Inertial and Hodge–Tate weights of crystalline representations

Let K be an unramified extension of Qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}_p$$\end{document} and ρ:GK→GLn(Z¯p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho :G_K \rightarrow {\text {GL}}_n(\overline{{\mathbb {Z}}}_p)$$\end{document} a crystalline representation. If the Hodge–Tate weights of ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} 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 ρ¯=ρ⊗Z¯pF¯p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\rho }} = \rho \otimes _{\overline{{\mathbb {Z}}}_p} \overline{{\mathbb {F}}}_p$$\end{document}. 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. B Robin Bartlett robinbartlett18@mpim-bonn.mpg.de 1 Max Plank Institute for Mathematics, Vivatgasse 7, 53111 Bonn, Germany 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 Sect. 2 we show how to attach to any continuous ρ : 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 F p (k,F p ) with λ τ = (λ 1,τ ≤ · · · ≤ λ n,τ ).
Throughout Hodge-Tate weights are normalised so that the cyclotomic character has weight − 1. Theorem 1.0.1 Let ρ : G K → GL n (Z p ) be a crystalline representation. For each τ ∈ Hom F p (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 [9]. When n = 2 and p = 2 the result is due to Wang [14]. In this paper we extend their methods to higher dimensions.
As already mentioned, when λ n,τ −λ 1,τ ≤ p−1, Theorem 1.0.1 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 Theorem 1.0.1 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 Q p , with Hodge-Tate weights (− p − 1, 0), whose reduction modulo p have the form ρ = ( χ cyc * 0 χ cyc ), 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 [5]. 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) Therefore, if W(r ) inert ⊂ (Z n + ) Hom(F,C) denotes the subset containing those (λ κ ) with λ κ + (0, 1, . . . , n − 1) ∈ Inert(r v ) then Theorem 1.0.1 implies Corollary 1.0.2 Let r : G F → GL n (F p ) be irreducible and continuous. Let W(r ) aut denote the set of weights λ ∈ (Z n + ) Hom(F,C) such that there exists an RACSDC automorphic representation Π of GL n (A F ) which is unramified at p, has weight λ, and is such that r ι, p (Π ) ∼ = r . Then where for * ∈ {aut, inert}, W(r ) * ≤ p−n+1 is the subset containing (λ κ ) ∈ W(r ) * with λ n,κ − λ 1,κ ≤ p − n + 1.
We point out that, while Corollary 1.0.2 involves only distinct Hodge-Tate weights, due to the regularity assumptions on our automorphic representations, Theorem 1.0.1 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 semi-simple for each v | p then, under a Taylor-Wiles hypothesis, the inclusion in the Corollary 1.0.2 is an equality. This follows from [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 Sect. 2 we define the set Inert(ρ) and in Sect. 3 we give some elementary results on filtered modules. In Sect. 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 Sect. 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 Sect. 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 6.7.1) is proved in Sect. 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 [9] 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 1.0.1. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1] and the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London, and the Max Planck Institute for Mathematics, Bonn.

Notation
Throughout we let k denote a finite field of characteristic p > 0 and write In the introduction we took K = K 0 ; however some of our constructions are valid for arbitrary finite extensions so now allow K to denote a totally ramified extension of K 0 of degree e, with ring of integers O K . At certain points it will be necessary to assume K = K 0 .
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 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 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 Inert(ρ) from the introduction.

Tame ramification
Let K ur and K t denote the maximal unramified and maximal tamely ramified extension of K respectively. Set I t = Gal(K t /K ur ). As in [12,Proposition 2] there is an isomorphism where, in the limit, l runs over finite extensions of k with transition maps given by the 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 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 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:F p ] = χ . After [12,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 a Card(l × )th root of π . We shall denote this character again by ω 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 [12,Proposition 4]. Since I t is abelian of order prime to p, V | I t is a sum of F × p -valued 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 σ γ ). 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 and corresponding to the stabiliser of χ in G k . By the orbit-stabiliser theorem Frobenius reciprocity gives a non-zero map V | H → Ind H I t χ . If L/K is the unramified extension corresponding to H then, since the image of H in G k stabilises χ , this character can be extended to H as in Lemma 2.1.1. Thus Ind 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 F p (l ζ ,F p ) such that Any such collection of r θ,ζ defines a weight λ = (λ τ ) τ ∈Hom F p (k,F p ) via λ τ = {r θ,ζ | θ | k = τ }. Define Inert(ρ) to be the set of λ obtained in this way. We remark that for a given ζ there always exists a unique tuple (r θ,ζ ) θ∈Hom F p (l ζ ,F p ) as above such that each r θ,ζ ∈ [0, p − 1] with not all r θ,ζ equal to p − 1. However if we drop the restriction that r θ,ζ ∈ [0, p − 1] then there will be many different such tuples.
It is easy to check that Inert(ρ) depends only on ρ ss | I t .

G K ∞ -representations
Then restriction defines an isomorphism Gal Proof Since K ∞ /K is totally wildly ramified we have K ∞ ∩ K t = K . The lemma then follows from Galois theory.

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 The module gr(A) admits an obvious structure of a ring and each gr(M) admits the structure of a module over gr(A).

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 The modules ker( f ) ⊂ M and im( f ) ⊂ N are each equipped with the induced filtration. The modules coker( f ) and coim( f ) are equipped with the quotient filtration, coming from N and M respectively.

Proof
The following diagram commutes and has exact rows.
Since M → N is an isomorphism of A-modules the leftmost and central vertical arrows are injective. For (1) use the snake lemma to obtain an exact sequence 0 → ker c → coker(a) → coker(b) → coker(c). One proves F i M → F i N is surjective by increasing induction on i; using as the base case the fact that (2) argue as in [13,Proposition 6].  Proof Argue as in [13, Corollary 1] 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 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 −r i 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−r i A. It follows that m = (a r i −n a i )(a −r i g i ) and so, since (a r i −n )F n−r i A ⊂ A, we are done.

Filtered vector spaces
Finally we give criteria to determine when two filtrations on a vector space are the same.

Lemma 3.3.1 Suppose A = k is a field and let V be an k-vector space equipped with two discrete filtrations
with equality if and only if G = F.
The desired inequality follows. This inequality is an equality if and only if dim

Corollary 3.3.3 Suppose A = k is a field and let
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 0 → gr(M) → gr(N ) → gr(coker( f )) → 0 is exact, and so 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).

Etale '-modules
First we recall the description of G K ∞ -representations given by etale ϕ-modules.
Definition 4.1.1 Let O C be the inverse limit of the system 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 defines a valuation on C for which it is complete. The field C is also algebraically closed. Further, the action of G K on O C induces a continuous action of G K on O C and C .
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 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 the 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. Proof The embedding O E → W (C ) reduces modulo p to an inclusion of k((u)) in C . The completion of K ∞ is a perfectoid field in the sense of [11], whose tilt is the completed perfection of k((u)) ⊂ C . It follows from [11,Theorem 3.7] that the action of G K ∞ on C identifies G K = G k((u)) . Let O E ur be the p-adic completion of the Cohen ring (i.e. the discrete valuation ring of characteristic zero with uniformizer p) with residue field k((u)) sep . Then O E ur may be identified as a subring of W (C ) stable under the action of G K ∞ and ϕ. The proposition with T (M et ) replaced by 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.
], the product running over τ ∈ Hom F p (k, F) (we abusively write τ also for its extension to an embedding τ : which becomes an isomorphism after inverting τ (E).

Torsion Breuil-Kisin modules
is an exact sequence of G K ∞ -representations then there exists a unique exact sequence This gives rise to a second composition series which evidently has different irreducible factors as the composition series above. This phenomenon is related to the fact that 0 → M 1 → M → M/M 1 → 0, while not itself ϕ-equivariantly split, becomes so after inverting u.

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.   (1) implies (2) 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 as

Strong divisibility with coefficients
We reproduce the previous subsection allowing O-coefficients.
and where two f -tuples of matrices satisfy • The multiset {r i,τ } is the multiset Weight τ (M).
• The M which satisfy Lemma 5.3.4 correspond to classes represented by an f -tuple of matrices (A τ ) such that each A τ = C τ Λ τ .
denote the full subcategory whose objects are strongly divisible when viewed as objects of Mod BK k .

Subquotients
We now show Mod SD k and Mod SD k (O) are closed under subquotients.  Using the second exact sequence of Remark 5.4.1 we obtain the following commutative diagram with exact rows.

The map N → P is strict when viewed as a map of filtered modules if and only
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.  Proof Consider the following commutative diagram.

It remains to show that if
The left and right vertical arrows are isomorphisms by assumption. Since N → P is strict, part (1)    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 ⊗ F p F-modules. Thus it decomposes into exact sequences

Irreducible objects
Provided F is sufficiently large, irreducible F-representations of G K and G K ∞ are induced from characters, see Lemma 2.1.2. In this section and the next we investigate the extent with which this is true for objects of Mod SD k (O). Throughout assume k ⊂ F.

Rank ones
where r τ ∈ Z and where (x) = xe τ 0 + τ =τ 0 e τ for some x ∈ F × .  Proof This is [9,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.

Induction and restriction
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.
for any continuous G K ∞ -representation V on a finitely generated Z p -module. As (6.2.5) is functorial in V , Yoneda's lemma provides the isomorphism ι N . As (6.2.5) is functorial in N we see that ι N is functorial. and Proof By functoriality both f * and f * preserve O-actions. Note that the inclusion (1) and (2) then follow by verifying the second condition of Lemma 5.3.4.

Approximation by induced Breuil-Kisin modules
We consider the situation from Notation 6.2.1. Thus L/K is a finite unramified extension, corresponding to an extension l/k of residue fields, and L ∞ = L(π 1/ p ∞ ). We also have the map f : S → S L . When T (M) is irreducible and F is sufficiently large T (M) is induced from a character. Thus, Lemma 6.3.1 produces an inclusion M → f * N with N of rank one. Lemma 6.1.1 allows us to describe N explicitly. In this case we would like to know which submodules of f * N arise in this way. The following example shows that there are non-trivial (i.e. M = f * N ) possibilities.

An example
Take K = Q p and let L/K be of degree 5 with residue extension l/k. Let N ∈ Mod SD l (O) be the rank one object defined by Here we have fixed θ ∈ Hom F p (l, F) and 1 ≤ n ≤ p, 0 ≤ x ≤ p. Let M ⊂ f * N be the sub-module generated over F [[u]] by e θ•ϕ 4 , e θ•ϕ 3 + e θ•ϕ , e θ•ϕ 2 , ue θ•ϕ , e θ . One computes that This shows that M ∈ Mod SD k (O). One checks that M = f * N for any rank one N ⊂ N .

Irreducibility and strong divisibility
Let L/K , l/k and L ∞ /K ∞ be as in Notation 6.2.1; we obtain f : S → S L . Let N ∈ Mod SD l (O) be the rank one object given by Note this N is as in Lemma 6.1.1, except we've fixed x = 1. This is to simplify notation (it will be easy to reduce from the general case to this one). The following proposition describes which Breuil-Kisin modules embed into f * N as in Lemma 6.3.1. Let us show this implies (2).

Finishing the proof of Proposition 6.5.1
Let N be as in the previous subsection and suppose that Assume that M satisfies conditions (1), (2) and (3) from Proposition 6.5.1. We are going to prove that M ∈ Mod SD k (O). Along the way we shall describe the weights of M in terms of the r θ . Construction 6.6.1 For a fixed λ ∈ Hom F p (l, F) define an ordering on Hom F p (l, F) by asserting that Using this ordering we define X ⊂ Hom F p (l, F) by θ / ∈ X ⇔ there exists α κ ∈ F such that e θ + κ< λ θ α κ e κ ∈ M (6.6.2) Clearly X depends upon the choice of λ. in which the sum runs over κ ∈ X satisfying (i) κ < λ θ (ii) r κ ≡ r θ modulo p and (iii) κ| k = θ | k . In particular, the element lies in M θ| k .
(2) As θ / ∈ X , there exists e θ + α κ e κ ∈ M with the sum running over κ < λ θ . Arguing inductively one shows there exists such a sum running only over those κ < λ θ with κ ∈ X . There can be at most one sum of this form; indeed their difference would be a sum as in (1) and so would be zero. Condition (3) of Proposition 6.5.1 therefore implies the sum may be taken to run over κ additionally satisfying (ii). As M = τ ∈Hom F p (k,F) M τ we also have (iii). with 0 ≤ I < [l : F p ] and ι ∈ Hom F p (l, F). We say (6.6.5) is minimal if there exists no ι ∈ Hom F p (l, F) together with an F-linear combination e ι + 0< j≤J α j e ι •ϕ j ∈ M such that J < I . Note that for a fixed ι there can exist at most one minimal sum as in (6.6.5); if there were two their difference would have shorter length.
Note that when there exists a θ such that e θ ∈ M then the minimal elements are simply scalar multiples of e θ for any θ with e θ ∈ M. Lemma 6.6.6 If (6.6.5) is a minimal sum then r ι•ϕ j = r ι whenever α j = 0 and j ≤ I .
Proof Uniqueness of minimal elements and condition (3) of Proposition 6.5.1 implies r ι ≡ r ι•ϕ i modulo p. Since each r ι•ϕ j ∈ [0, p] this will be an equality, except possibly if r ι = 0 or p. In this case set where γ j = 0 if r ι•ϕ j = p and γ j = 1 if r ι•ϕ j = 0. Then ϕ(z) equals u p times (6.6.5) and so condition (1) of Proposition 6.5.1 implies z ∈ M. Thus, either all γ i = 0 or all equal 1, otherwise we would obtain an element of M contradicting the minimality of (6.6.5).

Proof (End of the proof of Proposition 6.5.1)
We have to show M is strongly divisible. Fix λ as in Proposition 6.6.7 and for θ ∈ Hom F p (l, F) set f θ = e θ + α κ e κ as in Lemma 6.
To see this let W ⊂ M τ be the subspace they span. It is easy to see that if θ | k = τ then ue θ ∈ W . It therefore suffices to show any α θ e θ ∈ M τ with α θ ∈ F is in W . We see that α θ e θ − θ / ∈X α θ f θ is an F-linear combination of e θ with θ ∈ X , and is contained in M. Such a linear combination must be zero (cf. the proof of Lemma 6.6.3) so W = M τ , as claimed.
For each θ we now construct elements onto u r θ h θ (note the term u r κ =0 α κ e κ•ϕ appears only if r θ = p). As r θ > 0 this displayed sum is contained in M by condition (1) of Proposition 6.5.1. We claim this displayed sum is equal to To see this note that, by (1) of Proposition 6.6.7, if r κ = 0 then κ • ϕ ∈ X and if r κ•ϕ / ∈ X then r κ > 0. From this it follows that the difference between these two sums, which is an element of M, is an F-linear combination of e κ with κ ∈ X . This difference is therefore zero, and so ϕ(g θ•ϕ ) = u r θ h θ . As h θ ∈ M θ| k we have (1) and (3) of Proposition 6.6.7 implies r θ = 1, so we have to show ϕ(g θ•ϕ ) = h θ . Proposition 6.6.7 tells us κ ∈ X and κ • ϕ −1 / ∈ X implies r κ•ϕ −1 = 0, while if κ ∈ X and κ • ϕ −1 ∈ X then r κ•ϕ −1 ∈ [0, 1]. Using these two facts we see that the difference between ϕ(g θ•ϕ ) and h θ is an F-linear combination of e κ with κ ∈ X . Since this difference is contained in M it must be zero. As g θ•ϕ ∈ M θ•ϕ| k we have h θ ∈ M θ| k .

Putting everything together
Applying what we've shown so far in this subsection gives: Proposition 6.7.1 Let M ∈ Mod SD k (O) with T (M) irreducible. Then there exist integers r θ indexed over θ ∈ Hom F p (l, F) such that (i): for some unramified character ψ and for L ∞ = L(π 1/ p ∞ ) with L an unramified extension K , and such that (ii): Weight τ (M) = { r θ | θ | k = τ } Proof Lemma 6.3.1 produces a rank one N ∈ Mod SD k (O), which we assume is as in Lemma 6.1.1, together with an embedding M → f * N . We want to apply the results of Sects. 6.5 and 6.6, so we require the x ∈ F × appearing in the definition of N to equal 1. Let us explain how to reduce to this case. Let ur x ∈ Mod SD k (O) be the rank one object given by Thus, if the proposition holds for M it holds for M, and so we may assume x = 1. Applying Corollary 6.6.10, Weight τ (M) = {r θ + ps θ•ϕ − s θ | θ | k = τ }. On the other hand, χ = T (N ) and this equals θ ω r θ + ps θ•ϕ −s θ θ by Proposition 6.1.3. Therefore, we can take r θ = r θ + ps θ•ϕ − s θ .

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

Crystalline representations and Breuil-Kisin modules
As in [6] let B dR denote Fontaine's ring of p-adic periods, and B crys ⊂ B dR the ring of crystalline periods. As in [7] a p-adic representation V of G K is crystalline if D crys (V ) := (V ⊗ Q p B crys ) G K has K 0 -dimension equal to dim Q p V . The inclusion B crys ⊗ K 0 K ⊂ B dR induces an equality D crys (V ) K := D crys (V ) ⊗ K 0 K = (V ⊗ Q p B dR ) G K which allows us to equip D crys (V ) K with the filtration Here B + dR ⊂ B dR is the discrete valuation ring with field of fractions B dR , and t is any choice of uniformiser. Proof This is the main result of [10]. The formulation we give here is taken from [4,Theorem 4.4].

Proof of main theorem
We can now give the proof of the theorem in the introduction. Assume K = K 0 . Recall that if ρ : G K → GL n (F p ) is a continuous representation then in Definition 2.1.3 we defined the set Inert(ρ). Theorem 7.2.1 Let K = K 0 . Let ρ : G K → GL n (Z p ) be crystalline and suppose that HT τ (ρ) = (λ 1,τ ≤ · · · ≤ λ n,τ ) with λ n,τ − λ 1,τ ≤ p. Then Since this is true for each i we deduce (λ τ ) ∈ Inert(ρ).