix}) ,

where the \overline{V}_{i} := \overline{U}_{i}/\overline{U}_{i−1} are the graded pieces of the filtration. For each i, suppose that we are given a \overline{ℤ}_{p}-representation V_{i} of G_{K} lifting \overline{V}_{i}. Then there exist unramified characters ψ_{1}, …, ψ_{ℓ} with trivial reduction such that \overline{r} may be lifted to a representation r of the form

r = (\begin{matrix}V_{1} ⊗ ψ_{1} & … & ∗ \\ & ... & ⋮ \\ & & V_{ℓ} ⊗ ψ_{ℓ} \end{matrix}) .

More precisely, r is equipped with an increasing filtration \{U_{i}\} by \overline{ℤ}_{p}-direct summands such that U_{i}/U_{i−1} ≅ V_{i}⊗ψ_{i} and r⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ \overline{r} induces U_{i}⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ \overline{U}_{i}, for each 1 ≤ i ≤ ℓ. In fact, there are infinitely many choices of characters (ψ_{1}, …, ψ_{ℓ}) for which this is true, in the strong sense that for any 1 ≤ i ≤ ℓ, if (ψ_{1}, …, ψ_{i−1}) can be extended to an ℓ-tuple of characters for which such a lift exists, then there are infinitely many choices of ψ_{i} such that (ψ_{1}, …, ψ_{i}) can also be extended to such an ℓ-tuple.

Proof. We proceed by induction on ℓ, the case ℓ = 1 being trivial. From the induction hypothesis, we can find ψ_{1}, …, ψ_{ℓ−1} so that \overline{U} := \overline{U}_{ℓ−1} can be lifted to some

U := (\begin{matrix}V_{1} ⊗ ψ_{1} & … & ∗ \\ & ... & ⋮ \\ & & V_{ℓ−1} ⊗ ψ_{ℓ−1} \end{matrix}) .

as in the statement of the theorem. It suffices to prove that for each such choice of ψ_{1}, …, ψ_{ℓ−1}, there exist infinitely many choices of ψ_{ℓ} for which \overline{r} lifts to an extension of V_{ℓ} ⊗ ψ_{ℓ} by U as in the statement of the theorem. Choose the field E/ℚ_{p} large enough so that U and V_{ℓ} are realisable over O_{E}, and so that \overline{r} is realisable over the residue field of E. Suppose that F/E is a finite extension with ramification degree e(F/E) > (dim \overline{V}_{ℓ})(dim \overline{U}), write O for the integers of F and 𝔽 for its residue field, and let ψ : G_{K} → O^{×} be an unramified character such that 0 < v_{E}(ψ(Frob_{K}) − 1) < 1/(dim \overline{V}_{ℓ})(dim \overline{U}). In the remainder of this argument, when we write U and V_{ℓ} we will mean their (chosen) realisations over O, and similarly \overline{U} and \overline{V}_{ℓ} will mean their realisations over 𝔽 obtained by reduction from U and V_{ℓ}. Extensions of V_{ℓ}⊗ψ by U correspond to elements of H^{1}(G_{K}, Hom_{O}(V_{ℓ}⊗ψ, U)), while \overline{r} corresponds to an element c of Ext_{𝔽[G_{K}]}^{1}(\overline{V}_{ℓ}, \overline{U}), which we identify with H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{ℓ}, \overline{U})). By hypothesis (together with the remark about base change immediately following Definition 2.1.2) the class c is annihilated by H_{ur}^{1}(G_{K}, Hom_{𝔽}(\overline{U}, \overline{V}_{ℓ}(1))) under Tate local duality. Taking the cohomology of the exact sequence

0 → Hom_{O}(V_{ℓ} ⊗_{O} ψ, U) →^{ϖ} Hom_{O}(V_{ℓ} ⊗_{O} ψ, U) → Hom_{𝔽}(\overline{V}_{ℓ}, \overline{U}),

we have in particular an exact sequence

H^{1}(G_{K}, Hom_{O}(V_{ℓ} ⊗_{O} ψ, U)) → H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{ℓ}, \overline{U})) →^{δ}

→^{δ} H^{2}(G_{K}, Hom_{O}(V_{ℓ} ⊗_{O} ψ, U)),

so it is enough to check that that c ∈ ker(δ) except for finitely many choices of ψ. From Tate duality, we have the dual map

H^{0}(G_{K}, Hom_{O}(U, V_{ℓ}(1) ⊗_{O} ψ) ⊗ F/O) →^{δ^{∨}} H^{1}(G_{K}, Hom_{𝔽}(\overline{U}, \overline{V}_{ℓ}(1)).

As ker(δ)^{⊥} = im(δ^{∨}), it is enough to show that im(δ^{∨}) is contained in H_{ur}^{1}(G_{K}, Hom_{𝔽}(\overline{U}, \overline{V}_{ℓ}(1))) except, again, for possibly finitely many choices of ψ. Letting X = Hom_{O}(U, V_{ℓ}(1)), we first claim that (X ⊗_{O} O(ψ))^{G_{K}} = 0 for all but finitely many choices of ψ. Indeed, if

(X ⊗_{O} O(ψ))^{G_{K}} = Hom_{O[G_{K}]}(U, V_{ℓ}(1) ⊗_{O} ψ) ≠ 0

then we must have W ≅ Z(1)⊗_{O} ψ for some Jordan–Hölder factor W of U and Z of V_{ℓ}. This can happen for only finitely many choices of ψ (by determinant considerations applied to each of the finitely many pairs W, Z). Now we are done by the following proposition. □

Proposition 2.1.9. Let F/ℚ_{p} be a finite extension with ring of integers O and residue field 𝔽. Let X be an O[G_{K}]-module that is free of finite rank as an Omodule. Suppose that there is a field lying E lying between F and ℚ_{p} such that X is realisable over O_{E} and with ramification index e(F/E) > rank_{O}(X). Let ψ : G_{K} → O^{×} be an unramified character such that 0 < v_{E}(ψ(Frob_{K}) − 1) < 1/ rank_{O}(X). Assume further that (X ⊗_{O} O(ψ))^{G_{K}} = 0. Then the image of

δ^{∨} : H^{0}(G_{K}, (X ⊗_{O} O(ψ)) ⊗_{O} F/O) → H^{1}(G_{K}, X ⊗_{O} 𝔽)

is equal to the subspace of unramified classes, and in particular depends only on X ⊗_{O} 𝔽, and not on X, F, or ψ.

Proof. The statement is unchanged upon replacing E with the maximal unramified extension E^{ur} of E contained in F. We are therefore reduced to the case where F/E is totally ramified (so that in particular 𝔽 is also the residue field of E).

Let X_{O_{E}} be a realisation of X over O_{E}. Write \overline{X} = X_{O_{E}} ⊗_{O_{E}} 𝔽 = X ⊗_{O} 𝔽 and X_{ψ} = X_{O_{E}} ⊗_{O_{E}} O(ψ). The inclusion ι : X_{O_{E}} ↪ X_{ψ} sending x ↦ x ⊗ 1 is a map of O_{E}-modules inducing an isomorphism of 𝔽[G_{K}]-modules \overline{X} ≅ X_{ψ} ⊗_{O} 𝔽. Moreover for any g ∈ G_{K} and x ∈ X_{O_{E}} we have g · ι(x) = ψ(g)(ι(g · x)), so that the map ι is at least I_{K}-linear. Define α = ψ(Frob_{K})^{−1} − 1 and write N = Frob_{K} − 1, which acts on \overline{X}^{I_{K}} with kernel ker(N) = \overline{X}^{G_{K}}. We have an isomorphism

H^{1}(G_{K}/I_{K}, \overline{X}^{I_{K}}) ≅ \overline{X}^{I_{K}} /N\overline{X}^{I_{K}}

induced by evaluation at Frob_{K}. Note that any class in this quotient space has a representative in ∪_{i=0}^{∞} ker(N^{i}), as can be seen for example by writing \overline{X}^{I_{K}} = \overline{Y} ⊕ \overline{Z} with N nilpotent on \overline{Y} and invertible on \overline{Z}. Hence to see that the image of δ^{∨} contains all unramified classes, it suffices to exhibit for \overline{f} ∈ ∪_{i=0}^{∞} ker(N^{i}) an element e\overline{f} ∈ (X_{ψ} ⊗_{O} F/O)^{G_{K}} such that δ^{∨}(e\overline{f}) = [\overline{f}] in

H^{1}(G_{K}/I_{K}, \overline{X}^{I_{K}}). Suppose then that \overline{f} ∈ ∪_{i=0}^{∞} ker(N^{i}) is nonzero. Let i ≥ 0 be the largest integer such that N^{i}\overline{f} ≠ 0, and let \overline{f}_{i} := \overline{f}. For each 0 ≤ j ≤ i let f_{j} ∈ X_{O_{E}} be a lift of N^{i−j}\overline{f}_{i}, and define

f^{∗} = ∑_{j=0}^{i} α^{j} · ι(f_{j}) ∈ X_{ψ}.

Since \overline{f}_{j} ∈ \overline{X}^{I_{K}}, it follows that for g ∈ I_{K} we have g(f_{j}) ≡ f_{j} (mod ϖ_{E}X_{O_{E}}) with ϖ_{E} ∈ O_{E} a uniformiser, and so also g(f^{∗}) ≡ f^{∗} (mod ϖ_{E}X_{ψ}). Now let us compute (Frob_{K} − 1)(f^{∗}). Noting that (Frob_{K} − 1)f_{j} ≡ f_{j−1} (mod ϖ_{E}X_{O_{E}}), with f_{−1} := 0, and recalling that (1 + α)(Frob_{K} · ι(x)) = ι(Frob_{K} · x), we have

(1 + α)(Frob_{K}(f^{∗}) − f^{∗}) = ∑_{j=0}^{i} α^{j}ι(Frob_{K}(f_{j})) − (1 + α) ∑_{j=0}^{i} α^{j}ι(f_{j})

= ∑_{j=0}^{i} α^{j}ι((Frob_{K} − 1)f_{j}) − ∑_{j=0}^{i} α^{j+1}ι(f_{j})

≡ ∑_{j=0}^{i} α^{j}ι(f_{j−1}) − ∑_{j=0}^{i} α^{j+1}ι(f_{j}) (mod ϖ_{E}X_{ψ})
≡ −α^{i+1}ι(f_{i}) (mod ϖ_{E}X_{ψ}).

Note that N^{i+1}\overline{f}_{i} = 0 and N^{i}\overline{f}_{i} ≠ 0, so that i + 1 ≤ dim_{𝔽} \overline{X}^{I_{K}} ≤ rank_{O} X. Therefore v_{E}(α^{i+1}) < 1, and we deduce that g(f^{∗}) ≡ f^{∗} (mod α^{i+1}X_{ψ}) for all g ∈ G_{K}, or in other words f^{∗} ⊗ α^{−i−1} ∈ (X_{ψ} ⊗ F/O)^{G_{K}}. Furthermore, if c\overline{f} := δ^{∨}(f^{∗} ⊗ α^{−i−1}) ∈ H^{1}(G_{K}, \overline{X}), then c\overline{f}(g) is by definition the image in \overline{X} of α^{−i−1}(g(f^{∗}) − f^{∗}). So on the one hand c\overline{f} is unramified (because v_{E}(α^{i+1}) < v_{E}(ϖ_{E}) = 1), while on the other hand c\overline{f}(Frob_{K}) = −\overline{f}_{i}. Thus we can take e\overline{f} := −f^{∗} ⊗ α^{−i−1}, and we have shown

that H^{1}(G_{K}/I_{K}, \overline{X}^{I_{K}}) ⊂ im δ^{∨}. On the other hand, since X_{ψ}^{G_{K}} is assumed to be trivial, we have that (X_{ψ} ⊗ F/O)^{G_{K}} is of finite length; so if ϖ_{F} is a uniformiser of F, then

dim(im δ^{∨}) = dim((X_{ψ} ⊗ F/O)^{G_{K}} /ϖ_{F})
= dim((X_{ψ} ⊗ F/O)^{G_{K}} [ϖ_{F}]) = dim \overline{X}^{G_{K}} = dim(ker N).

On the other hand dim(ker N) = dim(coker N) = dim H^{1}(G_{K}/I_{K}, \overline{X}^{I_{K}}), and the result follows. □

Theorem 2.1.8 implies the following result on the existence of certain potentially crystalline Galois representations.

Proposition 2.1.10. Suppose that K/ℚ_{p} is a finite extension. Consider a representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) that is peu ramifiée with respect to the increasing filtration \{\overline{U}_{i}\}, so that \overline{r} may be written as

\overline{r} = (\begin{matrix}\overline{V}_{1} & … & ∗ \\ & ... & ⋮ \\ \overline{V} & ℓ \end{matrix}) ,

where the \overline{V}_{i} := \overline{U}_{i}/\overline{U}_{i−1} are the graded pieces of the filtration. For each i, suppose that we are given a \overline{ℤ}_{p}-representation V_{i} of G_{K} lifting \overline{V}_{i} such that:

• each V_{i} is potentially crystalline, and

• for each 1 ≤ i < ℓ and each κ : K ↪ \overline{ℚ}_{p}, every element of HT_{κ}(V_{i+1})

is strictly greater than every element of HT_{κ}(V_{i}). Then \overline{r} may be lifted to a potentially crystalline representation r of the form

r = (\begin{matrix}V_{1} ⊗ ψ_{1} & … & ∗ \\ & ... & ⋮ \\ & & V_{ℓ} ⊗ ψ_{ℓ} \end{matrix}) ,

where each ψ_{i} is an unramified character with trivial reduction, and if K′/K is a finite extension such that each V_{i}|_{G_{K′}} is crystalline, then r|_{G_{K′}} is also crystalline. In fact, there are infinitely many choices of characters (ψ_{1}, …, ψ_{ℓ}) for which this is true, in the strong sense that for any 1 ≤ i ≤ ℓ, if (ψ_{1}, …, ψ_{i−1}) can be extended to an ℓ-tuple of characters for which such a lift exists, then there are infinitely many choices of ψ_{i} such that (ψ_{1}, …, ψ_{i}) can also be extended to such an ℓ-tuple.

Proof. This follows from Theorem 2.1.8 along with standard facts about extensions of de Rham representations. Indeed, by [Nek93, Prop. 1.28(2)] and our assumption on the Hodge–Tate weights of the V_{i}, the representation r|_{G_{K′}} is semistable for any r as in Theorem 2.1.8 and any K′ as above. Then by repeated application of the third part of [Nek93, Prop. 1.24(2)], as well as [Nek93, Prop. 1.26], this semistable representation is guaranteed to be crystalline as long as there is no G_{K′}-equivariant surjection (V_{j} ⊗ V_{i}^{∗})(ψ_{j}ψ_{i}^{−1}) ↠ ε for any j < i. Once ψ_{1}, …, ψ_{i−1} have been determined, this can be arranged by avoiding finitely many possibilities for ψ_{i}. □

We give two sample applications of Proposition 2.1.10. The following Corollary will be used in forthcoming work of Arias de Reyna and Dieulefait (in the special case where \overline{r} is Fontaine–Laffaille and the Hodge type λ is 0).

Corollary 2.1.11. Fix an integer n ≥ 1. Then there is a finite extension K′/K, depending only on n, with the following property: if \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) is peu ramifiée and λ = (λ_{κ,i}) ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,\overline{ℚ}_{p})}, then \overline{r} has a potentially diagonalisable lift r : G_{K} → GL_{n}(\overline{ℤ}_{p}) that is regular of Hodge type λ, with the property that r|_{G_{K′}} is crystalline.

Proof. Write \overline{r} as in Proposition 2.1.10 with \overline{V}_{i} irreducible for all i, and set d_{i} = dim_{\overline{𝔽}_{p}} \overline{U}_{i}. By Proposition 2.1.10 and [BLGGT14, Lem. 1.4.3], it is enough to show that there is a finite extension K′/K depending only on n, with the property that we may lift each \overline{V}_{i} to a potentially crystalline representation V_{i}, such that for all i, κ the set HT_{κ}(V_{i}) is equal to {λ_{κ,n−j} + j : j ∈ [d_{i−1}, d_{i} − 1]}, with the additional property that V_{i}|_{G_{K′}} is isomorphic to a direct sum of crystalline characters. This is immediate from Lemma 2.1.12 below. □

Lemma 2.1.12. Let d ≥ 1 be an integer. Let K_{d} be the unramified extension of K of degree d, and define L to be any totally ramified extension of K_{d} of degree |k_{d}^{×}|, where k_{d} is the residue field of K_{d}. Let \overline{r} : G_{K} → GL_{d}(\overline{𝔽}_{p}) be an irreducible representation. Then for any collection of multisets of d integers \{h_{κ,1}, …, h_{κ,d}\}, one for each continuous embedding κ : K ↪ \overline{ℚ}_{p}, there is a lift of \overline{r} to a representation r : G_{K} → GL_{d}(\overline{ℤ}_{p}), such that r|_{G_{L}} is isomorphic to a direct sum of crystalline characters, and for each κ we have HT_{κ}(r) = {h_{κ,1}, …, h_{κ,d}}.

Proof. Since \overline{r} is irreducible, we can write \overline{r} ≅ Ind_{G_{K_{d}}}^{G_{K}} \overline{ψ}, and \overline{ψ} : G_{K_{d}} → \overline{𝔽}_{p}^{×} is

a character. Choose a crystalline character χ : G_{K_{d}} → \overline{ℚ}_{p}^{×} with the property that for each continuous embedding κ : K ↪ \overline{ℚ}_{p} we have

⋃_{κ̃|_{K}=κ} HTκ̃(χ) = {h_{κ,1}, …, h_{κ,d}},

where the union is taken as multisets. (That such a character exists is wellknown; see e.g. [Ser79, §2.3, Cor. 2].) If we let θ : G_{K_{d}} → \overline{ℤ}_{p}^{×} be the Teichmüller lift of \overline{ψ}\overline{χ}^{−1}, then we may take r := Ind_{G_{K_{d}}}^{G_{K}} χθ, which has the correct Hodge– Tate weights by [GHS15, Cor. 7.1.3]. (Note that ^{g}θ|_{G_{L}} is unramified for any g ∈ G_{K}.) □

As a second application of Proposition 2.1.10, we show that each peu ramifié representation has a crystalline lift of some Serre weight.

Corollary 2.1.13. Suppose that K/ℚ_{p} is a finite extension, and that \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) is peu ramifiée. Then \overline{r} has a crystalline lift of some Serre weight.

Proof. When \overline{r} is irreducible, this is straightforward from [GHS15, Thm B.1.1]. (One only has to note that when \overline{r} is irreducible, an obvious lift of \overline{r} in the terminology of [GHS15, §7] is always an unramified twist of a true lift of \overline{r}.) In the general case, suppose that \overline{r} is peu ramifiée with respect to the filtration \{\overline{U}_{i}\}, and as usual set \overline{V}_{i} := \overline{U}_{i}/\overline{U}_{i−1}. By the previous paragraph, for each \overline{V}_{i} we are able to choose a crystalline lift V_{i} of some Serre weight. By an argument as in the fourth paragraph of the proof of [GHS15, Thm B.1.1] it is possible to arrange that every element of HT_{κ}(V_{i+1}) is strictly greater than every element of HT_{κ}(V_{i}), and that ⊕_{i}V_{i} is a crystalline lift of ⊕_{i}\overline{V}_{i} of some Serre weight. (This is just a matter of replacing each V_{i} with a twist by a suitably-chosen crystalline character of trivial reduction.) Now the Corollary follows directly from Proposition 2.1.10 (with K′ = K). □

2.2. Highly twisted lifts. In this section we give a criterion (Proposition 2.2.4) for checking that a representation is peu ramifiée, which we will apply to show in Section 2.3 that Fontaine–Laffaille representations are peu ramifiée.

Definition 2.2.1. Suppose that K/ℚ_{p} is a finite extension. Consider a representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}), let \overline{V} be the underlying \overline{𝔽}_{p}[G_{K}]-module of \overline{r}, and suppose that 0 = \overline{U}_{0} ⊂ \overline{U}_{1} ⊂ ⋯ ⊂ \overline{U}_{ℓ} = \overline{V} is an increasing filtration on \overline{V} by \overline{𝔽}_{p}[G_{K}]-submodules. Denote \overline{V}_{i} := \overline{U}_{i}/\overline{U}_{i−1} for 1 ≤ i ≤ ℓ, the graded pieces of the filtration. We say that \overline{r} admits highly twisted lifts with respect to the filtration \{\overline{U}_{i}\} if there exist \overline{ℤ}_{p}-lifts V_{i} of the \overline{V}_{i}, and a family of \overline{ℤ}_{p}-lifts V (ψ_{1}, …, ψ_{ℓ}) of \overline{V} indexed by a nonempty set Ψ of ℓ-tuples of unramified characters ψ_{i} : G_{K} → \overline{ℤ}_{p}^{×} with trivial reduction modulo 𝔪_{\overline{ℤ}_{p}}, having the following additional properties:

• Each V (ψ_{1}, …, ψ_{ℓ}) is equipped with an increasing filtration \{U(ψ_{1}, …, ψ_{ℓ})_{i}\} by \overline{ℤ}_{p}[G_{K}]-submodules that are \overline{ℤ}_{p}-direct summands.

• We have U(ψ_{1}, …, ψ_{ℓ})_{i}/U(ψ_{1}, …, ψ_{ℓ})_{i−1} ≅ V_{i} ⊗ ψ_{i} for each 1 ≤ i ≤ ℓ.

• The isomorphism V (ψ_{1}, …, ψ_{ℓ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ \overline{V} induces isomorphisms U(ψ_{1}, …, ψ_{ℓ})_{i} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ \overline{U}_{i} for each 1 ≤ i ≤ ℓ.

• U(ψ_{1}, …, ψ_{ℓ})_{i} depends up to isomorphism only on ψ_{1}, …, ψ_{i} (that is, it does not depend on ψ_{i+1}, …, ψ_{ℓ}).

• For each (ψ_{1}, …, ψ_{i}) that extends to an element of Ψ and for each ϵ > 0, there exists ψ_{i+1} such that (ψ_{1}, …, ψ_{i+1}) extends to an element of Ψ, with the further property that 0 < v_{ℚ_{p}}(ψ_{i+1}(Frob_{K}) − 1) < ϵ.

If moreover the set Ψ can be taken to be the set of all ℓ-tuples of unramified characters ψ_{i} : G_{K} → \overline{ℤ}_{p}^{×} with trivial reduction modulo 𝔪_{\overline{ℤ}_{p}}, we say that \overline{r} admits universally twisted lifts with respect to the filtration \{\overline{U}_{i}\}.

As with Definition 2.1.2, the preceding definition is most interesting in the case where the filtration \{\overline{U}_{i}\} is saturated, and so we make the following further definition.

Definition 2.2.2. We say that \overline{r} admits highly (resp. universally) twisted lifts if it admits highly (resp. universally) twisted lifts as in Definition 2.2.1 with respect to some saturated filtration.

Remark 2.2.3. It is natural to ask whether, if \overline{r} admits highly (resp. universally) twisted lifts with respect to some saturated filtration as in Definition 2.2.2, it admits highly (resp. universally) twisted lifts with respect to any such filtration. Proposition 2.2.4 below, in combination with Example 2.1.5, gives a negative answer to this question in the highly twisted case. In fact, Example 2.1.5 also shows that the above question has a negative answer in the universally twisted case. Suppose for simplicity that K/ℚ_{p} is unramified and that p > 2. Then \overline{r} in Example 2.1.5 admits universally twisted lifts for the first filtration considered there. To see this, we first note that the first

block \overline{U}_{2} = \binom{ω ∗_{1}}{1} admits universally twisted lifts for V_{1} = ε and V_{2} = 1

by Proposition 2.3.1 below, because \overline{U}_{2} is Fontaine–Laffaille. Since there is no nontrivial map \overline{U}_{2} → \overline{V}_{3}(1), one easily checks that r̄ admits universally twisted lifts for this filtration. However, \overline{r} does not admit universally twisted lifts for the second filtration considered in Example 2.1.5. This is because the

first block \overline{U}′_{2} = \binom{ω ∗_{2}}{1} does not admit universally twisted lifts (e.g. by

Proposition 2.2.4).

Proposition 2.2.4. Let K/ℚ_{p} be a finite extension, and let \{\overline{U}_{i}\} be an increasing filtration on the representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}). Then \overline{r} is peu ramifiée with respect to \{\overline{U}_{i}\} if and only if it admits highly twisted lifts with respect to \{\overline{U}_{i}\}.

Proof. An inspection of the proof of Theorem 2.1.8 already gives the “only if” implication (for any choice of V_{i}’s lifting \overline{V}_{i}). For the other direction, we assume that \overline{r} admits highly twisted lifts with respect to the filtration \{\overline{U}_{i}\} and some \overline{ℤ}_{p}-lifts V_{i} of the \overline{V}_{i}. We proceed by induction on ℓ, the length of the filtration. By the induction hypothesis we may assume that for all i < ℓ the class in H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{i}, \overline{U}_{i−1})) defined by \overline{U}_{i} is annihilated under Tate local duality by H_{ur}^{1}(G_{K}, Hom_{𝔽}(\overline{U}_{i−1}, \overline{V}_{i}(1))), and it remains to prove this for i = ℓ. Choose any (ψ_{1}, …, ψ_{ℓ−1}) that extends to an element of the set Ψ (as in Definition 2.2.1 for \overline{r}), and let U := U(ψ_{1}, …, ψ_{ℓ})_{ℓ−1} (where ψ_{ℓ} is any character such that (ψ_{1}, …, ψ_{ℓ}) ∈ Ψ); note that this is independent of ψ_{ℓ}.

Let S be the set of characters ψ : G_{K} → \overline{ℤ}_{p}^{×} such that (Hom_{\overline{ℤ}_{p}}(U, V_{ℓ}(1)) ⊗_{\overline{ℤ}_{p}} \overline{ℤ}_{p}(ψ))^{G_{K}} ≠ 0. As in the proof of Theorem 2.1.8 we see that S is finite. Let E/ℚ_{p} be a finite extension such that U and V_{ℓ} are realisable over O_{E}. It follows from the highly twisted lift condition on \overline{r} that there exists ψ_{ℓ} having the following properties:

(i) (ψ_{1}, …, ψ_{ℓ}) ∈ Ψ, (ii) ψ_{ℓ} ∉ S, and (iii) 0 < v_{E}(ψ_{ℓ}(Frob_{K}) − 1) < 1/(dim \overline{U}_{ℓ−1})(dim \overline{V}_{ℓ}).

Let F/E be a finite extension over which ψ_{ℓ} and V (ψ_{1}, …, ψ_{ℓ}) are both realisable. Write O for the ring of integers of F, and 𝔽 for its residue field. For the remainder of this proof, when we write U, V_{ℓ}, ψ_{ℓ} we will mean their chosen realisations over F, and similarly for \overline{U}, \overline{V}_{ℓ} over 𝔽 (obtained by reduction). Set X = Hom_{O}(U, V_{ℓ}(1)). As in the proof of Theorem 2.1.8, write δ for the connection map

H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{ℓ}, \overline{U})) →^{δ} H^{2}(G_{K}, Hom_{O}(V_{ℓ} ⊗_{O} ψ_{ℓ}, U)).

The existence of the lift V (ψ_{1}, …, ψ_{ℓ}) (i.e. the property (i) of ψ_{ℓ}) shows that the class c ∈ H^{1}(G_{K}, Hom_{𝔽}(\overline{V}_{ℓ}, \overline{U})) defining \overline{r} lies in ker(δ). On the other hand, the properties (ii) and (iii) of ψ_{ℓ} mean that Proposition 2.1.9 applies (with ψ_{ℓ} playing the role of ψ) to show that the dual map δ^{∨} has image H_{ur}^{1}(G_{K}, X⊗_{O}𝔽). Since c ∈ ker(δ) it is annihilated under Tate local duality by this image, and we deduce that \overline{r} is peu ramifiée. □

Corollary 2.2.5. Suppose that \overline{r} admits highly twisted lifts with respect to the filtration \{\overline{U}_{i}\}. Then \overline{r} satisfies the definition of admitting highly twisted lifts with respect to the filtration \{\overline{U}_{i}\} for any lifts V_{i} of the \overline{V}_{i}.

Proof. This is immediate from Proposition 2.2.4 along with the first sentence of its proof. □

Remark 2.2.6. The above corollary fails if we replace ‘highly twisted’ with ‘universally twisted’. For instance, consider Example 2.1.4(1) with K/ℚ_{p} unramified, the extension class ∗ peu ramifiée, and χ = 1. It admits universally twisted lifts if we set V_{1} = ε and V_{2} = 1. (This will follow from Proposition 2.3.1 below.) But it does not admit universally twisted lifts for V_{1} = ε^{p} and V_{2} = 1.

Remark 2.2.7. We do not know whether there exist representations that admit highly twisted lifts but not universally twisted lifts.

2.3. Fontaine–Laffaille representations. In this section we will prove that representations which admit a Fontaine–Laffaille lift also admit universally twisted lifts, and so by Proposition 2.2.4 are peu ramifiée. We begin by recalling the formulation of unipotent Fontaine–Laffaille theory in [DFG04, §1.1.2]. Throughout this section let K/ℚ_{p} be a finite unramified extension with integer ring O_{K}, and write Frob_{p} for the absolute geometric Frobenius on K.

Let O be the ring of integers in E, a finite extension of ℚ_{p} with residue field 𝔽. We assume that E is sufficiently large as to contain the image of some (hence any) continuous embedding of K into an algebraic closure of E. Fix an integer 0 ≤ h ≤ p−1, and let MF_{O}^{h} denote the category of finitely generated O_{K}⊗_{ℤ_{p}}Omodules M together with

• a decreasing filtration Fil^{s} M by O_{K} ⊗_{ℤ_{p}} O-submodules which are O_{K}-

direct summands with Fil^{0} M = M and Fil^{h+1} M = {0};

• and Frob_{p}^{−1} ⊗1-linear maps Φ^{s} : Fil^{s} M → M with Φ^{s}|_{Fil^{s+1} M} = pΦ^{s+1}

and ∑_{s} Φ^{s}(Fil^{s} M) = M.

We say that an object M of MF_{O}^{p−1} is étale if Fil^{p−1} M = M, and define MF_{O}^{p−1,u} to be the full subcategory of MF_{O}^{p−1} consisting of objects with no nonzero étale quotients. Such objects are said to be unipotent. Note that MF_{O}^{p−2} is a subcategory of MF_{O}^{p−1,u}. In the following paragraphs, let MF_{O} denote either MF_{O}^{h} (for 0 ≤ h ≤ p − 2) or MF_{O}^{p−1,u} (for h = p − 1). Let Rep_{O}(G_{K}) denote the category of finitely generated O-modules with a continuous G_{K}-action. There is an exact, fully faithful, covariant functor of O-linear categories T_{K} : MF_{O} → Rep_{O}(G_{K}). This is the functor denoted 𝕍 in [DFG04, §1.1.2]. The essential image of T_{K} is closed under taking subquotients. If M is an object of MF_{O}, then the length of M as an O-module is [K : ℚ_{p}] times the length of T_{K}(M) as an O-module. Let MF_{𝔽} denote the full subcategory of MF_{O} consisting of objects killed by the maximal ideal of O and let Rep_{𝔽}(G_{K}) denote the category of finite 𝔽-modules with a continuous G_{K}-action. Then T_{K} restricts to a functor MF_{𝔽} → Rep_{𝔽}(G_{K}). If M is an object of MF_{𝔽} and κ is a continuous embedding K ↪ \overline{ℚ}_{p}, we let FL_{κ}(M) denote the multiset of integers i such that gr^{i}M ⊗_{O_{K}⊗_{ℤp} O,κ⊗1} O ≠ {0} and i is counted with multiplicity equal to the 𝔽-dimension of this space. If M is a p-torsion free object of MF_{O} then T_{K}(M) ⊗_{ℤ_{p}} ℚ_{p} is crystalline and for every continuous embedding κ : K ↪ \overline{ℚ}_{p} we have

HT_{κ}(T_{K}(M) ⊗_{ℤ_{p}} ℚ_{p}) = FL_{κ}(M ⊗_{O} 𝔽).

Moreover, if Λ is a G_{K}-invariant lattice in a crystalline representation V of G_{K} with all its Hodge–Tate numbers in the range [0, h], having (when h = p − 1) no nontrivial quotient isomorphic to a twist of an unramified representation by ε^{−(p−1)}, then Λ is in the essential image of T_{K}. If some twist of \overline{r} : G_{K} → GL_{n}(𝔽) lies in the essential image of T_{K} on MF_{O}^{p−2}, we say that \overline{r} admits a Fontaine–Laffaille lift, while if some twist of \overline{r} lies in the essential image of T_{K} on MF_{O}^{p−1,u} we say that it admits a unipotent extended Fontaine–Laffaille lift. The proof of the following result is essentially the same as that of [BLGGT14, Lem. 1.4.2]. (We remark that [BLGGT14, §1.4] uses the formulation of Fontaine–Laffaille theory as [CHT08, §2.4.1], which in fact is equivalent to that of [DFG04, §1.1.2] (at least on MF_{O}^{p−2}), although this equivalence is not needed for the following argument.)

Proposition 2.3.1. Let K/ℚ_{p} be unramified. Consider a representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) with an increasing filtration \{\overline{U}_{i}\} such that \overline{U}_{0} = 0 and \overline{U}_{ℓ} = \overline{r}, so that \overline{r} may be written as

\overline{r} = (\begin{matrix}\overline{V}_{1} & … & ∗ \\ & ... & ⋮ \\ \overline{V} & ℓ \end{matrix}) ,

where the \overline{V}_{i} = \overline{U}_{i}/\overline{U}_{i−1} are the graded pieces of the filtration. Suppose that \overline{r} admits a Fontaine–Laffaille (resp. unipotent extended Fontaine– Laffaille) lift. Then \overline{r} admits universally twisted lifts with respect to the filtration \{\overline{U}_{i}\}; indeed, it admits universally twisted Fontaine–Laffaille (resp. unipotent extended Fontaine–Laffaille) lifts. In either case \overline{r} is peu ramifiée.

Remark 2.3.2. By duality, the same result holds when \overline{r} admits a nilpotent extended Fontaine–Laffaille lift, i.e., if some twist of \overline{r} lies in the essential image of T_{K} on MF_{O}^{p−1,n}, the full subcategory of MF_{O}^{p−1} whose objects admit no nonzero subobject M with Fil^{1} M = 0. We refer the reader to [GL14] for a further discussion of nilpotent Fontaine–Laffaille theory. Similar arguments (which we omit to keep the paper at a reasonable length) can be used to show that the same result holds when \overline{r} is finite flat (for arbitrary K/ℚ_{p}; in this case the argument uses Kisin modules).

Proof of Proposition 2.3.1. Since the truth of this proposition for \overline{r} evidently implies its truth for any twist of \overline{r} (using the fact that every character of G_{K} admits a crystalline lift), we reduce to the case that \overline{r} lies in the essential image of T_{K} on MF_{O}^{p−2} (or on MF_{O}^{p−1,u}, in the unipotent extended case). The case that each \overline{V}_{i} is one-dimensional is essentially found in [BLGGT14, Lem. 1.4.2] and, as previously remarked, we will follow the proof of that result closely. We can and do suppose that \overline{r} is defined over some finite field 𝔽, and we fix a finite extension E of ℚ_{p} with ring of integers O and residue field 𝔽. Let \overline{V} be the underlying 𝔽-vector space of \overline{r}, and let \overline{M} denote the object of MF_{𝔽} corresponding to \overline{V}, which exists by our assumption that \overline{r} has a (possibly unipotent extended) Fontaine–Laffaille lift. Then we have a filtration

\overline{M} = \overline{M}_{ℓ} ⊃ \overline{M}_{ℓ−1} ⊃ ⋯ ⊃ \overline{M}_{1} ⊃ \overline{M}_{0} = (0)

by MF_{𝔽}-subobjects such that \overline{M}_{i} corresponds to \overline{U}_{i} and so \overline{M}_{i}/\overline{M}_{i−1} corresponds to \overline{V}_{i}. Then we claim that we can find an object M of MF_{O} which is p-torsion free together with a filtration by MF_{O}-subobjects

M = M_{ℓ} ⊃ M_{ℓ−1} ⊃ ⋯ ⊃ M_{1} ⊃ M_{0} = (0)

and an isomorphism

M ⊗_{O} 𝔽 ≅ \overline{M}

under which M_{i} ⊗_{O} 𝔽 maps isomorphically to \overline{M}_{i} for all i. Write d_{i} := dim \overline{V}_{i}. We note first that \overline{M} has an 𝔽-basis \overline{e}_{i,κ} for i = 1, …, n and κ ∈ Hom_{ℚ_{p}}(K, \overline{ℚ}_{p}) such that

• the residue field k of K acts on \overline{e}_{i,κ} via κ;
• \overline{M}_{j} is spanned over 𝔽 by the \overline{e}_{i,κ} for i ≤ d_{1} + ⋯ + d_{j};

• and for each j, s there is a subset Ω_{j,s} ⊂ \{1, …, n\} × Hom_{ℚ_{p}}(K, \overline{ℚ}_{p}) such that \overline{M}_{j} ∩ Fil^{s} \overline{M} is spanned over 𝔽 by the \overline{e}_{i,κ} for (i, κ) ∈ Ω_{j,s}.

(Such a basis is easily constructed recursively in j. The case j = 1 is trivial, and it is straightforward to extend a basis of this kind for \overline{M}_{j−1} to one for \overline{M}_{j}.) We put Ω_{s} := Ω_{ℓ,s}. Then we define M to be the free O-module with basis e_{i,κ} for i = 1, …, n and κ ∈ Hom_{ℚ_{p}}(K, \overline{ℚ}_{p}).

• We let O_{K} act on e_{i,κ} via κ;

• we define M_{j} to be the O-submodule generated by the e_{i,κ} with i ≤ d_{1} + ⋯ + d_{j};

• and we define Fil^{s} M to be the O-submodule spanned by the e_{i,κ} for (i, κ) ∈ Ω_{s}.

We define Φ^{s} : Fil^{s} M → M by reverse induction on s. If we have defined Φ^{s+1} we define Φ^{s} as follows:

• If (i, κ) ∈ Ω_{s+1} then Φ^{s}e_{i,κ} = pΦ^{s+1}e_{i,κ}.

• If (i, κ) ∈ Ω_{s} − Ω_{s+1} then Φ^{s}e_{i,κ} is chosen to be any lift of \overline{Φ}^{s}\overline{e}_{i,κ} in ∑_{i′≤d_{1}+···+d_{j}} O · e_{i′,κ∘Frob_{p}}, where j is minimal such that i ≤ d_{1} + ⋯ +

d_{j}.

It follows from Nakayama’s lemma that M is an object of MF_{O}^{h}. When h = p−1, suppose that M → M′ is a nontrivial étale quotient of M. We can without loss of generality replace M′ with M′ ⊗_{O} 𝔽; but then the map M → M′ would factor through \overline{M}, contradicting the assumption that \overline{M} is an object of MF_{O} (and not just MF_{O}^{p−1}). It follows that M is also an object of MF_{O}. In the same way we see that {M_{i}} is an increasing filtration of subobjects of M in MF_{O}. It is immediate that M verifies the desired property that M_{i} ⊗_{O} 𝔽 maps isomorphically to \overline{M}_{i} under the isomorphism M ⊗_{O} 𝔽 ≅ \overline{M}. Set V_{i} := T_{K}(M_{i}/M_{i−1}) ⊗_{O} \overline{ℤ}_{p}. We claim that for this choice of V_{i}, the conditions of Definition 2.2.1 are satisfied. Since Fontaine–Laffaille theory is compatible in an obvious fashion with extension of scalars from E to a finite extension of E, we can and do suppose that the characters ψ_{i} are valued in O^{×}. Then the objects of MF_{O} corresponding to the desired lifts V (ψ_{1}, …, ψ_{ℓ}) are obtained from M by rescaling the maps Φ^{s}. More precisely, if we let M(ψ_{1}, …, ψ_{ℓ}) be defined from M by rescaling Φ^{s}e_{i,κ} by ψ_{j}(Frob_{K}) for (i, κ) ∈ Ω_{j,s} \ Ω_{j−1,s}, then one can take V (ψ_{1}, …, ψ_{ℓ}) = T_{K}(M(ψ_{1}, …, ψ_{ℓ})) ⊗_{O} \overline{ℤ}_{p}. (To establish the second bullet point in Definition 2.2.1, note from [DFG04, p. 670] that T_{K} is compatible with tensor products, and use (1) of loc. cit. to compute the Fontaine–Laffaille module corresponding to each ψ_{i}.) □

Corollary 2.3.3. Suppose that \overline{r} admits a (possibly unipotent extended) Fontaine–Laffaille lift. Then the conclusions of Theorem 2.1.8 and Proposition 2.1.10 hold for \overline{r} with respect to any separated, exhaustive increasing filtration \{\overline{U}_{i}\} on \overline{r}.

Corollary 2.3.4. Suppose that \overline{r} admits a (possibly unipotent extended) Fontaine–Laffaille lift. Then the conclusion of Corollary 2.1.11 holds for \overline{r}.

The following result will be used in [GHS15].

Corollary 2.3.5. Suppose that \overline{r} : G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{p}) admits a (possibly unipotent extended) Fontaine–Laffaille lift. Suppose also that

\overline{r} = (\begin{matrix}\overline{χ}_{1} & … & ∗ \\ & ... & ⋮ \\ \overline{χ} & n \end{matrix}) .

Suppose that h_{1} > ⋯ > h_{n} are integers such that \overline{χ}_{i}|_{I_{ℚp}} = ω^{h_{i}}. Then \overline{r} has a crystalline lift of the form

r = (\begin{matrix}χ_{1} & … & ∗ \\ & ... & ⋮ \\ & & χ_{n} \end{matrix}) ,

where χ_{i}|_{I_{ℚp}} = ε^{h_{i}}.

Proof. This is immediate from Corollary 2.3.3, taking the V_{i} to be appropriate unramified twists of ε^{h_{i}}. □

3. de Rham lifts by global methods

3.1. Potential automorphy and globalisation. In this section, we make use of (global) potential automorphy techniques to produce potentially crystalline lifts. Ultimately, these results rely on those of [BLGGT14], but the actual global results we need are those of [EG14, App. A]. The key idea is as follows: by the methods of [BLGGT14] and [Cal12], we can often realise \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) as the restriction to a decomposition group of \overline{ρ}, the reduction mod p of the p-adic Galois representation associated to an automorphic representation on some unitary group. Then the existence of congruences between automorphic representations of different weights and types produces lifts of \overline{r} of the corresponding Hodge and inertial types. To keep this paper from becoming longer than necessary, and to avoid obscuring the relatively simple arguments that we need to make, we will not recall the precise definitions of the spaces of automorphic forms that we work with; the details may be found in [EG14] (and the papers referenced therein). Suppose from now until the end of Lemma 3.1.1 that:

• p ∤ 2n, and

• \overline{r} has a potentially diagonalisable lift of some type (λ\overline{r}, τ\overline{r}).

Then in particular Conjecture A.3 of [EG14] holds for \overline{r}, so that by [EG14, Cor. A.7], there is a CM field F with maximal totally real field F^{+}, and an irreducible representation \overline{ρ} : G_{F^{+}} → G_{n}(\overline{𝔽}_{p}) (where G_{n} is the algebraic group defined in [CHT08, §2.1]) which is automorphic in the sense of [EG14, Def. 5.3.1], and which globalises \overline{r} in the sense that for each place v | p of F^{+} we have that v splits in F and that there is a place ṽ of F lying over v such that Fṽ ≅ K and \overline{ρ}|_{G_{Fṽ}} ≅ \overline{r}. The above data will remain fixed throughout this section. Suppose that for each place v | p of F^{+} we fix a representation of GL_{n}(O_{K}) on a finite \overline{ℤ}_{p}-module W_{v}. Via the isomorphisms ιṽ of [EG14, §5.2], we can regard W := ⊗_{\overline{ℤ}_{p},v|p}W_{v} as a representation of G(O_{F^{+},p}), where G is a certain unitary group. Then there is a space of algebraic modular forms S(U, W), as in [EG14, §5.2]. (In fact, [EG14] works with coefficients in the ring of integers of some finite extension of ℚ_{p}, rather than with \overline{ℤ}_{p}-coefficients, but this makes no difference for the arguments we are making here.) In particular, for any (λ_{v}, τ_{v})_{v|p} a space of automorphic forms S_{λ,τ} (U, \overline{ℤ}_{p}) is defined in [EG14, §5.2] for certain sufficiently small compact open subgroups U ⊂ G(𝔸_{F^{+}}^{∞}) which are hyperspecial at p, corresponding to taking each W_{v} to be σ(λ_{v}, τ_{v}). Examining the proof of [EG14, Cor. A.7], we see that in fact we have S_{λ\overline{r},τ\overline{r}}(U, \overline{ℤ}_{p})_{𝔪} ≠ 0, where 𝔪 is as in [EG14, Def. 5.3.1], and in a mild abuse of notation we write (λ_{\overline{r},v}, τ_{\overline{r},v}) = (λ\overline{r}, τ\overline{r}) for all v | p. (This says that there is an automorphic representation π of weight λ\overline{r} and type τ\overline{r}, whose associated p-adic Galois representation ρ_{π} lifts \overline{ρ}; this representation ρ_{π} is the representation ρ constructed in [EG14, Lem. A.5].)

Lemma 3.1.1. Keep the notation and assumptions of the preceding discussion.

(1) If for some choice of (λ_{v}, τ_{v})_{v|p} we have S_{λ,τ} (U, \overline{ℤ}_{p})_{𝔪} ≠ 0, then for each v | p, \overline{r} has a potentially crystalline lift of type (λ_{v}, τ_{v}).

(2) S_{λ,τ} (U, \overline{ℤ}_{p})_{𝔪} ≠ 0 if and only if there are Serre weights F_{v} of GL_{n}(k) such that

• S(U, ⊗_{\overline{𝔽}_{p},v|p}F_{v})_{𝔪} ≠ 0, and
• for all v | p, F_{v} is a Jordan–Hölder factor of \overline{σ}(λ_{v}, τ_{v}).

Proof. (1) is immediate from [EG14, Prop. 5.3.2]. (We remind the reader that S_{λ,τ} (U, \overline{ℤ}_{p}) is torsion-free.) In the case that τ is trivial, (2) is [BLGG15, Lem. 2.1.6], and the proof goes over unchanged to the general case. □

Theorem 3.1.2. Suppose that p ∤ 2n, and that \overline{r} has a potentially diagonalisable lift of some regular weight. Then the following hold.

(1) There exists a finite extension K′/K (depending only on n and K, and not on \overline{r}) such that \overline{r} has a lift r : G_{K} → GL_{n}(\overline{ℤ}_{p}) of Hodge type 0 that becomes crystalline over K′.

(2) The representation \overline{r} has a crystalline lift of some Serre weight.

Proof. We begin with the proof of (2), since the argument is much shorter. Let r be the given potentially diagonalisable lift, and as above, write (λ\overline{r}, τ\overline{r}) for the type of r. By Lemma 3.1.1(2), there are Jordan–Hölder factors F_{a_{v}} of \overline{σ}(λ\overline{r}, τ\overline{r}) (possibly varying with v) such that S(U, ⊗_{\overline{𝔽}_{p},v|p}F_{a_{v}})_{𝔪} ≠ 0. Let λ_{v} be a lift of a_{v} for each v, and let τ_{v} be trivial for each v. Applying Lemma 3.1.1(2) again, we see that S_{λ,1}(U, \overline{ℤ}_{p})_{𝔪} ≠ 0. By Lemma 3.1.1(1), \overline{r} has a crystalline lift of Hodge type λ_{v} for each v | p; any such lift will do. Turn now to (1). As in the previous part we get S(U, V)_{𝔪} ≠ 0 for some irreducible representation V = ⊗_{v|p}V_{v} of G(O_{F^{+}} ⊗ ℤ_{p}) over \overline{𝔽}_{p}. Let T ⊂ B ⊂ GL_{n} denote the subgroups of diagonal and upper-triangular matrices, as algebraic groups over ℤ. Consider V_{v} as a representation of GL_{n}(O_{Fṽ}) =: K_{v} via ιṽ. Let I_{v} ⊂ K_{v} denote the preimage of B(kṽ) ⊂ GL_{n}(kṽ). Then we can choose a character \overline{χ}_{v} : I_{v} → \overline{𝔽}_{p}^{×} such that V_{v}|_{I_{v}} ↠ \overline{χ}_{v}. Let q := #k. We claim that for any s ≥ 1 such that q^{s−1} ≥ n, we can find a (smooth) lift χ_{v} = χ_{1,v} ⊗ ⋯ ⊗ χ_{n,v} : T (O_{Fṽ}) → \overline{ℤ}_{p}^{×} of \overline{χ}_{v}|_{T (O_{Fṽ})} = \overline{χ}_{1,v} ⊗ ⋯ ⊗ \overline{χ}_{n,v} such that the \{χ_{i,v}\}_{i=1}^{n} are pairwise distinct and χ_{i,v}|_{1+ϖs ; ṽO_{Fṽ}} = 1 for all i. Indeed, recalling that Fṽ ≅ K, write O_{Fṽ}^{×}/(1 + ϖs ; ṽO_{Fṽ}) ≅ k^{×} × H (via the Teichmüller splitting), where H is abelian of order q^{s−1}. Then each \overline{χ}_{i,v}|_{k^{×}} lifts uniquely to \overline{ℤ}_{p}^{×}, whereas \overline{χ}_{i,v}|_{H} = 1 and can be lifted arbitrarily to \overline{ℤ}_{p}^{×}. Hence it is enough to note that # Hom(H, \overline{ℤ}_{p}^{×}) = #H = q^{s−1} ≥ n, and this proves the claim. For the rest of the proof, we fix such a choice of s and χ_{v}. Now, [Roc98, §3] (applied with a standard Chevalley basis such that U_{α,0} = U_{α}∩GL_{n}(O_{Fṽ}) for all roots α) provides a pair (J_{χ_{v}}, ρ_{χ_{v}}) consisting of a compact open subgroup J_{χ_{v}} ⊂ I_{v} that contains T (O_{Fṽ}) and a smooth character ρ_{χ_{v}} : J_{χ_{v}} → \overline{ℤ}_{p}^{×} such that ρ_{χ_{v}}|_{T (O_{Fṽ})} = χ_{v}. By construction, \overline{ρ}_{χ_{v}} is the restriction of \overline{χ}_{v} to J_{χ_{v}}, so by Frobenius reciprocity we get a K_{v}-equivariant map V_{v} ↪ Ind_{J_{χv}}^{K_{v}} (\overline{ρ}_{χ_{v}}).

In particular, S (U, ⊗_{v|p} (Ind_{J_{χv}}^{K_{v}} ρ_{χ_{v}}))_{𝔪} ≠ 0. Using Deligne–Serre lifting we get an automorphic representation π of G(𝔸_{F^{+}}) with associated Galois representation ρ_{π} lifting \overline{ρ}|_{G_{F}} : G_{F} → GL_{n}(\overline{𝔽}_{p}) such that (i) π_{∞} ≅ 1 and (ii) Hom_{K_{v}}(Ind_{J_{χv}}^{K_{v}} ρ_{χ_{v}}^{−1}, π_{v}) ≠ 0 (again via ιṽ) for any v | p. Applying [Roc98, Thm 7.7] (noting there are no restrictions on p, cf. [CHT08, Lem. 3.1.6]), we deduce that π_{v} is a subquotient of Ind_{B(Fṽ)}^{G(Fṽ)}(χ̃_{v}^{−1}) for some χ̃_{v} : T (Fṽ) → \overline{ℚ}_{p}^{×} extending χ_{v}. (Note that J_{χ_{v}} = J_{χ_{v}^{−1}}.) Since the characters {χ_{i,v}}_{i=1}^{n} are pairwise distinct, the Bernstein–Zelevinsky irreducibility criterion implies that π_{v} ≅ Ind_{B(Fṽ)}^{G(Fṽ)}(χ̃_{v}^{−1}). It follows that rec(π_{v}) has N = 0 and on inertia is of the form χ_{1,v}^{−1} ⊕ ⋯ ⊕ χ_{n,v}^{−1} via the local Artin map, where rec denotes the local Langlands correspondence, normalised as in [EG14] (i.e., as in [HT01]). Using Lemma 3.1.4 below there exists a finite extension K′/K depending only on n and K such that rec(π_{v})|_{I_{K′}} is trivial. Applying local-global compatibility at p to ρ_{π}, we deduce that for any v | p, the representation ρ_{π}|_{G_{Fṽ}} provides a desired lift of \overline{ρ}|_{G_{Fṽ}} ≅ \overline{r}. □

Remark 3.1.3. The argument in the proof above shows that if χ_{i,v} : O_{F_{v}}^{×} → \overline{ℚ}_{p}^{×} are pairwise distinct smooth characters of O_{F_{v}}^{×} (or equivalently of I_{F_{v}}), then Ind_{J_{χv}}^{K_{v}} ρ_{χ_{v}} is a K_{v}-type corresponding to ⊕_{i=1}^{n}χ_{i,v} under the inertial Langlands correspondence, i.e. [EG14, Conj. 4.1.3] holds with σ(⊕_{i=1}^{n}χ_{i,v}) ≅ Ind_{J_{χv}}^{K_{v}} ρ_{χ_{v}}^{−1}.

Lemma 3.1.4. Suppose K/ℚ_{p} is a finite extension and s ≥ 1. Then there exists a finite extension L/K such that any smooth character χ : W_{K} → ℂ^{×} that is trivial on the ramification subgroup G_{K}^{s} satisfies χ|_{I_{L}} = 1.

Proof. By local class field theory there exists a finite extension M_{s}/K^{nr} that is independent of χ such that χ|_{G_{Ms}} = 1. (We can take M_{s}/K abelian such that K^{ab}/M has Galois group 1 + ϖ_{K}^{s}O_{K}, with ϖ_{K} a uniformiser of K.) Then we choose L/K finite such that M_{s} is contained in L · K^{nr} = L^{nr}. This implies χ|_{I_{L}} = 1. In fact, this argument shows that we can take L/K of degree q^{s−1}(q − 1), where q = #k. □

Our final result may be viewed as a “weak Breuil–Mézard”-type statement.

Theorem 3.1.5. Suppose that p ≠ n, that K/ℚ_{p} is unramified, and that \overline{r} has a crystalline lift of weight F for some extended FL weight F. If F is a Jordan– Hölder factor of \overline{σ}(λ, τ) for some λ, τ, then \overline{r} has a potentially crystalline lift of type (λ, τ).

Proof. Choose a ∈ (X_{1}^{(n)})^{Hom(k,\overline{𝔽}_{p})} such that F ≅ F_{a}. The conditions that p ≠ n and \overline{r} has a crystalline lift of weight F_{a} with a an extended FL weight imply that p > n; so either p ∤ 2n, as we have assumed throughout this section, or else p = 2 and n = 1. First suppose that p ∤ 2n. By the main result of [GL14] any crystalline representation of extended FL weight is potentially diagonalisable. Let λ′ be the lift of a (uniquely defined, as K/ℚ_{p} is unramified). Since a_{\overline{κ},1}−a_{\overline{κ},n} ≤ p−(n−1) for each \overline{κ}, F_{a} = L_{a} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} (see §1.2.3). By hypothesis, we can apply the constructions from the paragraphs preceding Lemma 3.1.1 with λ\overline{r} = λ′ and τ\overline{r} = 1 to deduce that S_{λ′,1}(U, \overline{ℤ}_{p})_{𝔪} ≠ 0. By Lemma 3.1.1(2), S(U, ⊗_{\overline{𝔽}_{p},v|p}F_{a})_{𝔪} ≠ 0. Applying Lemma 3.1.1(2) with (λ_{v}, τ_{v}) = (λ, τ) for each v, we see that S_{λ,τ} (U, \overline{ℤ}_{p})_{𝔪} ≠ 0, and the result follows from Lemma 3.1.1(1). On the other hand, the case n = 1 is an easy consequence of local class field theory: σ(τ)^{∨} is obtained from τ by local class field theory, so that the locally algebraic characters of K^{×} extending σ(λ, τ) correspond to de Rham characters of type (λ, τ), while \overline{r}|_{I_{K}} corresponds to F_{a}. □

References

[BLGG15] Thomas Barnet-Lamb, Toby Gee, and David Geraghty, Serre weights for U(n), J. Reine Angew. Math. (to appear) (2015).

[BLGGT14] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Potential automorphy and change of weight, Ann. of Math.

(2) 179 (2014), no. 2, 501–609.

[Cal12] Frank Calegari, Even Galois Representations and the Fontaine-Mazur conjecture II, J. Amer. Math. Soc. 25 (2012), no. 2, 533–554.

[CEG^{+}16] Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas, and Sug Woo Shin, Patching and the p-adic local Langlands correspondence, Camb. J. Math. 4 (2016), no. 2, 197–287.

[CHT08] Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Pub. Math. IHES 108 (2008), 1–181.

[DFG04] Fred Diamond, Matthias Flach, and Li Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 663–727.

[EG14] Matthew Emerton and Toby Gee, A geometric perspective on the Breuil-Mézard conjecture, J. Inst. Math. Jussieu 13 (2014), no. 1, 183–223.

[GHS15] Toby Gee, Florian Herzig, and David Savitt, General Serre weight conjectures, preprint, 2015.

[GL14] Hui Gao and Tong Liu, A note on potential diagonalizability of crystalline representations, Math. Ann. 360 (2014), no. 1-2, 481–487.

[Her09] Florian Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116.

[HT01] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich.

[Jan03] Jens Carsten Jantzen, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003.

[Kis08] Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513–546.

[Mul13] Alain Muller, Relèvements cristallins de représentations galoisiennes, Université de Strasbourg Ph.D. thesis, 2013.

[Nek93] Jan Nekovář, On p-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91, Progr. Math., vol. 108, Birkhäuser Boston, Boston, MA, 1993, pp. 127–202.

[Ram02] Ravi Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 (2002), no. 1, 115–154.

[Roc98] Alan Roche, Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. École Norm. Sup.

(4) 31 (1998), no. 3, 361–413.

[Ser79] Jean-Pierre Serre, Groupes algébriques associés aux modules de Hodge-Tate, Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III, Astérisque, vol. 65, Soc. Math. France, Paris, 1979, pp. 155–188.

Toby Gee Department of Mathematics Imperial College London London SW7 2AZ, UK toby.gee@imperial.ac.uk Florian Herzig Department of Mathematics University of Toronto herzig@math.toronto.edu

Tong Liu Department of Mathematics Purdue University tongliu@math.purdue.edu David Savitt Department of Mathematics Johns Hopkins University savitt@math.jhu.edu