ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL TONG LIU ABSTRACT. For p 3 an odd prime and a nonnegative integer r p 2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the antiequivalence of categories between the category of strongly divisible lattices of weight r and the category of Galois stable Z p -lattices in semi-stable p-adic Galois representations with Hodge-Tate weights in {0,..., r}. CONTENTS 1. Introduction 2 2. Preliminary and the Main Result 3 2.1. Semi-stable Galois representations and weakly admissible modules 4 2.2. Breuil s theory on filtered (ϕ, N)-modules over S 5 2.3. The Main Theorem 6 3. Construction of Quasi-Strongly Divisible Lattices 8 3.1. (ϕ, N )-modules. 9 3.2. A functor from Mod ϕ,n to MF (ϕ, N). /O 10 3.3. Finite ϕ-modules of finite height and finite Z p -representations of G. 12 3.4. G -stable Z p -lattices in a semi-stable Galois representation 14 3.5. Fully faithfulness of T st. 17 4. Cartier Dual and a Theorem to Connect M with T cris (M) 18 4.1. Structure of filtration of quasi-strongly divisible lattice. 18 4.2. Cartier dual on Mod ϕ. 20 4.3. Application to Galois representations 20 5. The Proof of Lemma 3.5.3 23 5.1. G-action on A cris S D 24 5.2. A Q p -version of Theorem 4.3.4 25 5.3. Proof of the Main Theorem 27 References 29 1991 Mathematics Subject Classification. Primary 14F30,14L05. Key words and phrases. p-adic representations, Semi-stable, Strongly divisible lattices. 1
2 TONG LIU 1. INTRODUCTION Let k be a perfect field of characteristic p > 2, W(k) its ring of Witt vectors, K 0 = W(k)[ 1 p ], K/K 0 a finite totally ramified extension and e = e(k/k 0 ) the absolute ramification index. We are interested in understanding semi-stable p-adic Galois representations of G := Gal( K/K). An important result in this direction is proved by Colmez and Fontaine [CF00]: semi-stable p-adic Galois representations are classified by weakly admissible filtered (ϕ, N)-modules. Since G is compact, any continuous representation ρ : G GL n (Q p ) admits a G-stable Z p -lattice. It is thus natural to ask whether there also exists a corresponding integral structure on the side of filtered (ϕ, N)-modules. Fontaine and Laffaille [FL82] first attacked this question by defining W(k)-lattices in filtered (ϕ, N)-modules. Unfortunately, their theory only works for the case e = 1, N = 0 and Hodge-Tate weights in {0,..., p 2}. In the late 1990s, Breuil introduced the theory of filtered (ϕ, N)- modules over S to study semi-stable Galois representations ([Bre97], [Bre98b], [Bre99a]), where S is the p-adic completion of divided power envelop of W(k)[u] with respect to the ideal (E(u)), and E(u) is the Eisenstein polynomial for a fixed uniformizer π of K. Breuil proved that the knowledge of filtered (ϕ, N)-modules over S is equivalent to that of filtered (ϕ, N)-modules (See Theorem 2.2.1 for the precise statement). Furthermore, it turns out that there are integral structures, strongly divisible lattices, which naturally live inside filtered (ϕ, N)-modules over S. These structures allow for arbitrary ramification of K/K 0. For a strongly divisible lattice M, Breuil constructed a G-stable Z p -lattice T st (M) in a semi-stable Galois representation and raised the following conjecture (the main conjecture in [Bre02]): Conjecture 1.0.1. Fix a nonnegative integer r p 2, the functor T st establishes an antiequivalence of categories between the category of strongly divisible lattices of weight r and the category of G-stable Z p -lattices in semi-stable representations of G with Hodge-Tate weights in {0,..., r}. If r 1, the conjecture has been proved by Breuil in [Bre00] and [Bre02]. The case e = 1 was shown by Fontaine and Laffaille in [FL82] for crystalline representations. In [Bre99a], Breuil proved that there at least exists a strongly divisible lattice in the side of filtered (ϕ, N)-modules over S if er < p 1. Based on this result, Breuil [Bre99c] proved the case e = 1 for general semi-stable representations and Caruso [Car05] proved the Conjecture for er < p 1. Their ideas involve a weak version of Conjecture 1.0.1, see the end of 2.3 for details. In [Fal99], Faltings proved that the restriction of T st to the subcategory of filtered free strongly divisible lattices is fully faithful. In this paper, we give a complete proof for the above conjecture by using results of Kisin ([Kis05]). Let K = n 1 K( pn π), G = Gal( K/K ) and S = W(k)[[u]]. We equip S with the endomorphism ϕ which acts via Frobenius on W(k), and sends u to u p. Let Mod ϕ denote the category of finite free S-modules M equipped with a ϕ-semi-linear map ϕ M : M M such that the cokernel of S-linear map 1 ϕ M : S ϕ,s M M is killed by E(u) r. In [Kis05], Kisin proved that any G -stable Z p -lattice T in a semi-stable Galois representation comes from an object (M, ϕ) in Mod ϕ. Using the functor M S ϕ,s M provided by Breuil, Kisin s theory allows us to construct quasi-strongly divisible lattices, i.e, strongly divisible lattices without considering monodromy, to establish an anti-equivalence between
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 3 the category of quasi-strongly divisible lattices and the category of G -stable Z p - lattices in semi-stable Galois representations. Furthermore, we prove that a quasistrongly divisible lattice is strongly divisible if and only if the corresponding G - stable Z p -lattice is G-stable (see Theorem 3.5.4 for the more precise statement). Conjecture 1.0.1 then follows. The paper proceeds as follows. In 2, after briefly reviewing the theory of semistable p-adic Galois representations, filtered (ϕ, N)-modules over S and definition of (quasi-)strongly divisible lattices, we are then able to give a precise statement of our main theorem. 3 is devoted to review Kisin s theory from [Kis05], which allows us to construct quasi-strongly divisible lattices and establishes an anti-equivalence between the category of quasi-strongly divisible lattices and the category of G - stable Z p -lattices in semi-stable Galois representations; and the full faithfulness of T st follows from this. In the next two sections, we prove that a quasi-strongly divisible lattice is strongly divisible if and only if the corresponding G -stable Z p -lattice is G-stable. The idea is to use an extended version of a Falting s theorem (Theorem 5, [Fal99]). The proof of such a theorem (Theorem 4.3.4) mainly depends on the construction of the Cartier dual for quasi-strongly divisible lattices from [Car05], which we discuss in 4. In the last section, we combine our previous preparations to prove the essential surjectivity of T st. Acknowledgment: It is a pleasure to thank T. Arnold, C. Breuil, X. Caruso, B. Conrad and M. Kisin for very useful conversations and correspondences during the preparation of this paper. Our overwhelming debt to Mark Kisin will be obvious to readers. I would like to thank him in particular for pointing out me the possibility to prove the Main Conjecture by his result. The author wrote this paper as a post-doc of European Network AAG in Université de Paris-Sud 11. The author is grateful to Université de Paris-Sud 11 for its hospitality. 2. PRELIMINARY AND THE MAIN RESULT This paper discusses lots of categories and functors. However, we may summarize their relations and our main results as the following diagram: Mod ϕ,n D /O Mod ϕ,n D O MF(ϕ, N) Θ Z p Q p MF W (ϕ, N) D MF (ϕ, N) D MF W (ϕ, N) V st Rep st Q p (G) Rep Qp (G ) Mod ϕ,n Mod ϕ Mod ϕ,n Mod ϕ M S Mod ϕ T st T cris Here is a general explanation of the above diagram: T S Rep st Z p (G) Rep st Z p (G ) Rep Zp (G ) Rep Zp (G ) T cris Rep Zp (G )
4 TONG LIU Injection arrows symbolize fully faithful functors. The notations Rep st symbolize the categories of semi-stable representations with Hodge-Tate weights in {0,..., r}. The first column is about Kisin s theory on ϕ-modules over S. The second column is about classical modules in Fontaine s theory and the third about Breuil s theory on S-modules. These three theories can be connected by auxiliary categories in the first row (see 3.2). The last two columns are about the Galois sides. Note that representations of G (e.g., G -stable Z p -lattices inside semi-stable representations) can be more conveniently described by Kisin s theory (see 3.1 and 3.4). The second row is about the theory over Q p whereas the third row is about the theory over Z p, which also is the key result of this paper. Many important inputs depend on the last two rows where are about theories on Z p -representations of G (see 3.3 and 3.4). 2.1. Semi-stable Galois representations and weakly admissible modules. Fix an odd prime p. Recall that a p-adic representation is a continuous linear representation of G := Gal( K/K) on a finite dimensional Q p -vector space V and a p-adic representation V of G is called semi-stable ([Fon94b]) if: (2.1.1) dim K0 (B st Qp V) G = dim Qp V, where B st is the period ring constructed by Fontaine, see for example [Fon94a] or 2.2 for the construction. In [CF00] and [Fon94b], Fontaine and Colmez gives an alternative description of semi-stable p-adic representations. Recall that a filtered (ϕ, N)-module is a finite dimensional K 0 -vector space D endowed with: (1) a Frobenius semi-linear injection: ϕ : D D. (2) a linear map N : D D such that Nϕ = pϕn. (3) a decreasing filtration (Fil i D K ) i Z on D K := K K0 D by K-vector spaces such that Fil i D K = D K for i 0 and Fil i D K = 0 for i 0. If D is a one dimensional (ϕ, N)-module, and v D is a basis vector, then ϕ(v) = αv for some α K 0. We write t N (D) for the p-adic valuation of α (p-adic valuation of α does not depends on choice of v) and t H (D) the unique integer i such that gr i D K is non-zero. If D has dimension d > 1, then we write t N (D) = t N ( d D) and t H (D) = t H ( d D). Recall that a filtered (ϕ, N)-module is called weakly admissible if t H (D) = t N (D) and for any (ϕ, N)-submodule D D, t H (D ) t N (D ), where D K D K is equipped with the induced filtration. The aforementioned result of Colmez and Fontaine [CF00] is that the functor D st, : V (B st Qp V) G establishes an equivalence of categories between the category of semi-stable p-adic representations of G and the category of weakly admissible filtered (ϕ, N)-modules. In the sequel, we will instead use the contravariant functor D st (V) := D st, (V ), where V is the dual representation of V. The advantage of this is that the Hodge- Tate weights of V is exactly the i Z such that gr i D st (V) K 0. A quasi-inverse to D st is then given by : (2.1.2) V st (D) := Hom ϕ,n (D, B st ) Hom Fil (DK, K K0 B st ).
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 5 Convention 2.1.1. Here we use a little different notations from those in [Bre02] and [CF00]. D st here is D st in [Bre02] and [CF00]; V st here is Vst in [Bre02] and [CF00]. Also we will use T st to denote Tst in [Bre02] and [Bre99a] later. The reason for using such notations is that we will always use contravariant functors instead of covariant functors in this paper. Removing from the superscript looks more neat and convenient. A filtered (ϕ, N)-module is called positive if Fil 0 D = D. In this paper, we only consider positive filtered (ϕ, N)-modules. We denote the category of positive filtered (ϕ, N)-modules by MF(ϕ, N) and the category of positive weakly admissible filtered (ϕ, N)-modules by MF w (ϕ, N). 2.2. Breuil s theory on filtered (ϕ, N)-modules over S. Throughout the paper we will fix a uniformiser π O K, and E(u) W(k)[u] the Eisenstein polynomial of π. We denote by S the p-adic completion of the divided power envelope of W(k)[u] with respect to Ker(s), where s : W(k)[u] O K is the canonical surjection by sending u to π. For any positive integer i, let Fil i S S be the p-adic closure of the ideal generated by the divided powers γ j (u) = E(u)j j! for all j i. There is a unique map ϕ : S S which extends the Frobenius on W(k) and satisfies ϕ(u) = u p. We define a continuous W(k)-linear derivation N : S S such that N(u) = u. It is easy to check that Nϕ = pϕn and ϕ(fil i S) p i S for 0 i p 1, and we write ϕ i = p i ϕ Fil i S and c 1 = ϕ 1 (E(u)). Note that c 1 is a unit in S. Finally, we put S K0 := S Zp Q p and Fil i S K0 := Fil i S Zp Q p. Let MF (ϕ, N) be a category whose objects are finite free S K0 -modules D with: a ϕ SK0 -semi-linear morphism ϕ D : D D such that the determinant of ϕ D is invertible in S K0 (the invertibility of the determinant does not depend on the choice of basis). a decreasing filtration over D of S K0 -modules: Fil i (D), i Z, such that Fil 0 (D) = D and that Fil i S K0 Fil j (D) Fil i+j (D). a K 0 -linear map (monodromy) N : D D such that (1) for all f S K0 and m D, N( f m) = N( f )m + f N(m). (2) Nϕ = pϕn, (3) N(Fil i D) Fil i 1 (D). Let D MF(ϕ, N) be a filtered (ϕ, N)-module. We can associate an object D MF (ϕ, N) by the following: (2.2.1) D := S W(k) D and ϕ := ϕ S ϕ D : D D. N := N Id + Id N : D D Fil 0 (D) := D and by induction: Fil i+1 D := {x D N(x) Fil i D and f π (x) Fil i+1 D K } where f π : D D K is defined by λ x s(λ)x. For a D MF (ϕ, N), Breuil associated a Q p [G]-module V st (D). Several period rings have to be defined before we can describe this functor. Let R = lim O K/p where the transition maps are given by Frobenius. By the universal property of
6 TONG LIU Witt vectors W(R) of R, there is a unique surjective map θ : W(R) Ô K to the p-adic completion Ô K, which lifts the projection R O K/p = Ô K/p onto the first factor in the inverse limit. We denote by A cris the p-adic completion of the divided power envelope of W(R) with respect to the Ker(θ), and write B + cris := A cris[1/p]. For each n 0, fix π n K a p n -th root of π such that π p n+1 = π n. Write π = (π n ) n 0 R, and let [π] W(R) be the Teichmüller representation. We embed the W(k)-algebra W(k)[u] into W(R) by u [π]. Since θ([π]) = π this embedding extends to an embedding S A cris, and θ S is the map s : S O K sending u to π. The embedding is compatible with Frobenius endomorphisms. As usual,we denote by B + st the ring obtained by formally adjoining the element log[π] to B+ cris, and by B + the Ker(θ)-adic completion of W(R)[1/p]. Choose a generator t of dr Z p (1) A cris. Such t can be constructed by t := log([ɛ]) for ɛ = (ɛ i ) i 0 R, where ɛ i is a primitive p i -th root of unity such that ɛ p i+1 = ɛ i. We denote B + st [1/t] by B st. Let Âst be the p-adic completion of the P.D. polynomial algebra A cris X. We endow Âst with a continuous G-action, a Frobenius ϕ, a monodromy operator N and positive filtration Fil i as the following: For any g G, let ɛ(g) = g([π]) [π] A cris. We extend the natural G-action and Frobenius on A cris to Âst by putting g(x) = ɛ(g)x + ɛ(g) 1 and ϕ(x) = (1 + X) p 1. We define a monodromy operator N on Âst to be a unique A cris -linear derivation such that N(X) = 1 + X. For any i 0, we define Fil i  st = { a j γ j (X), a j A cris, lim a j = 0, a j Fil i j A cris, 0 j i}. j j=0 Finally, by 4.2 in [Bre97], we have an isomorphism S (Âst) G compatible with all structures given by u [π](1 + X) 1. Therefore, Âst is an S-algebra. For any D MF (ϕ, N), one can associate a Q p [G]-module V st (D) := Hom S,Fil,ϕ,N(D, Âst[1/p]). The following theorem is one of main results in [Bre97]: Theorem 2.2.1 (Breuil). The functor D : D S W(k) D defined in (and below) (2.2.1) induces an equivalence between the category MF(ϕ, N) and MF (ϕ, N) and there is a natural isomorphism V st (D) V st (D) as Q p [G]-modules. From now on, we always identify V st (D) with V st (D) as the same Galois representations, and denote MF w (ϕ, N) the essential image of D restricted to MF w (ϕ, N). 2.3. The Main Theorem. Theorem 2.2.1 shows that the knowledge of filtered (ϕ, N)-modules over S is equivalent to that of filtered (ϕ, N)-modules. It turns out that integral structures can be more conveniently defined inside filtered (ϕ, N)- modules over S. However, when working on integral p-adic Hodge theory via S-modules, the following technical restriction has to be always assumed. Assumption 2.3.1. Fix a positive integer r p 2. The filtration on the weakly admissible filtered (ϕ, N)-module D is such that Fil 0 D K = D K and Fil r+1 D K = 0. Equivalently, the Hodge-Tate weights of the semi-stable p-adic Galois representation under consideration are always contained in {0,..., r}.
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 7 Remark 2.3.2. (1) Conjecture 1.0.1 has been proved for r = 0 in 3.1, [Bre02]. So we only consider the case r > 0 from now on (r = 0 will cause a little trouble only in the end). (2) Up to the twist of the (ϕ, N)-module of a power of the cyclotomic character, all modules whose filtration length does not exceed r satisfy the above assumption. Following 2.2 in [Bre02], we define the integral structures inside D to correspond to the Galois stable Z p -lattices. Definition 2.3.3. Let D be a weakly admissible filtered (ϕ, N)-module satisfying Assumption 2.3.1 and D := D(D) MF w (ϕ, N). A quasi-strongly divisible lattice of weight r in D is an S-submodule M of D such that: (1) M is S-finite free and M[ 1 p ] D (2) M is stable under ϕ, i.e., ϕ(m) M. (3) ϕ(fil r M) p r M where Fil r M := M Fil r D. A strongly divisible lattice of weight r in D is a quasi-strongly divisible lattice M in D such that N(M) M. It will be more convenient and explicit to describe the category of (quasi- )strongly divisible lattices by projective limits of torsion objects. Let Mod ϕ,n denote the category whose objects are 4-tuples (M, Fil r M, ϕ r, N), consisting of (1) an S-module M (2) an S-submodule Fil r M M containing Fil r S M. (3) a ϕ-semi-linear map ϕ r : Fil r M M such that for all s Fil r S and x M we have ϕ r (sx) = (c 1 ) r ϕ r (s)ϕ r (E(u) r x). (4) a W(k)-linear morphism N : M M such that : (a) for all s S and x M, N(sx) = N(s)x + sn(x). (b) E(u)N(Fil r M) Fil r M. (c) the following diagram commutes: Fil r M ϕ r M (2.3.1) E(u)N Fil r M ϕ r M c 1 N Morphisms are given by S-linear maps preserving Fil r s and commuting with ϕ r and N. A sequence is defined to be short exact if it is short exact as a sequence of S-module, and induces a short exact sequence on Fil r s. We denote by Mod ϕ the category which forgets the operation N in the definition of Mod ϕ,n. Objects in Mod ϕ are called filtered ϕ-module over S. Let Mod FIϕ,N (resp. Mod FI ϕ ) be the full subcategory of Mod ϕ,n (resp. Mod ϕ ) consisting of objects such that (1) as an S-module M is isomorphic to i I S/p n i S, where I is a finite set and n i is a positive number. (2) ϕ r (M) generates M over S.
8 TONG LIU Finally we denote by Mod ϕ,n (resp. Mod ϕ ) the full subcategory of Mod ϕ,n (resp. Mod ϕ ) such that M is a finite free S-module and for all n, (M n, Fil r M n, ϕ r, N) Mod FI ϕ,n (resp. (M n, Fil r M n, ϕ r ) Mod FI ϕ ), where M n = M/p n M, Fil r M n = Fil r M/p n Fil r M, and ϕ r, N are induced by modulo p n. Note that Âst Mod ϕ,n. For any M Modϕ,N, define T st (M) := Hom Mod ϕ,n (M, Âst). Proposition 2.3.4 (Breuil). (1) If M is a quasi-strongly divisible lattice in D with D MF w (ϕ, N), then (M, Fil r M, ϕ r ) is in Mod ϕ where ϕ r := ϕ/p r. (2) The category of strongly divisible lattices of weight r is just Mod ϕ,n. In particular, for any M Mod ϕ,n, there exists a D MFw (ϕ, N) such that D(D) M Zp Q p as filtered (ϕ, N)-modules over S. Furthermore, T st (M) is a G-stable Z p -lattice in V st (D). Proof. (1) is a Proposition 2.1.3 in [Bre99a] and Theorem 2.2.3 in [Bre02] From now on, we use Mod ϕ,n to denote the category of strongly divisible lattices of weight r and regard Mod ϕ as a full subcategory of Modϕ, where Mod ϕ denote the category of quasi-strongly divisible lattices. Now we can state our Main Theorem: Theorem 2.3.5 (Main Theorem). If 0 r p 2, the functor M T st (M) establishes an anti-equivalence of categories between the category of strongly divisible lattices of weight r and the category of G-stable Z p -lattices in semi-stable p-adic Galois representations with Hodge-Tate weights in {0,..., r}. Remark 2.3.6. In fact, there exists a weak version of Conjecture 1.0.1: Fix a D inside MF w (ϕ, N). Consider the restriction of the functor T st, namely, T st D : {strongly divisible lattices in D} {G-stable Z p -lattices in V st (D)}. The weak version claims that all functors T st D are equivalences. It is obvious that Conjecture 1.0.1 implies the weak one. On the other hand, from the weak version, one can deduce the essentially surjectivity of T st. Therefore if the full faithfulness of T st has been known, then the weak version and the strong version are equivalent. [Car05] and [Bre98a] used this ideal to prove some special cases of Conjecture 1.0.1. 3. CONSTRUCTION OF QUASI-STRONGLY DIVISIBLE LATTICES Let T be a G-stable Z p -lattice in a semi-stable Galois representation V with Hodge-Tate weights in {0,..., r}. In this section, we will use the theory from [Kis05] to prove that there exists a quasi-strongly divisible lattice M Mod ϕ to correspond to T G. As we will see later, M provides the ambient module for the strongly divisible lattice corresponding to T.
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 9 3.1. (ϕ, N )-modules. We equip K 0 [[u]] with the endomorphism ϕ : K 0 [[u]] K 0 [[u]] which acts via the Frobenius on K 0, and sends u to u p. Suppose that I [0, 1) is a subinterval. We set O I the subring of K 0 [[u]] whose elements converge for all x K such that x I. Put O = O [0,1). By Lemma 2.1 in [Bre97], S can be identified as the subring of K 0 [[u]] whose elements have the following form (3.1.1) n=0 w i u i q(i)!, w i W(k), lim w i = 0, i where q(i) is the quotient in the Euclidean division of i by e. Therefore, for any real number µ satisfying p 1 (p 1)e < µ 1, we have natural inclusions S[1/p] O [0,µ) S K0 compatible with Frobenius. Set c 0 = E(0)/p K 0 and λ = ϕ n (E(u)/pc 0 ) O. We define a derivation N := uλ d du : O O and denote by the same symbol the induced derivation O I O I, for each I [0, 1). By a ϕ-module over O we mean a finite free O-module M, equipped with a ϕ-semi-linear, injective map ϕ : M M. A (ϕ, N )-module over O is a ϕ-module M over O, together with a differential operator N M over N. That is, for any f O and m M, we have N M ( f m) = N ( f )m + f N M (m). ϕ and N M are required to satisfy the relation NM ϕ = (1/c 0)E(u)ϕN M. We will usually write N for N M if this will cause no confusion. The category of (ϕ, N )- modules over O has a natural structure of a Tannakian category. We denote by Mod ϕ,n the category of (ϕ, N /O )-modules M of height r, in the sense that the cokernel of 1 ϕ : ϕ M M is killed by E(u) r for our fixed positive integer r, where ϕ M := O ϕ,o M. In 1.2 of [Kis05], Kisin constructed a functor D : Mod ϕ,n MF(ϕ, N). Let M /O be an object in Mod ϕ,n. Define the underlying K /O 0-vector space of D(M) is M/uM, and the operator ϕ and N are induced by ϕ, N on M. The construction of filtration on D(M) is somewhat not strait forward. First we define a decreasing filtration on ϕ M by Fil i ϕ M = {x ϕ M 1 ϕ(x) E(u) i M}. Fix any fixed real number µ such that p 1 e < µ < p 1 pe. Lemma 1.2.6 in [Kis05] showed that there exists a unique O [0,µ) -linear, ϕ-equivariant isomorphism (3.1.2) ξ : D(M) K0 O [0,µ) ϕ M O O [0,µ). The required filtration on D(M) K is defined to be the image filtration under the composite D(M) K0 O [0,µ) D(M) K0 O/E(u)O D(M) K0 K = D(M) K. Theorem 1.2.8 in [Kis05] shows that the functor D induces an exact equivalence between the category Mod ϕ,n and MF(ϕ, N). /O n=0
10 TONG LIU 3.2. A functor from Mod ϕ,n to MF (ϕ, N). Combining the functor D in 3.1 with /O the functor D in 2.2 together, we obtain a functor D D from Mod ϕ,n to MF (ϕ, N). /O But it will be convenient to give another description of D D for later use. Let M be an object in Mod ϕ,n /O. Define D O(M) = S K0 ϕ,o M, a ϕ SK0 -semi-linear endomorphism ϕ DO (M) := ϕ SK0 ϕ M (as usual, we will drop the subscript of ϕ DO (M) if no confusion will arise) and decreasing filtration on D O (M) by (3.2.1) Fil i (D O (M)) := {m D O (M) (1 ϕ)(m) Fil i S K0 O M}. Note that ϕ(λ) is a unit in S K0, we can define N on D O (M) by N := N 1 + p ϕ(λ) 1 N. We can naturally extend N from O to S K0. Note that for any f S K0 we have N(ϕ( f )) = p ϕ(λ) ϕ(n ( f )). Thus it is easy to check that N is a well-defined derivation of D O (M) over the derivation N of S K0 defined by N(u) = u d du. Proposition 3.2.1. N is well defined on D O (M) and (D O (M), ϕ, Fil i, N) is an object in MF (ϕ, N). Proof. Let D = D O (M). We check that Frobenius, filtration and monodromy defined on D satisfy the required properties listed in 2.2. Since E(u) r kills the cokernel of 1 ϕ : O ϕ,o M M, we see that the determinant of ϕ M is a divisor of E(u) rd, where d is the O-rank of M. Thus the determinant of ϕ D is a divisor of ϕ(e(u)) rd = p rd c rd 1, therefore is invertible in S K 0. Using (3.2.1), one easily checks that Fil i S K0 Fil j D Fil i+j D. Now it suffices to check that the monodromy N satisfies the required properties. To see Nϕ = pϕn, for any s S K0 and m M, we have Nϕ(s m) = N(ϕ SK0 (s) ϕ M (m)) = N(ϕ SK0 (s)) ϕ M (m) + p ϕ(λ) ϕ S K0 (s) N (ϕ M (m)) = pϕ SK0 (N(s)) ϕ M (m) + p ϕ(λ) = pϕ D (N(s) m + p ϕ(λ) s N (m)) = pϕ(n(s m)). To check that N(Fil i D) Fil i 1 D, note that ϕ(e(u)) ϕ(c 0 ) ϕ S K0 (s) ϕ M (N (m)) N (E(u) i ) = uie(u) i 1 E (u)λ = E(u) i ( uie (u) ϕ(λ) pc 0 ). Thus N (Fil i S K0 O M) Fil i S K0 O M. Now let x = i s i m i Fil i D. We claim that (3.2.2) E(u)(1 ϕ M )(N(x)) = c 0p ϕ(λ) N ((1 ϕ M )(x))
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 11 In fact, since E(u)N = c 0p ϕ(λ) N and N ϕ = E(u) c 0 ϕn, we have E(u)(1 ϕ M )(N(x)) = E(u)( N(s i ) ϕ M (m i ) + p ϕ(λ) s i ϕ M (N (m i ))) = = i c 0 p ϕ(λ) ( N (s i ) ϕ M (m i ) + s i N (ϕ M (m i ))) i c 0 p ϕ(λ) N ( s i ϕ M (m i )) i This proves the claim (3.2.2). Finally, to prove N(x) Fil i 1 D, it suffices to show that (1 ϕ M )(N(x)) Fil i 1 S K0 O M. But (3.2.2) has shown us that E(u)(1 ϕ M )(N(x)) Fil i S K0 O M. Then we reduce our proof to the following lemma: Lemma 3.2.2. Let x S (resp. A cris ). If E(u) j x Fil j+i S (resp. E([π]) j x Fil j+i A cris ) then x Fil i S (resp. x Fil i A cris ). Proof. We have a natural embedding S u [π] A cris B + with respect to filtration. dr By definition, Fil n B + dr = E([π])n B + dr for all n 0. Thus, if E([π])j x Fil i+j B + dr then x Fil i B +, as required. dr Corollary 3.2.3. The following equivalences of category commute: MF(ϕ, N) D MF (ϕ, N) D O D Mod ϕ,n /O Proof. Let M Mod ϕ,n and D = D /O O (M). Proposition 3.2.1 has shown that D O (M) MF (ϕ, N). By Theorem 2.2.1, there exists a unique D MF(ϕ, N) such that D O (M) = D(D). It suffices to check that D D(M). There exists an isomorphism i S : S K0 ϕ,o M D K0 S K0 in MF (ϕ, N). Modulo u both sides, we get a K 0 -linear isomorphism i : D(M) D. It is obvious that i is compatible with ϕ and N structures on both sides. To see that i is compatible with filtration, recall that the filtration on D(M) depend on the construction of the unique O [0,µ) -linear, ϕ-equivariant morphism ξ in (3.1.2): ξ : D(M) K0 O [0,µ) ϕ M O O [0,µ) where µ is any fixed real number such that p 1 e < µ < p 1 pe. Choose µ such that p 1 (p 1)e < µ < p 1 pe. By (3.1.1), O[0,µ) is a subring of S K0. Then we have an isomorphism ϕ M O O [0,µ) S K0 M O,ϕ S K0 = D O (M). So ξ O[0,µ) S K0 and i S induce an S K0 -linear, filtration compatible isomorphism (D(M) K0 O [0,µ] ) S K0 D K0 S K0.
12 TONG LIU Both sides define filtration on D(M) and D by modulo E(u) respectively. Therefore, filtration on D(M) and D coincides. 3.3. Finite ϕ-modules of finite height and finite Z p -representations of G. Recall that S = W(k)[[u]] with the endomorphism ϕ : S S which acts on W(k) via Frobenius and send u to u p. In this subsection, we first recall the theory in [Fon90] on finite ϕ-modules over S of finite height and associated finite Z p -representations of G. Then we study the relations between the finite ϕ-module over S of finite height and filtered ϕ-modules over S, and their associated finite representations of G. These results have been essentially done in [Bre98c] and 1.1 in [Kis04]. Denote by Mod ϕ the category of S-modules M equipped with a ϕ-semi-linear map ϕ M : M M such that the cokernel of the S-linear map: 1 ϕ M : S ϕ,s M M is killed by E(u) r. (We always drop subscript M of ϕ M if no confusion will arise.) We give Mod ϕ the structure of exact category induced by that on the abelian category of S-modules. We denote by Mod FI ϕ the full category of Mod ϕ consisting of those M such that as an S-module M is isomorphic to i I S/p n i S, where I is a finite set and n i is a positive integer. Finally we denote by Mod ϕ the full subcategory of Mod ϕ consisting of those M which are S-finite free. Recall that [π] W(R) constructed in 2.2. We embed S W(R) by u [π]. This embedding is compatible with Frobenius endomorphisms. Denote by O E the p-adic completion of S[ 1 u ]. Then O E is a discrete valuation ring with the residue field the Laurent series ring k((u)). We write E for the field of fractions of O E. If FrR denotes the field of fractions of R, then the inclusion S W(R) extends to O E W(FrR). Let E ur W(FrR)[1/p] denote the maximal unramified extension of E contained in W(FrR)[1/p], and O ur its ring of integers. Since FrR is easily seen to be algebraically closed, the residue field O ur /po ur is the separable closure of k((u)). We denote by Êur the p-adic completion of E ur, and by Ôur its ring of integers. Ê ur is also equal to the closure of E ur in W(FrR). We write S ur = Ôur W(R) W(FrR). We regard all these rings as subrings of W(FrR)[1/p]. Recall K = n 0 K(π n ) and G = Gal( K/K ). G naturally acts on S ur and Ôur and fixes the subring S W(R). Denote Rep Zp (G ) the category of continuous finite Z p -representations of G. For an M Mod FI ϕ, one can associate a finite Z p -representation of G by (B 1.8, [Fon90]): T S : M Hom S,ϕ (M, S ur [1/p] ur ). In B.1.8.4 [Fon90] and A.1.2 [Fon90], Fontaine has proved that the functor m T S : Mod FI ϕ Rep Z p (G ) is an exact functor. If M S/p n i S as finite S- modules, then T S (M) m Z/p n i Z as finite Z p -modules. As the consequence, if i=1 M Mod ϕ is a finite free S-module with rank d, define T S (M) = Hom S,ϕ (M, S ur ), then T S (M) is a continuous finite free Z p -representation of G with Z p -rank d. As in [Bre98c] or 1.1 [Kis04], we define a functor M S : Mod ϕ Mod ϕ as follows: we have a map of W(k)-algebra S S given by u u, so we regard S as i=1
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 13 an S-algebra. We will denote by ϕ the map S S obtained by composing this map with ϕ on S. Given an M Mod ϕ, set M = M S(M) := S ϕ,s M. One has the map 1 ϕ : S ϕ,s M S S M. Set Fil r M = {y M (1 ϕ)(y) Fil r S S M S S M} and define ϕ r : Fil r M M as the composite Fil r M 1 ϕ Fil r S S M ϕ r 1 S ϕ,s M = M. This gives M the structure of an object in Mod ϕ. We have the following result similar to Lemma 2.2.1 in [Bre98c] and Proposition 1.1.11 in [Kis04]. Proposition 3.3.1 (Breuil, Kisin). The functor M S : Mod ϕ Mod ϕ defined above induces an exact and fully faithful functor M S : Mod FI ϕ Mod FIϕ. This functor is an equivalence of categories between the full subcategories consisting of objects killed by p. Proof. Lemma 2.2.1 in [Bre98c] and Proposition 1.1.11 in [Kis04] proved the case r = 1. The idea of proof can be easily extended for 0 r p 2. In particular, the equivalence of subcategories consisting of p-torsion objects is again (almost) verbatim the proof of Theorem 4.1.1 in [Bre99a]. Corollary 3.3.2. The functor M S : Mod ϕ Mod ϕ induces an exact and fully faithful functor M S : Mod ϕ Modϕ. Remark 3.3.3. In fact, the functor M S can be proved to be an equivalence ([CL06]). Note that A cris is an object in Mod ϕ by defining ϕ r := ϕ/p r on Fil r A cris. For any M Mod ϕ, one can define a finite free continuous Z p-representation of G : (3.3.1) T cris : M Hom Mod ϕ (M, A cris ) as in 2.3.1 in [Bre99a]. Let M Mod ϕ and M = M S(M) Mod ϕ. For any f T S (M) = Hom S,ϕ (M, S ur ), consider the natural embedding ι : S ur A cris. It is easy to check that ϕ(ι f ) T cris (M) = Hom Mod ϕ A (M, cris ). Therefore, we get a natural map Hom S,ϕ (M, S ur ) Hom Mod ϕ (M S (M), A cris ). Lemma 3.3.4. The natural map T S (M) T cris (M S (M)) defined above is an isomorphism of finite free Z p -representations of G. Proof. This is the consequence of the fact that for any M Mod FI ϕ, the natural map (3.3.2) Hom S,ϕ (M, S ur [1/p] ur ) Hom Mod ϕ (M S (M), A cris [1/p]/A cris ) is an isomorphism of finite Z p [G ]-modules. Note that the left hand side of (3.3.2) is an exact functor on Mod FI ϕ. The right hand side is also an exact functor from the fact that Ext 1 A Mod(M, ϕ cris [1/p]/A cris ) = 0 for any M Mod FI ϕ (Lemma 2.3.1.3 in [Bre99a]). Thus by the standard dévissage, it suffices to prove (3.3.2) for the case that p kills M, and this is Proposition 4.2.1 in [Bre99b].
14 TONG LIU 3.4. G -stable Z p -lattices in a semi-stable Galois representation. A (ϕ, N)-module over S is a finite free ϕ-module M Mod ϕ, equipped with a linear endomorphism N : M/uM Zp Q p M/uM Zp Q p such that Nϕ = pϕn. We denote by Mod ϕ,n the category of (ϕ, N)-module over S, and by Mod ϕ,n Z p Q p the associated isogeny category 1. The following theorem is one of main results (cf. Corollary 1.3.15) in [Kis05]. Theorem 3.4.1 (Kisin). There exists a fully faithful -functor Θ from the category of positive weakly admissible filtered (ϕ, N)-modules MF w (ϕ, N) to Mod ϕ,n Z p Q p. Let M Mod ϕ,n and M = M S O. Then there exists a D MF w (ϕ, N) such that M = Θ(D) if and only if there exists a differential operator N on M such that (M, ϕ, N ) Mod ϕ,n, D(M) D in MF(ϕ, N) and N /O mod u = N on M/uM Zp Q p. Such N (if exists) is necessarily unique. Remark 3.4.2. (1) The above theorem is valid without any restriction of the maximal Hodge-Tate weight. But here we only consider the case that Hodge-Tate weights in {0,..., r} with r p 2. (2) The second paragraph of the above theorem is not the same as that of Corollary 1.3.15 in [Kis05]. But they are equivalent (See Lemma 1.3.10 and Lemma 1.3.13 in [Kis05]), and our description of Theorem 3.4.1 will be more convenient. Furthermore, Kisin proved (cf. Proposition 2.1.5 in [Kis05]) that there exists a canonical bijection (without restriction of maximal Hodge-Tate weights) (3.4.1) η : T S (M) Zp Q p Vst (D) which is compatible with the action of G on the two sides. For our purpose to connect strongly divisible lattices, we reconstruct (3.4.1) in a little different way. Let D MF w (ϕ, N) be a weakly admissible filtered (ϕ, N)-module under our Assumption 2.3.1, M = Θ(D) and (M, ϕ, N ) Mod ϕ,n as in the Theorem 3.4.1. /O Let D = D(D) (Recall D(D) := S W(k) D in 2.2). By Corollary 3.2.3, we have D = S K0 ϕ,o M = S K0 ϕ,s M = M S (M) Zp Q p, where M S is the functor defined in Corollary 3.3.2. Then we have a natural map of Z p [G ]-modules (3.4.2) Hom S,ϕ (M, S ur ) Hom Mod ϕ (M S (M), A cris ) Hom Mod ϕ B (D, + cris ). The first map is an isomorphism by Lemma 3.3.4. Recall that V st (D) = Hom Mod ϕ,n (D, Âst[1/p]). The canonical projection Âst A cris defined by sending γ i (X) to 0 induces a natural map: (3.4.3) Hom Mod ϕ,n (D, Âst[1/p]) Hom Mod ϕ B (D, + cris ). We claim that the above map is an bijection. Let us accept the claim and postpone the proof in Lemma 3.4.3. Recall that Theorem 2.2.1 has shown that there exists 1 Recall that if C is an additive category, then the associated isogeny category D has same objects and Hom D (A, B) = Hom C Z Q for all objects A and B.
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 15 a canonical isomorphism V st (D) V st (D) as Q p -representations of G. Therefore, combining (3.4.2) and (3.4.3), we have a natural injection η : T S (M) Zp Q p V st (D) of Q p [G ]-modules and thus dim Qp (V st (D)) rank S (M) = dim K0 (D). But an elementary argument (Prop. 4.5, [CF00]) showed that weak admissibility of D implies that dim Qp (V st (D)) has to be dim K0 (D). Hence the map η is a bijection. Lemma 3.4.3. The natural map defined in (3.4.3) is a bijection. Proof. We basically follow the idea of Lemma 2.3.1.1 in [Bre99a]. For any f Hom Mod ϕ,n (D, Âst[1/p]), let f 0 be its image of the map in (3.4.3). For any x D where D = D W(k) S, since N i (x) = 0 for i enough big, we can easily check that (3.4.4) f (x) = f 0 (N i (x))γ i (log(1 + X)), i=0 where γ i (x) = xi i! is the standard divided power. So if f 0 = 0, we have f = 0 because D generates D. Thus (3.4.3) is injective. To prove the surjectivity, let f 0 Hom Mod ϕ B (D, + ). For any y D, define cris f (y) = f 0 (N i (y))γ i (log(1 + X)). i=0 To see that f is well defined, note that f (y) converges in B + [[X]], and if x D cris then f (x) converges in Âst[1/p] because N i (x) = 0 for i enough big. By a standard computation, we can easily check that f : D B + [[X]] is S-linear. Therefore cris f : D Âst[1/p] is well defined. It suffices to check that f preserves Frobenius, monodromy and filtration. Since f 0 preserves all these structures, it is a strait forward calculation to check that f preserves Frobenius, monodromy and filtration, combining with the facts that ϕ(log(1 + X)) = p log(1 + X), N(log(1 + X)) = 1, N j (Fil i D) Fil i j D and log(1 + X) Fil 1 Â st. Remark 3.4.4. (1) Let V cris (D) := Hom Mod ϕ B (D, + ). The above lemma gives a cris natural transformation which makes the following diagram commutative: MF (ϕ, N) V cris Rep Qp (G ) MF (ϕ, N) V st Rep Qp (G) (2) From the above proof, we see that the lemma is always valid without any restriction of the maximal Hodge-Tate weight. One advantage of using (ϕ, N)-module over S is that we can classify all G - stable Z p -lattices inside the Galois representation. Lemma 3.4.5 (Kisin). (1) Let V be a semi-stable representation with Hodge-Tate weights in {0,..., r}. For any G -stable Z p -lattice T V, there always exists an N Mod ϕ such that T S(N) T.
16 TONG LIU (2) The functor T S : Mod ϕ Rep Z p (G ) is fully faithful. Proof. These are easy consequences of Lemma (2.1.15) and Proposition (2.1.12) in [Kis05]. Remark that the lemma is valid without restriction of r. Recall that Mod ϕ denote the category of quasi-strongly divisible lattices of weight r. Let M Mod ϕ be a quasi-strongly divisible lattice. By Definition 2.3.3, there exists a D MF w (ϕ, N) such that M D and D D(D) with D weakly admissible. Let V := V st (D) be the semi-stable Galois representation. Then we can associate a G -stable Z p -lattice in V as the following: M T cris (M) = Hom Mod ϕ (M, A cris ) Hom Mod ϕ B (D, + cris ) V st(d) = V. Recall that the isomorphism V st (D) Hom Mod ϕ B (D, + ) has been established in cris Lemma 3.4.3. Therefore T cris induces a functor from Mod ϕ to Repst Z p (G ), where Rep st Z p (G ) denotes the category of G -stable Z p -lattices in semi-stable Galois representations with Hodge-Tate weights in {0,..., r}. Proposition 3.4.6. The functor T cris induces an anti-equivalence between Rep st Z p (G ). Mod ϕ and Proof. We first prove the essential surjectivity of the functor. Let M = Θ(D) as in Theorem 3.4.1 and D = D(D). By corollary 3.2.3 and Theorem 3.4.1, we see that D = M S,ϕ S K0. Suppose that T V is a G -stable Z p -lattice. Then by lemma 3.4.5, there exists an N Mod ϕ, such that T T S(N). We claim that M Zp Q p N Zp Q p. In fact, since T S (M) and T S (N) are G -stable Z p -lattices in V, there exist G -equivariant maps f : T S (M) T S (N) and g : T S (N) T S (M) such that f g = p n Id. By full faithfullness of T S, there exists F : N M and G : M N such that F G = p n Id. Hence the claim follows. Now put N = M S (N). We see that N is a quasi-strongly divisible lattice in D, and by Lemma 3.3.4, T cris (N) = T. This proves that the functor is essential surjective. Let M, N Mod ϕ and f : T cris (N) T cris (M) a morphism of Z p [G ]-module. From the above proof, there exist M, N Mod ϕ such that T S(M) = T cris (M) and T S (N) = T cris (N). Since T S is fully faithful (Lemma 3.4.5 (2)), there exists f : M N a morphism in Mod ϕ such that T S(f) = f. Then by Lemma 3.3.2 and Lemma 3.3.4, we have T cris (M S (f)) = f. It suffices to show that M = M S (M) and N = M S (N). Therefore, we reduce the proof to the following Lemma 3.4.7. Let M, M be two quasi-strongly lattices contained in D. If T cris (M) = T cris (M ) then M = M. We postpone our proof of the Lemma after Lemma 5.3.1.
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 17 We may summarize our discussion in this subsection into the follow commutative diagram: Mod ϕ M S T S Mod ϕ T cris Rep Zp (G ) Mod ϕ T cris Rep st Z p (G ) 3.5. Fully faithfulness of T st. Now suppose that T is a G-stable Z p -lattice in a semistable Galois representation V. By Proposition 3.4.6, there exists a quasi-strongly divisible lattice M in D such that T cris (M) = T G and there exists an M Mod ϕ such that M = M S (M). Proposition 3.5.1. Notations as the above. If N(M) M, then (M, ϕ, Fil r M, N) is a strongly divisible lattice in D and T st (M) = T. Proof. M is clearly a strongly divisible lattice in D. It suffices to prove that T st (M) = T. By Proposition 2.3.4, T st (M) = Hom Mod ϕ,n (M, Âst) V st (D) V st (D) = V is a G-stable Z p -lattice. As in (3.4.3), the canonical projection Âst A cris defined by sending γ i (X) 0 induces a natural map (3.5.1) T st (M) = Hom Mod ϕ,n (M, Âst) Hom Mod ϕ A (M, cris ) = T cris (M). Then we have the following commutative diagram: Hom Mod ϕ,n (M, Âst) Hom Mod ϕ,n (D, Âst[1/p]) Hom Mod ϕ (3.5.1) (M, A cris ) (3.4.3) Hom Mod ϕ (D, B + cris ) T V Thus it suffices to show that (3.5.1) is an isomorphism of Z p -modules. This has been proved in 2.3.1, [Bre99a]. Corollary 3.5.2. The functor T st in the Main Conjecture 1.0.1 is fully faithful. Proof. Let M, M be strongly divisible lattices, D = M Zp Q p, D = M Zp Q p and T st (M), T st (M ) G-stable Z p -lattices in V st (D), V st (D ) respectively. Suppose that f : T st (M) T st (M ) is a morphism of Z p [G]-modules. Tensoring by Q p, there exists an f : D D such that V st (f) = f Zp Q p. It suffices to show that f(m ) M. Select an n such that p n f(m ) M. Then g := p n f is a morphism of strongly divisible lattices and T st (g) = p n f. Note that (3.5.1) is an isomorphism of Z p [G ]-modules. So if g is regarded as a morphism of quasi-strongly divisible
18 TONG LIU lattices, we have T cris (g) = T st (g) = p n f. On the other hand, by Proposition 3.4.6, T cris is fully faithful, there exists a morphism g : M M in Mod ϕ such that T cris (g ) = f. Therefore p n g = g = p n f. Then f = g and f(m ) = g (M ) M. Also we reduce the proof of the essential surjectivity of T st to the following Lemma 3.5.3. Notations as the above, If T is G-stable then N(M) M. We will devote the next two sections to prove this Lemma. Combining with Proposition 3.5.1, Corollary 3.5.2 and Proposition 3.4.6, we prove the Main Theorem (Theorem 2.3.5) and the following Theorem 3.5.4. The functor T cris induces an anti-equivalence between the category of quasi-strongly divisible lattices of weight r and the category of G -stable Z p -lattices inside semi-stable Galois representations with Hodge-Tate weights in {0,..., r}. Furthermore, a quasi-strongly divisible lattice M is strongly divisible if and only if T cris (M) is G-stable. 4. CARTIER DUAL AND A THEOREM TO CONNECT M WITH T cris (M) In this section, we extend a theorem of Faltings (cf. Theorem 5, [Fal99]) to a more general setting to connect filtered ϕ-modules over S with their associated Z p -representations of G. This theorem is one of technical keys to prove Lemma 3.5.3. For this purpose, we need more explicit structure of Fil r M and a notion of Cartier dual for M Mod ϕ. Luckily, such Cartier dual has been available from the thesis of Caruso [Car05]. In the following two section, we always regard W(k)[u] and S as subrings of A cris via u [π], and denote the identity matrix by I. 4.1. Structure of filtration of quasi-strongly divisible lattice. Lemma 4.1.1. Let A be a d d matrix with coefficients in W(k)[u]. Suppose that there exist matrices B and C with coefficients in S and Fil p S respectively such that AB = E(u) r I + C. Then (1) There exists a matrix B with coefficients in S such that AB = E(u) r I. (2) Let a i A cris for i = 1,..., d. If (a 1,..., a d )A is in Fil r A cris, then there exists b i A cris and c i Fil p A cris for i = 1,..., d such that (a 1,..., a d ) = (b 1,..., b d )B + (c 1,..., c d ). Proof. Note that for any f S, we can always write f = f 0 + f 1 with f 0 W(k)[u] and f 1 Fil p S. So B = B 0 + B 1 with B 0 s coefficients in W(k)[u] and B 1 s coefficients in Fil p S. Therefore, E(u) r I = AB 0 + C 1 with C 1 s coefficients in W(k)[u] Fil p S = E(u) p W(k)[u]. Thus C 1 = E(u) p C 2 with C 2 s coefficients in W(k)[u]. Now we have E(u) r I = AB 0 + E(u) p C 2. Since E(u) n 0 p-adically in S when n, I E(u) p r C 2 is invertible. Thus (4.1.1) E(u) r I = AB 0 (I E(u) p r C 2 ) 1. Let B = B 0 (I E(u) p r C 2 ) 1 and we settle (1). For (2), write (a 1,..., a d ) = (b 1,..., b d )+(c 1,..., c d ) with b i W(R) and c i Fil p A cris for i = 1,..., d. It suffices to prove that there exists b i A cris such that (b 1,..., b d ) = (b 1,..., b d )B. Note that (a 1,..., a d )A = (b 1,..., b d )A + (c 1,..., c d )A Fil r A cris.
ON LATTICES IN SEMI-STABLE REPRESENTATIONS: A PROOF OF A CONJECTURE OF BREUIL 19 Then (b 1,..., b d )A Filr A cris W(R) = E(u) r W(R). So there exists b i W(R) such that (b 1,..., b d )A = E(u)r (b 1,..., b d ). Multiplying by B on both sides, we get (b 1,..., b d )AB = E(u)r (b 1,..., b d )B. Finally, (b 1,..., b d ) = (b 1,..., b d )B as required. Proposition 4.1.2. Let M Mod ϕ. There exists α 1,..., α d Fil r M such that (1) Fil r M = (2) E(u) r M d Sα i + (Fil p S)M. i=1 d Sα i and (ϕ r (α 1 ),..., ϕ r (α d )) is a basis of M. i=1 Proof. Considering M/pM, by Proposition 2.2.1.3 in [Bre99a], M/pM has a base adaptée, i.e., there exists a basis (e 1,..., e d ) of M and α 1,..., α d Fil r M such that d (4.1.2) Fil r M/pFil r M = S 1 ᾱ i + Fil p S 1 (M/pM) i=1 such that (ᾱ 1,..., ᾱ d ) = (u r 1ē1,..., u r dēd ) with 0 r i er, where S 1 = S/pS and ᾱ i, d ē i is the image of α i, e i in M/pM respectively. Let M = Sα i + (Fil p S)M. Then M Fil r M. We claim that the natural map f : M/Fil p SM Fil r M/Fil p SM is surjective. To see the claim, note that S/Fil p S W(k)[u]/(E(u) p ) is Noetherian. By Nakayama s lemma, it suffices to show that f mod p is a surjection. Note that i=1 Fil r M/Fil p SM mod p = (Fil r M) 1 /(Fil p SM) 1 where (Fil r M) 1 = Fil r M/pFil r M and (Fil p SM) 1 = Fil p SM/pFil p SM. By (4.1.2), we see that f mod p is surjective and thus prove the claim. Then d (4.1.3) Fil r M = M = Sα i + (Fil p S)M. i=1 Let (α 1,..., α d ) = (e 1,..., e d )A where A is a d d matrix with coefficients in S. Write A = A 0 + A 1 with A 0 s coefficients in W(k)[u] and A 1 s coefficients in Fil p S. Replacing (α 1,..., α d ) by (e 1,..., e d )A 0, we can always assume that A s coefficients are in W(k)[u]. By (4.1.3), there exists d d matrices B, C with coefficients in S, Fil p S respectively such that E(u) r I = AB + C. Then by Lemma 4.1.1, there exists a B with d coefficients in S such that AB = E(u) r I. Therefore E(u) r M Sα i. Since ϕ r (Fil r M) generates M and one always has p ϕ r (Fil p S), we see that (ϕ r (α 1 ),..., ϕ r (α d )) is a basis of M. Let D MF (ϕ, N) be a filtered (ϕ, N)-module over S. Following 3 in [Bre97], we define r (4.1.4) Fil r (A cris S D) = Im(Fil r i A cris S Fil i D), i=0 where Im(Fil r i A cris S Fil i D) is the image of Fil r i A cris S Fil i D in A cris S D. We also define Fil r (A cris S M) = Fil r (A cris S D) (A cris S M). i=1
20 TONG LIU Corollary 4.1.3. Notations as Proposition 4.1.2, d Fil r (A cris S M) = A cris α i + Fil p A cris S M. i=1 Proof. Since we always have Fil r i S Fil i D Fil r D, it is easy to see that Fil r (A cris S D) = A cris S Fil r D. Then the corollary follows the fact that Fil i M = Fil i D M. By the above corollary, we can ϕ Acris -semi-linearly extend ϕ r of M to ϕ r : Fil r (A cris S M) A cris S M and we see that (A cris S M, Fil r (A cris S M), ϕ r ) is an object in Mod ϕ. 4.2. Cartier dual on Mod ϕ. In this subsection, we recall the construction of Cartier dual on Mod ϕ from [Car05]. Let M Modϕ. Define M := Hom S (M, S), and Fil r M := { f M f (Fil r M) Fil r S} ϕ r : Fil r M M, for all x Fil r M, ϕ r ( f )(ϕ r (x)) = ϕ r ( f (x)). Note that ϕ r ( f ) is well defined because ϕ r (Fil r M) generates M. Theorem 4.2.1 (Caruso). The functor M M induces an exact anti-equivalence on Mod ϕ and (M ) = M. Proof. Proposition V 3.3.1 in [Car05] proved the theorem on the category of strongly divisible lattices. But the same proof also works on Mod ϕ if we ignore monodromy. Example 4.2.2. Let S be the Cartier dual of S. Then S is the S-rank-1 quasi-strongly divisible lattice with Fil r S = S and ϕ r (1) = 1. 4.3. Application to Galois representations. Let M Mod ϕ and M its Cartier dual. The canonical perfect pairing M M S in the construction of Cartier dual is compatible with filtration and Frobenius on both sides. Taking Cartier dual on both sides, note that (M ) M by Theorem 4.2.1, we have a map and i induces a pairing i : S M (M ) M M (4.3.1) ĩ : Hom Mod ϕ (M, A cris ) Hom Mod ϕ (M, A cris ) Hom S (S, A cris ). Lemma 4.3.1. The above pairing induces a perfect paring of Z p -representations of G : (4.3.2) T cris (M) T cris (M ) T cris (S ) = Z p (r). Proof. This has been essentially proved in Chapter 5, 4 of [Car05]. The proof consists of two steps. The first step is to check the image of ĩ is in Hom Mod ϕ (S, A cris ). This is basically a direct check by the definition of Cartier dual. The proof of the perfectness of the paring (4.3.2) is non-trivial. It suffices to show that the pairing is perfect by modulo p, and the proof is contained in the proof of Theorem V.