Inertial and Hodge--Tate weights of crystalline representations

Let $K$ be an unramified extension of $\mathbb{Q}_p$ and $\rho\colon G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{Z}}_p)$ a crystalline representation. If the Hodge--Tate weights of $\rho$ differ by at most $p$ then we show that these weights are contained in a natural collection of weights depending only on the restriction to inertia of $\overline{\rho} = \rho \otimes_{\overline{\mathbb{Z}}_p} \overline{\mathbb{F}}_p$. Our methods involve the study of a full subcategory of $p$-torsion Breuil--Kisin modules which we view as extending Fontaine--Laffaille theory to filtrations of length $p$.


Introduction
Let K/Q p be a finite unramified extension with residue field k. In this paper we show that if the Hodge-Tate weights of a crystalline representation ρ of G K are sufficiently small then these weights are encoded in an explicit way by the reduction of ρ modulo p. Using Fontaine-Laffaille theory this is known for Hodge-Tate weights differing by at most p − 1; we will treat weights differing by at most p. Our techniques are local and involve the study of a full subcategory of p-torsion Breuil-Kisin modules, which we view as extending (p-torsion) Fontaine-Laffaille theory to filtrations of length p.
To state our result let Z n + denote the set of (λ 1 , . . . , λ n ) ∈ Z n with λ 1 ≤ . . . ≤ λ n . In Section 2 we show how to attach to any continuous ρ : G K → GL n (F p ) a subset Inert(ρ) ⊂ (Z n + ) Hom Fp (k, Fp) This subset depends only on the restriction to inertia of the semi-simplification of ρ, and does so in an explicit fashion. We typically write an element of Inert(ρ) as (λ τ ) τ ∈Hom Fp (k,Fp) with λ τ = (λ 1,τ ≤ . . . ≤ λ n,τ ). Throughout Hodge-Tate weights are normalised so that the cyclotomic character has weight −1.
Theorem A. Let ρ : G K → GL n (Z p ) be a crystalline representation. For each τ ∈ Hom Fp (k, F p ) let λ τ ∈ Z n + denote the τ -Hodge-Tate weights of ρ. If λ n,τ − λ 1,τ ≤ p for all τ then (λ τ ) τ ∈ Inert(ρ) When n = 2 and p > 2 the result is a theorem of Gee-Liu-Savitt [11]. When n = 2 and p = 2 the result is due to Wang [17]. In this paper we extend their methods to higher dimensions.
As already mentioned, when λ n,τ − λ 1,τ ≤ p − 1 the Theorem A is a straightforward consequence of Fontaine-Laffaille theory, so the main content of our result is that it applies to Hodge-Tate weights differing by p. On the other hand the Theorem A does not hold if the condition λ n,τ − λ 1,τ ≤ p is relaxed. For example, there exist irreducible two dimensional crystalline representations ρ of G Qp with Hodge-Tate weights (−p − 1, 0), whose reduction modulo p have the form ρ = ( χcyc * 0 χcyc ), see [3,Théorème 3.2.1]. Here χ cyc denotes the cyclotomic character. It is easy to check that (−p − 1, 0) is not an element of Inert(ρ).
Our motivation comes from the weight part of (generalisations of) Serre's modularity conjecture. As a corollary of our result we can prove some new cases of weight elimination for mod p representations associated to automorphic representations on unitary groups of rank n. To be more precise let F be an imaginary CM field in which p is unramified and fix an isomorphism ι : Q p ∼ = C. Attached to any RACSDC (regular, algebraic, conjugate self dual, and cuspidal) automorphic representation Π of GL n (A F ) there is a continuous irreducible r ι,p (Π) : G F → GL n (Q p ), cf. the main result of [6]. If Π is unramified above p then r ι,p (Π) is crystalline above p, and if λ = (λ κ ) κ ∈ (Z n + ) Hom(F,C) is the weight of Π then the κ-Hodge-Tate weights 1 of r ι,p (Π) equal λ κ + (0, 1, . . . , n − 1) We point out that while the Corollary B involves only distinct Hodge-Tate weights, due to the regularity assumptions on our automorphic representations, Theorem A does not require such distinctness.
If r is assumed to arise from some potentially diagonalisable RACSDC automorphic representation (a notion introduced in [2]) and if we assume r v is semisimple for each v | p then, under a Taylor-Wiles hypothesis, the inclusion in the Corollary B is an equality. This follows from e.g. [1,Theorem 3.1.3].
To conclude this introduction we briefly explain our proof of the theorem; let us do this by sketching the content of the various sections in this paper. In the first two sections we recall some basic notions; in Section 2 we define the set Inert(ρ) and in Section 3 we give some elementary results on filtered modules. In Section 4 we recall the notion of a Breuil-Kisin module, and recall how to associate to them Galois representations. Breuil-Kisin modules killed by p admit a natural set of weights and in Section 5 we define what it means for a p-torsion Breuil-Kisin module to be strongly divisible; it's weights must be contained in [0, p] and a certain explicit condition on its ϕ must be satisfied. We view the category of strongly divisible Breuil-Kisin modules Mod SD k as an extension of p-torsion Fontaine-Laffaille theory to filtrations of length p. We establish two important properties of Mod SD k . The first main property (Proposition 5.4.7) is shown in Section 5 and states that Mod SD k is stable under subquotients, and that weights behave well along short exact sequences. The second main property (Proposition 6.4.1) is proved in Section 6 and concerns the structure of simple objects in M ∈ Mod SD k . We show that for such M the weights of M coincide with the inertial weights of the associated Galois representation. These two properties mirror the situation for Fontaine-Laffaille theory. However, unlike in Fontaine-Laffaille theory, it is not the case that simple M ∈ Mod SD k are determined by their weights together with their associated Galois representation. This complicates the proofs considerably. Thus, while there are similarities between Mod SD k and Fontaine-Laffaille theory in some respects, the former category is more complicated, reflecting the fact that the reduction of crystalline representations with Hodge-Tate weights in [0, p] is genuinely more subtle than for weights in the Fontaine-Laffaille range. In the final section we recall a theorem of Gee-Liu-Savitt [11] which relates Mod SD k with the reduction modulo p of those crystalline representations with Hodge-Tate weights contained in [0, p]. Using this, and the two properties of Mod SD k described above, it is straightforward to deduce Theorem A.
Let C denote the completion of an algebraic closure K of K and let O C be its ring of integers, with residue field k. We write G K = Gal(K/K) and v p for the valuation on C normalised so that v p (p) = 1.
We fix a uniformiser π ∈ K and a compatible system π 1/p n ∈ K of p n -th roots of π. Many constructions in this paper depend upon these choices. Set K ∞ = K(π 1/p ∞ ) and G K∞ = Gal(K/K ∞ ).
Let µ p n (K) denote the group of p n -th roots of unity in K and write Z p (1) for the free rank one Z p -module lim ← − µ p n (K) Let χ cyc : G K → Z × p denote the character though which G K acts on Z p (1). Let E/Q p denote a finite extension with ring of integers O and residue field F. We assume throughout that K 0 ⊂ E. This will be our coefficient field in which the representations we consider will be valued.
If A is any ring of characteristic p we let ϕ : A → A denote the homomorphism x → x p . If A is perfect (i.e. ϕ is an automorphism) we let W (A) denote the ring of Witt vectors of A and write ϕ : W (A) → W (A) for the automorphism lifting ϕ on A.

Inertial Weights
In this section we recall the structure of irreducible torsion representations of G K and G K∞ . We then define the set W (ρ) inert from the introduction.
2.1. Tame ramification. Let K ur and K t be the maximal unramified and maximal tamely ramified extension of K respectively. Set I t = Gal(K t /K ur ). As in [15,Proposition 2] there is an isomorphism where in the limit l runs over finite extensions of k with transition maps given by norm maps. This isomorphism sends σ → (s(σ) l ) l where s(σ) l is the image in the residue field of K t of the Card(l × )-th root of unity σ(π 1/ Card(l × ) )/π 1/ Card(l × ) ∈ K t Here π 1/ Card(l × ) is any Card(l × )-th root of π; s(σ) l does not depend upon any of these choices. Via s we define the fundamental character Note this is a power of ω l and ω θ•ϕ = ω p θ .
Proof. Since 1 → I t → Gal(K t /K) → G k → 1 is split, χ extends to Gal(K t /K) if and only if χ is stable under the conjugation action of G k on I t . Via s this action is given by the natural action of G k on lim ← − l × , and so χ extends if and only if χ p [k:Fp ] = χ. After [15,Proposition 5] this is equivalent to asking that χ be a power of ω k , thus a product as in the lemma.
In particular we see each ω l extends to a character of G L where L/K is the unramified extension with residue field l. Such an extension is well defined only up to twisting by an unramified character. Our fixed choice of uniformiser π ∈ K allows us to define a canonical choice of extension by sending σ ∈ G L onto the image in the residue field of the element σ(π 1/ Card(l × ) )/π 1/ Card(l × ) ∈ K t where π 1/ Card(l × ) is an Card(l × )-th root of π. We shall denote this character again by ω l : G L → F × p . Also, for θ ∈ Hom Fp (l, F p ) we write ω θ = θ • ω l , as characters of G L . For an extension L/K write Ind K L V in place of Ind Proof. As V is irreducible the G K -action factors through G = Gal(K t /K) by [15,Proposition 4]. Since I t is abelian of order prime to p, V | I t is a sum of F × pvalued characters. If γ ∈ G k and χ : I t → F × p is a character define a new character by χ (γ) (σ) = χ(γ −1 σγ). If I t acts on v ∈ V | I t by χ then I t acts on γ(v) by χ (γ) ; thus G k acts on the set of χ appearing in V | I t . Fix χ appearing in V | I t and let H ⊂ G be the normal subgroup containing I t , corresponding to the stabiliser of χ in G k . By the orbit-stabiliser theorem [G : H] ≤ dim Fp V .

Frobenius reciprocity gives a non-zero map
The inequality of the first paragraph implies [G : H] = dim Fp V and so this map is an isomorphism.
Definition 2.1.3. Let ρ be a continuous representation of G K on an n-dimensional F p -vector space. After Lemma 2.1.2 there exist continuous characters ζ : with each summand irreducible. Let l ζ /k denote the residue field of L ζ . After Lemma 2.1.1 there are integers (r θ,ζ ) θ∈Hom Fp (l ζ ,Fp) such that Any such collection of r θ,ζ defines a weight λ = (λ τ ) τ ∈Hom Fp (k,Fp) via λ τ = {r θ,ζ | θ| k = τ }. Define Inert(ρ) to be the set of λ obtained in this way.
It is easy to check that W(ρ) inert depends only on ρ ss | I t .
Then restriction defines an isomorphism Proof. Since K ∞ /K is totally wildly ramified we have K ∞ ∩ K t = K. The lemma then follows from Galois theory.
Corollary 2.2.2. Restriction describes an equivalence between the category of semi-simple continuous G K -representations on finite dimensional F p -vector spaces, and the analogous category of G K∞ -representations.
Proof. The action of G K on such a semi-simple representation factors through Gal(K t /K) as Gal(K/K t ) is pro-p. Likewise the action of G K∞ factors through Gal(K t ∞ /K ∞ ), and so the result follows from Lemma 2.2.1.

Filtrations
This section contains some elementary results on filtered modules; they will be useful later. Consider a commutative ring A and a collection of ideals (F i A) i∈Z satisfying Then the category Fil(A) of filtered A-modules consists of A-modules M equipped with a collection of A-sub-modules (F i M ) i∈Z satisfying Morphisms are maps f :

Strict maps.
If M is an object of Fil(A) and N ⊂ M is an A-sub-module the induced filtration on N is that given by F i N = N ∩ F i M . If f : M → N is a surjective A-module homomorphism the quotient filtration on N is that given by   Proof. The following diagram commutes and has exact rows.
If M is complete and N is Hausdorff then the same is true with (2) replaced by (2 ′ ) gr(M ) → gr(N ) → gr(coker(f )) is exact for all i; Proof. It is straightforward to check (without any conditions on M and N ) that (2) is equivalent to gr i coim(f ) → gr i im(f ) being injective for all i, that (2 ′ ) is equivalent to this map being surjective for all i, and that (3) is equivalent to this map being an isomorphism for all i. Thus  Proof. Argue as in [16,Corollary] using the second part of Lemma 3.1.5.

Adapted Bases.
We now put ourselves in the following situation. Let a ∈ A be a nonzerodivisor and equip A with the a-adic filtration (so F i A = a i A). Let M be a finite free A-module and let N ⊂ M [ 1 a ] be a finitely generated A-sub- Lemma 3.2.1. Suppose that A is complete. Give N/a the quotient filtration and suppose that a finite collection (g i ) of elements of N is given along with integers (r i ) such that g i ∈ F ri N . If the images of g i in gr ri (N/a) form a gr(A/a) = A/a-basis of gr(N/a) then the (g i ) form a basis of N and the (a −ri g i ) form a basis of M .
Proof. The induced filtration on the kernel aN of N → N/a is given by Thus gr(N )/a = gr(N/a) where a ∈ gr(A) denotes the homogeneous element of degree 1 represented by a ∈ A. It is then easy to see (e.g. using the graded version of Nakayama's lemma) that the images of the g i in gr(N ) generate this module over gr(A). Since ∩ i a i gr(A) = 0 they are also gr(A)-linearly independent. As N is finitely generated N is Hausdorf and so we may apply Corollary 3.1.6 to deduce that the (g i ) form an A-basis of N and that As the g i are A-linearly independent the (a −ri g i ) are A-linearly independent. To show they generate M take m ∈ M and n large enough that a n m ∈ N . Then a n m ∈ F n N and so a n m = a i g i with a i ∈ F n−ri A. It follows that m = (a ri−n a i )(a −ri g i ) and so, since (a ri−n )F n−ri A ⊂ A, we are done.

Filtered Vector
Spaces. Finally we give criteria to determine when two filtrations on a vector space are the same.
Conversely if one of f or g is strict then equality implies the sequence is exact in Fil(k).
Proof. As P is discrete we can apply Lemma 3.3.1 to deduce that with equality if and only if g is strict. If f is strict Lemma 3.1.5 tells us that The lemma follows when we assume f is strict. If g is strict one argues similarly, applying Lemma 3.3.1 to the map M → ker(g).

Breuil-Kisin
with transition maps x → x p . This is a perfect integrally closed ring of characteristic p. There is a multiplicative identification O C ♭ = lim ← − O C (the limit again taken with respect to the transition maps x → x p ) given by . Both rings are equipped with a Z p -linear endomorphism ϕ; on A inf this is the usual Witt vector Frobenius and on S it is given by a i u i → ϕ(a i )u ip . The system π 1/p n defines an element π ♭ = (π, π 1/p , . . .) ∈ O C ♭ and we embed S → A inf by mapping u → [π ♭ ] (where [·] denotes the Teichmuller lifting). This embedding is compatible with ϕ.
By functoriality there are ϕ-equivariant G K -actions on A inf = W (O C ♭ ) and W (C ♭ ) lifting those modulo p.
When there is no risk of confusion we shall write ϕ in place of ϕ M et . Let Mod et K denote the abelian category of etale ϕ-modules.
admits a Z p -linear action of G K∞ (given by the trivial action on M et and natural G K∞ -action on W (C ♭ )). This describes a functor from Mod et K to the category of finitely generated Z p -modules equipped with a continuous Z p -linear G K∞ -action.
The completion of K ∞ is a perfectoid field in the sense of [14], whose tilt is the completed perfection of k((u)) ⊂ C ♭ . It follows from [14,Theorem 3.7] that the action of the equality follows by taking ϕ-invariants.
. When there is no risk of confusion we write ϕ in place of ϕ M . Let Mod BK K denote the abelian category of Breuil-Kisin modules.
More generally we use this notation whenever A is any ring equipped with a Frobenius ϕ and M is an A-module equipped with a map ϕ M :

Coefficients.
In practice we are interested in representations valued in extensions of Z p . For this reason we introduce a variant of Mod BK K .
], the product running over τ ∈ Hom Fp (k, F) (we abusively write τ also for its extension to an embedding τ : ]-algebra. By the above ϕ M restricts to a map

Similarly
is an exact sequence of G K∞ -representations then there exists a unique exact sequence 3 also implies that composition series for M are in bijection with composition series for T (M ).
Warning 5.1.5. It is not the case that the set of irreducible factors of a composition series is independent of the choice of composition series.

Strong Divisibility.
In this subsection we define a full-subcategory Mod SD k ⊂ Mod BK k which we view as an extension of p-torsion Fontaine-Laffaille theory to filtrations of length p.
We equip this k-vector space with the quotient filtration.
Proof. Suppose M k → M ϕ k is an isomorphism of filtered modules. We can find integers r i and elements g i ∈ F ri M whose images in gr(M k ) form a k-basis. As the induced map gr(M k ) → gr(M ϕ k ) is an isomorphism it follows that the images of To prove (2) implies (1) we use the f i to give explicit descriptions of the filtration on M ϕ k . Since ϕ(M ) generates M ϕ over k [[u]] every m ∈ M ϕ can be written ] since the f i form a basis of M . Hence ) and Λ τ = diag(u ri,τ ).
The first is just the exact sequence (1) The map N → P is strict when viewed as a map of filtered modules if and only if 0 → M k → N k → P k → 0 is an exact sequence in Fil(k) in the sense of Notation 3.3.2. (2) is equivalent to M ϕ k → N ϕ k being strict, which is equivalent to N ϕ k → P ϕ k being strict. Proof. Note that M → N is strict as a map of filtered modules. To see this Using the second exact sequence of Remark 5.4.1 we obtain the following commutative diagram with exact rows.
The previous paragraph shows that if N → P is strict then the left and middle columns are exact, and so the right column is exact also. Conversely if the right column is exact then one proves the middle column is exact by increasing induction on i (for small enough i the left column will be zero). This proves (1). The same argument but with the diagram replaced with the diagram obtained by considering the first exact sequence of Remark 5.4.1 proves (2) also. It remains to show that if M ϕ k → N ϕ k or N ϕ k → P ϕ k is strict then 0 → M ϕ k → N ϕ k → P ϕ k → 0 is exact. It suffices to show that i∈Weight(M) i + i∈Weight(P ) i = i∈Weight(N ) i after  Proof. Consider the following commutative diagram.
The left and right vertical arrows are isomorphisms by assumption. Since N → P is strict, part (1) of Lemma 5.4.2 implies the bottom row is exact. Thus gr i (N ϕ k ) → gr i (P ϕ k ) is surjective and so N ϕ k → P ϕ k is strict by Lemma 3.1.5. Part (3) of Lemma 5.4.2 then implies the top row is exact. We conclude that N k → N ϕ k is an isomorphism in Fil(k).
Lemma 5.4.4. Let 0 → M → N → P → 0 be an exact sequence in Mod BK k . Suppose that N satisfies the equivalent conditions of Lemma 5.2.5 and that M k → N k is strict. Then N → P is strict and M and P also satisfy the equivalent conditions of Lemma 5.2.5.
Proof. The following diagram of objects in Fil(k) commutes.
As maps of k-vector spaces the horizontal arrows are injective and the vertical arrows are isomorphisms. By assumption the maps M k → N k and N k → N ϕ k are strict. It follows that M ϕ k → N ϕ k and M k → M ϕ k are strict also. The following is also a commutative diagram in Fil(k).
As maps of k-vector spaces the vertical maps are isomorphisms and the horizontal arrows are surjections. By assumption the leftmost vertical arrow is strict. Using part (3) of Lemma 5.4.2, M ϕ k → N ϕ k being strict implies N ϕ k → P ϕ k is strict. It follows that P k → P ϕ k and N k → P k are strict. Thus M and P are as in Lemma 5.2.5 and after (1) of Lemma 5.4.2 we know N → P is strict. We have to show α is injective for every i. For injectivity of α when i < p we argue as follows. As Weight(N ) ⊂ [0, p], and because N k ∼ = N ϕ k , we have gr i (N k ) = 0 for i < 0. Hence gr i (N ) = gr i−p (N ) for i < 0. This implies gr i (N ) = 0 for i < 0 because for small enough i, F i N = N . Using the diagram we deduce that gr i (M ) = 0 for i < 0 also, and that for i < p we have gr i (M ) = gr i (M k ) and gr i (N ) = gr i (N k ). This proves α is injective when i < p.
For injectivity of α when i ≥ p it suffices to show F i N k = 0 for i > p (because then F i M k = 0 for i > p so α is just the zero map when i > p and when i = p, α is the inclusion F i M k → F i N k ). Let us prove this is the case. Since Weight(N ) ⊂ [0, p] we have gr i (N k ) = 0 for i > p; it suffices to show F i N k = 0 for i >> p. But N k is both Hausdorff (being a quotient of N , which is Hausdorff) and a finite dimensional k-vector space, this forces F i N k to vanish for large i. So we are done.
Putting all this together we deduce the following.
Proof. This is immediate from Proposition 5.4.6. In particular we point out that the exact sequence in (1) of Proposition 5.4.6 is functorial and so is an exact sequence of k ⊗ Fp F-modules. Thus it decomposes into exact sequences 0 → M ϕ k,τ → N ϕ k,τ → P ϕ k,τ → 0 which shows Weight τ (N ) = Weight τ (M ) ∪ Weight τ (P ).

Irreducible Objects
Provided F is sufficiently large, irreducible F-representations of G K and G K∞ are induced from characters (Lemma 2.1.2). In this section we investigate the extent with which this is true for objects of Mod SD k (O). Throughout assume k ⊂ F.
where r τ ∈ Z and where (x) = xe τ0 + τ =τ0 e τ for some x ∈ F × .  Here ψ x denotes the unramified character sending the geometric Frobenius to x, and the ω τ are the characters defined in the paragraph after the proof of Lemma 2.1.1.
Proof. This is [11,Proposition 6.7]. However note that in loc. cit. they contravariantly associate a G K∞ -representation to Breuil-Kisin module; this is why the character appearing here is the inverse of that in loc. cit.   Proof. The exact sequence Since the quotient N/M = P is free over k [[u]] this intersection equals M ⊗ k[[u]] u p/p−1 O C ♭ . Thus the crystalline G K -action on N restricts to a crystalline G Kaction on M . This implies the crystalline Example 6.2.6. If N is as in Lemma 6.1.1 then an easy calculation shows that the following describes a continuous ϕ- For this to be a crystalline G K -action we need to check that η(σ) Θτ −1 ∈ u p/p−1 O C ♭ . This follows from Lemma 6.2.7 below. As there is at most one way to extend a continuous character χ : G K∞ → F × to G K this is the unique crystalline G K -action on N .
which we claim is an isomorphism. It suffices to check the natural map ϕ * f * M → f * ϕ * M is an isomorphism, and this follows because the commutative diagram is a pushout.
commutes for all M ∈ Mod BK K . The top horizontal arrow is obtained from the identification in Lemma 6.3.3 by taking ϕ-invariants, and the lower horizontal arrow is given by Frobenius reciprocity. Hom GK ∞ (V, T (f * N )) → Hom GK ∞ (V, Ind K∞ L∞ T (N )) for any continuous G K∞ -representation V on a finitely generated Z p -module. As (6.3.5) is functorial in V Yoneda's lemma provides the isomorphism ι N . As (6.3.5) is functorial in N we see that ι N is functorial.
and for each θ ∈ Hom Fp (l, F) we have and Proof. By functoriality both f * and f * preserve O-actions. Note that the (1) and (2) for some x ∈ F × and r θ ∈ Z. In Example 6.2.6 it was shown N admits a crystalline G L -action. An identical calculation shows that the same formula σ(e θ ) = η(σ) Θ θ e θ , Θ θ = We give the proof over the next three subsections. Also, in Subsection 6.8 we give an example showing it is not necessarily true that M = f * M ′ . Before this we record a corollary of the proposition. 6.5. Proof of Proposition 6.4.1: a first approximation. We begin by providing a first approximation of M by a rank one object N ∈ Mod BK l (O). For this subsection we need only that M ∈ Mod SD k (O) has T (M ) ∼ = Ind K∞ L∞ χ for some character χ : G L∞ → F × (if p = 2 we will also need that χ is not an unramified twist of the trivial character). In particular we shall not use that T (M ) is irreducible, or that M admits a crystalline G K -action.
There M → f * N which after applying T induces an isomorphism. Thus (6.5.1) becomes an isomorphism after inverting u; in particular it is injective. As in Lemma 6.1.1 we may suppose Assume from now on that x = 1 in (6.5.2). Since M → f * N is an isomorphism after inverting u we can define integers δ θ ∈ Z ≥0 minimal amongst those satisfying u δ θ e θ ∈ M Let P ⊂ N be the rank one object of Mod BK l (O) generated by the u δ θ e θ . We claim P ∈ Mod SD l (O) and Weight θ (P ) ⊂ Weight θ| k (M ). After Proposition 5.4.7 it suffices to give an injection P → f * M with torsion-free cokernel. The map 3 u δ θ e θ → e θ (u δ θ e θ ⊗ 1) is such an injection. This is a morphism of Breuil-Kisin modules and has torsionfree cokernel by the definition δ θ . As Weight θ (P ) = {r θ + pδ θ•ϕ − δ θ } we conclude which implies δ θ ∈ [0, 1] if p > 2, and δ θ ∈ [0, 2] if p = 2. If p = 2 and δ θ•ϕ = 2 then, as r θ + pδ θ•ϕ − δ θ ∈ [0, p], we must have δ θ = 2 and r θ = 0. Thus r θ = 0 for all θ ∈ Hom Fp (l, F) and so χ is the trivial character. At the start of this subsection we assumed when p = 2 this was not the case, and so δ θ ∈ [0, 1] when p = 2 also.
At this point we have proved enough to deduce the following. Unlike the previous subsection we will require that T (M ) be irreducible and we will use that M admits a crystalline G K -action. This allows us to invoke the following lemma.  0 in this ring). Recall in the previous subsection we show δ θ ∈ [0, 1], therefore ue θ ∈ M for every θ. It follows that Remark 6.6.3. As well as the previous lemma we make repeated use of the observations: • If m ∈ M then ϕ(m) ∈ M .
• If m ∈ M is such that ϕ(m) ∈ u p+1 M then m ∈ uM . The first follows because Weight(M ) consists of positive integers. The second follows because gr i (M k ) = 0 for i > p and so We shall use that X satisfies the following two properties.
(1) If for θ ∈ X there exist α θ ∈ F such that θ∈X α θ e θ ∈ M then all α θ = 0. (2) If θ ∈ X then there exists a unique e θ + α κ e κ ∈ M, α κ ∈ F with the sum running over κ ∈ X such that r θ ≡ r κ modulo p and such that θ| k = κ| k . In particular the sum lies in M θ| k .
Proof. If in (1) not all the α θ = 0 there will be a largest i such that α θi = 0 and θ i ∈ X; then e θi + θ∈X,θ =θi α θ α θ i e θ ∈ M which contradicts the fact that θ i ∈ X. Uniqueness in (2) follows from (1) so we only need to show existence. We claim there exists α κ ∈ F such that e θ + κ∈X α κ e κ ∈ M ; note if this is the case then by Lemma 6.6.1 and the fact that M = M τ there will exist a sum as in (2). Write θ = θ i . We prove the claim by induction on i. If i is minimal amongst i with θ i ∈ X then in (6.6.5) each θ j ∈ X so the claim holds. For general i and any sum as in (6.6.5) some θ j may not be contained in X, however our inductive hypothesis then implies there exist β κ ∈ F such that α θj e θj + κ∈X β κ e κ ∈ M , and so the claim holds in this case also. Lemma 6.6.7. Suppose θ 0 is chosen so that (1) if θ ∈ X and θ • ϕ ∈ X then r θ > 0, (2) if θ ∈ X and θ • ϕ ∈ X then r θ = 0. Then the conclusion of Proposition 6.4.1 holds.
The next two lemmas finish the proof. The first step is to show ue θ ∈ W for each θ. If θ is as in (C4) then this is obvious. It is also obvious if θ is as in (C3) and e θ•ϕ ∈ M for then f θ = ue θ . If θ is as in (C1) then uf θ = ue θ + uα κ e κ where the sum runs over κ ∈ X such that if κ • ϕ ∈ X then e κ•ϕ ∈ M ; by the previous two sentences we deduce ue θ ∈ W . At this point we've shown ue κ ∈ W if κ • ϕ ∈ X. If θ is as in (C3) but with e θ•ϕ ∈ M then We know the u rκ e κ ∈ W when r κ > 0 so we have that Split this sum up as If κ • ϕ ∈ X and κ ∈ X then κ is as in (C1) and so f κ = e κ + ι∈X α ι e ι ∈ W . It follows there are β ι ∈ F such that ue θ + ι∈X β ι e ι ∈ W However then ι∈X β ι e ι ∈ M since ue θ ∈ M , which implies by Lemma 6.6.6(1) that all β ι = 0. Thus ue θ ∈ W . At this point we know ue θ ∈ W except if θ is as in (C2), i.e. θ ∈ X and θ • ϕ ∈ X. In this case and so that ue θ ∈ W follows from all the cases we have previously worked out.
To finish the proof note that if Q ⊂ N is the F-vector space spanned by the e κ with κ ∈ X then Q ∩ M = 0 by Lemma 6.6.6(1). If θ is as in (C1) then e θ − f θ ∈ Q by definition. Using this and the fact that ue θ ∈ W for all θ we see additionally that if θ is as in (C2) with r θ = 0, then there exists w ∈ W such that e θ − w ∈ Q This is also true if θ is as in (C2) with r θ > 0 since then e θ − f θ ∈ Q. Thus if θ ∈ X there exists w ∈ W such that e θ − w ∈ Q. Now take an arbitrary element z = α θ e θ ∈ M . We need to show it lies in W . We can assume α θ ∈ F because we know ue θ ∈ W for all θ. By the above we can find w ∈ W such that z − w ∈ Q; however since z − w ∈ M we conclude that z = w. Thus the (f θ ) generate M which proves the lemma. First we show ue θ•ϕ ∈ W for all θ. If θ is as in (C4) then this is clear. It is also clear if θ is as in (C3) and e θ•ϕ ∈ M because in this case g θ•ϕ = e θ•ϕ . If θ is as in (C1) then with each κ ∈ X, so using the two previous cases we deduce ue θ•ϕ ∈ W . In particular we've checked ue θ•ϕ ∈ W whenever θ • ϕ ∈ X. If θ is as in (C3) with e θ•ϕ ∈ M , or as in (C2) with r θ = 0 then ue θ•ϕ ∈ W since ug θ•ϕ = ue θ•ϕ + κ∈X uα κ e κ , and each ue κ ∈ W by the above. At this point the only remaining case is when θ is as in (C2) with r θ > 0. If θ is as in (C2) with r θ > 0 then with each κ ∈ X. As we've shown above that if κ ∈ X then ue κ•ϕ ∈ W we deduce that ue θ•ϕ ∈ W . This completes the proof that ue θ ∈ W for all θ.
We finish the proof just as in the previous lemma. Let Q ⊂ N be the F-span of the e κ with κ ∈ X, so that Q ∩ M = 0. If θ is as in (C3) then e θ•ϕ − g θ•ϕ ∈ Q by construction. Using this and the fact that ue θ ∈ W for all θ we also see that if θ is as in (C2) with r θ > 0 then there exists a w ∈ W such that e θ•ϕ − w ∈ Q. This is also true if θ is as in (C2) with r θ = 0, for then g θ•ϕ − e θ•ϕ ∈ Q. This shows that if θ • ϕ ∈ X then there exists a w ∈ W such that e θ•ϕ − w ∈ Q Thus for any general element Z = α θ e θ ∈ M there exists w ∈ W such that Z − w ∈ Q ∩ M . We conclude Z = w ∈ W which finishes the proof. 6.7. Verifying the hypothesis in Lemma 6.6.7. Continue with the notation of the previous subsection. Proof. We must show θ ∈ X and θ • ϕ ∈ X implies r θ = 0. If θ = θ [l:Fp]−1 then as r θ [l:Fp ]−1 = 0 by assumption there is nothing to prove. Now suppose θ = θ i with i = [l : F p ] − 1 so that θ i+1 is defined. As θ i+1 ∈ X, δ θi+1 = 1 and so r θi + p − δ θi ∈ [0, p]. Hence r θi ∈ [0, 1]. Suppose for a contradiction that r θi = 1. As θ i ∈ X there exists α θi ∈ F such that z := e θi + 0≤j<i α θj e θj ∈ M As in Lemma 6.6.6(2) we can assume α θj = 0 unless r θj ≡ r θi = 1 modulo p. Thus r θj = 1 when α θj = 0. If z ′ = e θi+1 + α θj e θj+1 then ϕ(uz ′ ) = u p+1 z and so z ′ ∈ M by Remark 6.6.3. Therefore θ i+1 ∈ X which is a contradiction. with α θ•ϕ j ∈ F. If this set is empty then X = Hom Fp (l, F) and so the conditions of Lemma 6.6.7 would be trivially verified. Thus we may consider the smallest I such that a sum as in (6.7.3) exists. For a given θ, there can be at most one such sum; if there were two their difference would have length < I and so must be zero. We now show that for any sum as in (6.7.3), r θ = r θ•ϕ j whenever α θ•ϕ j = 0. Lemma 6.6.1 implies r θ ≡ r θ•ϕ j modulo p whenever α θ•ϕ j = 0. As r θ , r θ•ϕ j ∈ [0, p] this congruence will be an equality except possibly if r θ = 0 or p. In this case set where γ j = 0 if r θ•ϕ j = p and γ j = 1 if r θ•ϕ j = 0, and likewise γ = 0 if r θ = p and γ = 1 if r θ = 0. Then ϕ(uz) equals u 2p multiplied by (6.7.3), and so by Remark 6.6.3, z ∈ M . If all the γ j , γ are equal then all the r θj are equal and our claim follows. If γ = 1 and not all γ j = 1 then z −ue θ•ϕ − γj =1 uα θ•ϕ j e θ•ϕ j+1 ∈ M is non-zero, contradicting the minimality of I. Thus, not all the γ, γ j being equal implies r θ = p. Since r θ + pδ θ•ϕ − δ θ ∈ [0, p] we must have δ θ•ϕ = 0, and so e θ•ϕ ∈ M . Therefore I = 0 and (6.7.3) reads e θ ∈ M and the claim is immediate. Now fix a sum as in (6.7.3). We shall show, under the hypotheses of the lemma, that there exist β θ•ϕ j+1 ∈ F such that (6.7.4) e θ•ϕ + 0<j≤I β θ•ϕ j+1 e θ•ϕ j+1 ∈ M and such that β θ•ϕ j+1 = α θ•ϕ j if α θ•ϕ j = 0. Let us explain why this implies T (M ) is reducible. Note that possibly β θ•ϕ j+1 = 0 while α θ•ϕ j = 0. Actually this cannot happen because if it did then applying this construction to θ • ϕ, and then θ • ϕ 2 , and so on till we get to θ • ϕ [l:Fp]−1 , would yield another sum as in (6.7.3) with more non-zero terms, which contradicts uniqueness. Thus (6.7.4) must be equal to e θ•ϕ + 0<j≤I α θ•ϕ j e θ•ϕ j+1 ; the second paragraph of the proof then implies r θ•ϕ s = r θ•ϕ I+s for every s ≥ 0. If ω is the character through which G K∞ -acts on T (N ) then ω = θ ω −r θ θ = θ ω −r θ θ•ϕ I = ω p I . If I > 0 this contradicts the irreducibility of T (M ) = Ind K L ω. If I = 0 then we've just shown e θ ∈ M for every θ. In this case the hypothesis of the lemma do not hold since X = ∅ and the conditions of Lemma 6.6.7 are trivially verified.
6.8. An example. We conclude this section by giving an example of M as in Proposition 6.4.1 with M = f * N . Take K = Q p and let L/K be of degree 5. We shall exhibit M as a sub-module of f * N where N is the rank one object of Mod SD l (O) given by N = l[[u]] ⊗ Fp F, ϕ N (1) = u x e θ•ϕ 4 + u n e θ•ϕ 3 + e θ•ϕ 2 + u n e θ•ϕ + e θ Here we have fixed a θ ∈ Hom Fp (l, F) and n, x are integers satisfying 1 ≤ n ≤ p, 0 ≤ x ≤ p.

Crystalline Representations
In this section we state the key results which relate Mod SD k (O) with crystalline representations. We then give a proof of the theorem from the introduction 7.1. Crystalline Representations and Breuil-Kisin modules. As in [8] let B dR denote Fontaine's ring of p-adic periods, and B crys ⊂ B dR the ring of crystalline periods. As in [9] a p-adic representation V of G K is crystalline if  Proof. This is the main result of [12]. The formulation we give here is taken from [4,Theorem 4.4].
Notation 7.1.2. A crystalline O-lattice is a G K -stable O-lattice inside a continuous representation of G K on a finite dimensional E-vector space which is crystalline when viewed as a Q p -representation. By functoriality M → T (M ) restricts to a functor from the category of crystalline O-lattices into Mod BK K (O). Definition 7.1.3. If V is a crystalline representation on an E-vector space then D crys (V ) is a free module over K 0 ⊗ Qp E of rank dim E V and so D crys (V ) K is a free K 0 ⊗ Qp E-module of rank e dim E V . If K 0 ⊂ E then as in