FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS

Ann. Inst. Fourier, Grenoble Article à paraître. FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS by Tong LIU (*) Abstract. Let p be an odd prime, K a finite extension of Q p and G := Gal(Q p /K) the Galois group. We construct and study filtration structures associated torsion semi-stable representations of G. In particular, we prove that two semi-stable representations share the same p-adic Hodge-Tate type if they are congruent modulo p n with n c, where c is a constant only depending on K and the differences between the maximal and minimal Hodge-Tate weights of two representations. As an application, we reprove a part of Kisin s result: the existence of a quotient of the universal Galois deformation ring which parameterizes semi-stable representations with a fixed p-adic Hodge-Tate type. Résumé. Soient p un nombre premier impair, K une extension finie de Q p et G := Gal(Q p /K) son groupe de Galois absolu. Nous construisons et étudions différentes filtrations associées aux représentations semi-stables de G. Nous démontrons en particulier que deux représentations semi-stables de G ont le même type de Hodge Tate si elles sont congrues modulo p n avec n c, où c est une constante dépendant uniquement de K et des différences entre les plus grands et les plus petits poids de Hodge-Tate des deux représentations. Comme application, nous redémontrons une partie d un résultat de Kisin portant sur l existence d un quotient de l anneau des déformations universelles paramétrisant les représentations semi-stables dont le type de Hodge-Tate est fixé. 1. Introduction Let k be a perfect field of characteristic p 3, W (k) its ring of Witt vectors, K 0 = W (k)[1/p], K/K 0 a finite totally ramified extension, G := Keywords: semi-stable representations, filtration. Math. classification: Primary 14F30,14L05. (*) This materials is based upon work supported by National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. This paper is written when the author visit the Institute for Advanced Study. The author is grateful to IAS for its support and hospitality. The author also thanks an anonymous referee for pointing out a mistake for the first version of the paper. The author is partially supported by NSF grant DMS-0901360.

2 Tong LIU Gal(K/K). The aim of this paper to study filtration structure attached to torsion semi-stable representations. If V is a semi-stable representation of G then V can be naturally attached to filtration structure because semi-stable representations are classified by filtered (ϕ, N)-modules via classical p-adic Hodge theory. Since V always admits integral structures and torsion structures, i.e., G-stable Z p -lattices and torsion representation obtained by quotients of such lattices, it is natural to ask if we can associate similar filtration to those integral and torsion structures. If K is unramified and V is crystalline with Hodge-Tate weights in {0,..., p 2} then one can attach such integral and torsion structure via Fontaine-Laffaille theory [7]. The aim of this paper is to construct and study such structures in a more general setting, in particular, without restriction of ramification and Hodge-Tate weights. More precisely, fix an integer r 0. Let Rep st,r Q p denote the category of semi-stable representations of G with Hodge-Tate weights in {0,..., r}, Rep st,r Z p denote the category of G-stable Z p -lattices inside representations which are objects in Rep st,r Q p and Rep st,r tor denote the category of p-power torsion representations T such that there exist lattices Λ 1, Λ 2 Rep st,r Z p satisfying Λ 1 Λ 2 and T Λ 2 /Λ 1. The objects in Rep st,r tor are also called torsion semi-stable representations with Hodge-Tate weights in {0,..., r}. In [13], for any Λ Rep st,r Z p with V := Q p Zp Λ, one can construct a W (k)-lattice M st (Λ) D st (V ) = (B st Qp V ) G which is ϕ-stable and N-stable, where V is the Q p -dual of V (our convention is always slightly different from the traditional convention up to duals, see Convention 2.1 for details). Note that D K := K K0 D st (V ) has a natural filtration structure Fil i D K induced from (B dr Qp V ) G. Now set M K := O K W (k) M st (Λ). It is natural to define that Fil i M K := M K Fil i D K. For any T Rep st,r let j : Λ 1 Λ 2 Rep st,r Z p be the inclusion of two lattices such that T Λ 2 /Λ 1. Since M st is a contravariant functor, there exists a W (k)-linear map M st (j) : M st (Λ 2 ) M st (Λ 1 ). In fact, M st (j) is an injection (see Corollary 3.2.4 in [13]). So we define M st,j (T ) := M st (Λ 1 )/M st (Λ 2 ) and associate a filtration structure on M st,j (T ) K := O K W (k) M st,j (T ) via Fil i M st,j (T ) K = q K (Fil i M st (Λ 1 ) K ), where q K is the natural projection q K : M st (Λ 1 ) K M st,j (T ) K. Note the above construction does depend on the choice of pair of lattices j : Λ 1 Λ 2 such that T Λ 2 /Λ 1. However we prove that there exists a constant c only depending on r and K such that the construction of Fil i M st,j (T ) K is independent on" the choice of j up to a p c -power (see Theorem 2.3 for the precise statement). tor, ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 3 In the last section, we use our theory to understand p-adic Hodge-Tate type. It turns out that the p-adic Hodge-Tate type can be read from p c - torsion level of the representation with c a constant only depending on K and r. More precisely, we proved the following theorem. Theorem 1.1. Assume that K is a finite extension over Q p. Let E be a finite extension of Q p and ρ i : G GL d (O E ) for i = 1, 2 two Galois representations such that V i := E OE ρ i is semi-stable with Hodge-Tate weights in {0,..., r}. There exists a constant c only depending on K and r such that if ρ 1 ρ 2 mod p n with n c then V 1 and V 2 has the same p-adic Hodge-Tate type. In fact, we proved a more general result Theorem 4.18, which allows us to recover a part of the main theorem in [9]. Let E be a finite extension of Q p with the residue field F, V F : G GL d (F) the Galois representation such that the universal deformation ring R VF of V F exists. It turns out that R VF is a complete noetherian local O E -algebra and any ring homomorphism x : R VF A with A an O E -algebra defines a Galois representation x : G GL d (A). Theorem 1.2. Fix a p-adic Hodge-Tate type v. There exists a quotient RV v F of R VF such that for a finite E-algebra B, a map x : R VF [ 1 p ] B factors though RV v F if and only if x is semi-stable with p-adic Hodge-Tate type v. We remark that our construction is different from that of Kisin: We construct a sub-functor D v of the Galois deformation functor D whose deformation admits a lift which is a semi-stable Galois representation with the p-adic Hodge-Tate type v. We prove that D v is pro-representable by R v V F if D is pro-representable by R VF and then the above theorem follows Theorem 4.18. It seems that we can fully recover Kisin s result at least for p > 2 if we also consider Galois type in D v. But we decide not to consider the refined result because we do not see any further advantage (except it looks more natural) of our construction comparing with that of Kisin. 2. Filtration encoded in p-adic Hodge data 2.1. Preliminary and definitions Recall k is 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 with degree e and G := Gal(K/K). Throughout this paper, we fix a uniformiser TOME 00 (XXXX), FASCICULE 0

4 Tong LIU π K with the Eisenstein polynomial E(u) W (k)[u] and a non-negative integer r 0. We denote by Rep st,r Q p the category of semi-stable representations of G whose Hodge-Tate weights are in {0,..., r}, and by Rep st,r Z p G-stable Z p -lattices in representations which are in Rep st,r Q p a filtered (ϕ, N)-module is a W (k)-module M endowed with: the category of. By definition, a ϕ W (k) -semilinear map: ϕ : M M; a W (k)-linear map N : M M such that Nϕ = pϕn; a decreasing filtration (Fil i M K ) i Z on M K := O K W (k) M by O K - submodules such that Fil i M K = M K for i 0 and Fil i M K = {0} for i 0. This definition is slightly different from that traditionally used in [6] because we need treat torsion representations. Morphisms between filtered (ϕ, N)-modules are W (k)-linear maps preserving all structures. We denote by M(ϕ, N, Fil) the category of filtered (ϕ, N)-modules. A filtered (ϕ, N)- module over K 0 is a filtered (ϕ, N)-module D such that D is a finite dimensional K 0 -vector space; ϕ D is an injection (hence a bijection); Fil i D K are K-vector subspaces of D K := K K0 D. By [3] and [5], the functor D st(v ) : V (B st Qp V ) G induces an equivalence between the category Rep st,r Q p and the category of weakly admissible filtered (ϕ, N)-modules over K 0 satisfying Fil (r+1) D K = D K and Fil 0 D K = 0. See [3] for the definition of weak admissibility. In the sequel, we will instead use the contravariant functor D st (V ) := Dst(V ), where V is the dual representation of V, because contravariant functors are more convenient in the integral theory. So let us remind the readers the problem of notations. Convention 2.1. Here we use slightly different conventions from those in [3], where D st defined here is denoted by Dst. Since contravariant functors instead of covariant functors dominate this paper, use D st to denote the contravariant functor will be more convenient. For any finite Z p -module (Q p -module) V, we use V to denote its Z p -dual (Q p -dual). In particular, if V is killed by some p-power, V = Hom Zp (V, Q p /Z p ). We will define p-adic Hodge structures such as Frobenius, monodromy on many different rings and modules. To distinguish them, we sometime add subscripts to ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 5 indicate where those structures are defined. For example, ϕ M is the Frobenius defined on M. We always drop these subscripts if no confusions arise. Throughout this paper, we reserve ϕ and N for various types of Frobenius and monodromy respectively. We denote γ i (x), M d (A) and Id for the standard divided power xi i!, the ring of d d-matrices with coefficients in ring A and the identity map respectively. Let A be a finite Z p -algebra and M an A-module. We always denote M K := O K Zp M, which is an A K := O K Zp A-module. Let D be a filtered (ϕ, N)-module over K 0. Following [13], a lattice M in D is a W (k)-submodule M of D such that M is W (k)-finite free and M[ 1 p ] D M is stable under ϕ, N, i.e., ϕ(m) M, N(M) M. There is a natural filtration structure on M K := O K W (k) M defined by Fil i M K := M K Fil i D K. Hence M is an object in M(ϕ, N, Fil). We use L r (ϕ, N, Fil) to denote the full subcategory of M(ϕ, N, Fil) whose objects are lattices in filtered (ϕ, N)-modules over K 0 satisfying Fil 0 D K = D K and Fil r+1 K D K = 0, and M r (ϕ, N, Fil) to denote the full subcategory of M(ϕ, N, Fil) whose objects are finite W (k)-modules M such that Fil 0 M K = M K and Fil r+1 M K = 0. Apparently, L r (ϕ, N, Fil) is a full subcategory of M r (ϕ, N, Fil). Let M r tor(ϕ, N, Fil) denote the full category of M r (ϕ, N, Fil) whose objects is killed by some p-power. For any O K - module L with decreasing filtration Fil i L, we define the graded module gr i L := Fil i L/Fil i+1 L. Now recall Theorem 2.1.3 in [13], we have Theorem 2.2. There exists a faithful functor M st from the category Rep st,r Z p to the category L r (ϕ, N, Fil). Moreover, let M st Zp Q p denote the functor M st associated to the isogeny categories. Then there is a natural isomorphism between M st Zp Q p and D st. If er < p 1 then M st is exact and fully faithful. Now let us construct a torsion version of M st via the above theorem as in 3 in [13]. We denote Rep st,r tor the category whose objects are torsion semi-stable representations with Hodge-Tate weights in {0,..., r}, in the sense that, for any T Rep st,r tor, there exist G-stable Z p-lattices Λ Λ in a V Rep st,r Q p such that T Λ /Λ as Z p [G]-modules. We call the pair Λ Λ a lift of T. Obviously, for any T Rep st,r tor, the lift is always not unique. A morphism between two lifts j : L L (lifting T ) and j : L L (lifting T ) is a morphism ˆf : L L in Rep st,r Z p such that ˆf(L) L. ˆf induces a TOME 00 (XXXX), FASCICULE 0

6 Tong LIU morphism f : T T in Rep st,r tor. We call ˆf a lift of f (with respect to lifts j and j). Let j : Λ Λ be a lift of T. By Theorem 2.2, we get a morphism M st (j) : M st (Λ ) M st (Λ) in L r (ϕ, N, Fil). Corollary 3.2.4 in [13] showed that M st (j) is injective. Now write j := M st (j) and set M st,j (T ) := M st (Λ)/ j(m st (Λ )). Then M := M st,j (T ) has Frobenius ϕ and monodromy N induced from M st (Λ). Now let us define filtration structure on M K := O K W (k) M. Let q K : M st (Λ) K M K be the natural projection where M st (Λ) K := O K W (k) M st (Λ). We define (2.1.1) Fil i M K := q K (Fil i (M st (Λ) K )) M K, for any i Z. Now M st,j (T ) is an object in M r tor(ϕ, N, Fil). As usual, one can define gr i M K := Fil i M K /Fil i+1 M K. One can prove that q K (gr i (M st (Λ) K )) = gr i M K for any i Z (see Corollary 3.1). Now we can state one of the main results: Theorem 2.3. There exists a positive integer constant c only depending on E(u) and r such that the following statement holds: for any morphism f : T T in Rep st,r tor and any lift j, j of T, T respectively, there exists a morphism g : M st,j (T ) M st,j (T ) in M r tor(ϕ, N, Fil) such that (1) if there exists a morphism of lifts ˆf : j j which lifts f then g = p c M st, ˆf (f). (2) let f : T T be a morphism in Rep st,r tor with j a lift of T and g : M st,j (T ) M st,j (T ) the morphism in M r tor(ϕ, N, Fil) attached to f, j and j. If there exists a morphism of lifts ĥ : j j which lifts f f then g g = p 2c M st, ĥ (f f ). Corollary 2.4. Notations as above, assume that f : T T is an isomorphism and f = f 1 : T T is the inverse map. Then g g Mst,j(T ) = p 2c Id Mst,j(T ) and g g Mst,j (T ) = p 2c Id Mst,j (T ). Moreover, for any i Z, g g gr i (M st,j(t ) K ) = p 2c Id gr i (M st,j(t ) K ) and g g gri (M st,j (T ) K ) = p 2c Id gri (M st,j (T ) K ). Remark 2.5. There are two differences between Theorem 3.1.1 in [13] and the above theorem expect p > 2 here. First, the maps g and g here not only preserve (ϕ, N)-structures (we ignore G K -structures because we only discuss semi-stable representations here, not potentially semi-stable representations as in [13]) but also filtration. In fact, we will see that g = p α g and g = p α g for g and g in Theorem 3.1.1 in [13] with α a constant only depending on E(u) and r. That is, to preserve filtration, we need to ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 7 multiply p α to original g and g. Consequently, the second difference, our constant c is always larger than c in Theorem 3.1.1 in [13], and c is more complicated, depending on not only e and r but also E(u) and r. We use very similar strategy to prove the above theorem. The only difficulty is to deal with filtration which is much involved than other structures for torsion representations. Our main tool is Breuil modules. The next subsection is devoted to study various different filtration structures attached to Breuil modules. 2.2. Filtration on Breuil modules Recall the fixed uniformiser π K with Eisenstein polynomial E(u). Put S := W (k)[[u]. S is equipped with a Frobenius endomorphism ϕ via u u p and the natural Frobenius on W (k). A ϕ-module (over S) is an S-module M equipped with a ϕ-semi-linear map ϕ M : M M. A morphism between two objects (M 1, ϕ 1 ), (M 2, ϕ 2 ) is an S-linear morphism compatible with the ϕ i. We denote by S the p-adic completion of the divided power envelope of W (k)[u] with respect to the ideal generated by E(u). Write S K0 := S[ 1 p ]. There is a unique map (Frobenius) ϕ : S S which extends the Frobenius on S. We write N S for the W (k)-linear derivation on S such that N S (u) = u. Let Fil i S denote the ideal which is the p-adic completion of the ideal generated by E(u)j j! for j i. Both S and S can be regarded as subrings of K 0 [[u]. Set I + S = S uk 0 [[u]. Let M be an S-module of finite type and recall that r is a fixed nonnegative integer. In this subsection, filtration Fil i M of M for i Z are submodules of M satisfying the following filtration conditions: Fil 0 M = M and Fil r+1 M Fil 1 SM. Fil i+1 M Fil i M and Fil i SFil j M Fil i+j M. An operator N on M is called a monodromy operator on M if N is a W (k)-linear map N : M M satisfying N(sx) = N S (s)x + sn(x) for all s S and x M. N is said to satisfy Griffiths Transversality if N(Fil i+1 M) Fil i M for all i Z. Using operator N and Fil i M, one can define another two filtration on M. Set F i M = Fil i M for i r and F i M = M for i 0; For 1 i < r, we inductively (start from i = r 1) define F i M to be the S-submodule generated by N(F i+1 M), F i+1 M and Fil i SM. Let M K be M/Fil 1 SM. Then M K is a finite O K -module. Let f π be the natural projection f π : TOME 00 (XXXX), FASCICULE 0

8 Tong LIU M M K. Then Fil i M K := f π (Fil i M) defines a natural filtration on M K. Define F i M = M for i 0 and F i inductively by the following formula: F i+1 M := {x M f π (x) Fil i+1 M K, N(x) F i M}. Since N satisfies Griffiths Transversality, we have F i M Fil i M F i M for all i Z. Lemma 2.6. F i M and F i M satisfy the filtration condition. Proof. It is easy to check by induction that F i M satisfy the filtration condition. For F i M, it is easy to check that the proof reduces to the statement Fil i SN(F j M) F i+j 1 M, which we will prove by reverse induction on j. It is clear the statement holds for j 0 or j r because Fil i M satisfies Griffiths Transversality. Now suppose that the statement holds for j = l + 1 r. Consider the case j = l. By the construction of F i, it suffices to check that Fil i SN 2 (F l+1 M) F i+l 1 M. Let s Fil i S and x F l+1 M. By induction, sn(x) is in F i+l M. Therefore then N(sN(x)) = N(s)N(x) + sn 2 (x) is in F i+l 1 M by the construction of F i. Note that N(s) Fil i 1 S (set Fil 0 S = S here), then the induction implies that N(s)N(x) is in F i+l 1 M. Hence sn 2 (x) is in F i+l 1 M. The following proposition shows that these three different filtration are not very different. Proposition 2.7. There exist constants c 1 and c 2 only depending on E(u) and r, such that p c1 Fil i M F i M and p c2 F i M Fil i M for 0 i r. Proof. It is easy to see that N S (E(u)) and E(u) are relatively prime in K 0 [u]. So there exists a constant γ such that p γ S is contained in the ideal generated by N S (E(u)) and E(u). Set β (i) 0 = 0 for all i and α 0 = 0. Define recursively β (i) l = v p (r i l + 1) + γ + max{β (i) l 1, α i 1} with 1 l r i. Finally set α i = β (i) r i. We will prove by induction on i that p αi Fil r i M F r i M for 0 i r. By definition, the statement is trivial when i = 0. Now assume that i = j 1 the statement is true, that is, p αj 1 Fil r j+1 M F r j+1 M. Set s = r j. For any x Fil s M, by Griffiths Transversality, E(u) s l N s l (x) Fil s M for 0 l r j. We show by induction on l that p β(j) l E(u) s l N s l (x) F s M. If l = 0 then E(u) s N s (x) E(u) s M F s M by definition. Now assume that the statement is valid for l 1, that is, p β(j) l 1 E(u) s l+1 N s l+1 (x) F s M. Write y = E(u)E(u) s l N s l (x) Fil s+1 M and then note that ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 9 p αj 1 N(E(u) s l+1 N s l (x)) = N(p αj 1 y) is in F s M by the induction on i. On the other hand, N(E(u) s l+1 N s l (x)) = (s l + 1)N(E(u))E(u) s l N s l (x) + E(u) s l+1 N s l+1 (x). By induction on l, we conclude that p λ (s l+1)n(e(u))e(u) s l N s l (x) F s M where λ = max{α j 1, β (j) l 1 }. Note that E(u)E(u) s l N s l (x) pαj 1 F s M. So p λ+γ+vp(s l+1) E(u) s l N s l (x) F s M. Thus we prove that p β(j) l E(u) s l N s l (x) F s M. So p β(j) r j x F s M, and then p αj Fil j M F j M. Now it suffices to set c 1 := max{α 0,..., α r }. Now let us prove by induction on i that there exists constant µ i depending on E(u) and r such that p µi F i M Fil i M. By definition, F 0 M = Fil 0 M. So µ 0 can be assigned to 0. Now assume that statement is valid for i = j 1. That is, there exists µ j 1 such that p µj 1 F j 1 M Fil j 1 M. Now let us consider the case i = j. For any x F j M, we have f π (x) Fil j M K. Hence there exists a y Fil j M, g Fil 1 S and z M such that x = y + gz. Note that there exists a constant λ only depending on r such that for any g S p λ g = g 0 + g 1 with g 0 W (k)[u] and g 1 Fil r S. So there exists a y Fil r M, z M such that p λ x = y + E(u)z. Now we claim that there exist constants ν l such that p ν l f π (N l (z)) Fil j 1 l M K for 0 l j 1. Accept the claim for a while and set ν = Max l {ν l }. We see that f π (N l (p ν z)) Fil j 1 l M K for 0 l j 1. By definition of F i M, we easily see (by reverse induction on l starting from l = j 1) that N l (p ν z) F j 1 l M. In particular, p ν z F j 1 M. By induction, we have p ν+µj 1 z Fil j 1 M. So set µ j = µ j 1 + ν + λ, we have p µj x = p ν+µj 1 y + E(u)p ν+µj 1 z Fil j M. Now it suffices to show that there exist ν l such that p ν l f π (N l (z)) Fil j 1 l M K for 0 l j 1. We prove by induction on l. Let l = 0. Note that N(p λ x) = N(y) + E(u)N(z) + N(E(u))z is in F j 1 M. Also, by Griffiths Transversality, we see that N(y) is in Fil j 1 M (note that j r). Note that f π (E(u)N(z)) = 0. So we have f π (N(E(u))z) Fil j 1 M K. Let δ be the least integer not less than v p (N(E(u))(π)). We see that p δ f π (z) Fil j 1 M K. For a general l, we have N l+1 (p λ x) = N l+1 (y) + l+1 m=0 ( l + 1 m ) N l+1 m (E(u))N m (z). By definition of F i M and Griffiths Transversality, we have N l+1 (p λ x) F j 1 l M and N l+1 (y) Fil j 1 l M. So applying f π to the above equation, TOME 00 (XXXX), FASCICULE 0

10 Tong LIU noting that f π (E(u)) = 0, we have l ( ) l + 1 f π (N l+1 m (E(u)))f π (N m (z)) Fil j 1 l M K. m m=0 By induction, for m = 0,..., l 1, we have p νm f π (N m (z)) Fil j 1 m M K Fil j 1 l M K. Let ν = max{ν m m = 0,..., l 1.}. We conclude that p ν (l + 1)f π (N(E(u)))f π (N l (z)) Fil j 1 l M K. By setting ν l = ν + v p (l + 1) + δ we have p ν l f π (N l (z)) Fil j 1 l M K. This completes the induction and proves the claim. 2.3. Filtration from Kisin modules Now let us study how the Frobenius on Breuil modules interacts with filtration. In particular, we discuss filtration built from Frobenius of Kisin modules. For this, we define the following: A filtered ϕ-module M over S is an S-module M with (1) a ϕ S -semi-linear morphism ϕ M : M M. (2) a decreasing filtration Fil i M M satisfying the filtration conditions. A filtered ϕ-module D over S K0 or a Breuil module D is a filtered ϕ- module over S such that (1) D is finite S K0 -free and the determinant of ϕ D is invertible in S K0 (2) There exists a monodromy operator N : D D satisfying Griffiths Transversality and N D ϕ D = pϕ D N D. Let D be a filtered (ϕ, N)-module over K 0. Following [1], we can associate a Breuil module as following: D = S K0 D, ϕ D := ϕ S ϕ D, N D = N S Id + Id N 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 the natural projection defined by D D/Fil 1 SD D K. Conversely, given a Breuil module D, one can recover D via D := D/I + SD; ϕ D := ϕ D mod I + SD; N D := N D mod I + SD and Fil i D K := f π (Fil i D). The main theorem in [1] showed that the functor D D(D) is an equivalence of categories between the category of filtered (ϕ, N)-modules over K 0 with Fil 0 D K = D K and the category of Breuil modules. Now let us recall Kisin module and its basic properties as in [10] and [12]. Recall S := W (k)[[u] with a Frobenius endomorphism ϕ via u u p ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 11 and the natural Frobenius on W (k), and the category of ϕ-modules over S. Denote by Mod ϕ,r /S the category of ϕ-modules of height r, in the sense that M is S-finite type and the cokernel of ϕ is killed by E(u) r, where ϕ is the S-linear map 1 ϕ : S ϕ,s M M. By definition, a finite free Kisin module (of height r) is a ϕ-module (of height r) M such that M is finite S-free. A torsion Kisin module (of height r) is a ϕ-module M such that M is killed by some p-power and there exists an injective morphism L L of two finite free Kisin modules of height r satisfying M L /L. When we mention Kisin module (of height r) in the remaining of the paper, we mean either finite free Kisin module of height r or torsion Kisin module of height r. Let K := K( pn π) and G = Gal(K/K ). There exists a functor n=0 T S from the category of Kisin modules to the category of Z p [G ]-modules: If M is finite free then T S (M) := Hom S,ϕ (M, W (R)) and if M is p-power torsion then T S (M) := Hom ϕ,s (M, Q p /Z p Zp W (R)), where W (R) is an S-algebra with a natural Frobenius and a natural G-action. Though T S has many nice properties, we do not need them in this paper. The readers are refereed to [10] and [4] for the construction of W (R) and more discussion of T S. Let (M, ϕ) be a Kisin module. Following 5.3 in [10], we can define a functor M S from the category of Kisin modules to the category of filtered ϕ-modules over S as the following: Define M S (M) = S ϕ,s M and ϕ MS (M) := ϕ S ϕ M ; Note that 1 ϕ : M S (M) S S M is an S-linear map. Set (2.3.1) F i M S (M) := {m M S (M) (1 ϕ)(m) Fil i S S M}. Lemma 2.8. Assume that M is finite S-free and write M = M S (M). Then F i M satisfies the filtration condition defined in the previous subsection. Proof. All other requirements are easily to verified by the definition except that F r+1 M Fil 1 SM. Let {e 1,..., e d } be an S-basis of M. Assume that x = i a i e i is in F r+1 M with a i S. We have to show that a i Fil 1 S. Let A be a matrix in M d (S) such that (ϕ(e 1 ),..., ϕ(e d )) = (e 1,..., e d )A. Since (1 ϕ)(x) = i a iϕ(e i ) is in Fil r+1 S S M, we conclude that Aα = β where α, β are n 1 matrices, coefficients of α are a i, and coefficients of β are in Fil r+1 S. The fact that M has E(u)-height r means that there exists a matrix B M d (S) such that AB = BA = E(u) r I d. Hence Bβ = BAα = E(u) r α still has coefficients in Fil r+1 S, and then α has all its coefficients in Fil 1 S. TOME 00 (XXXX), FASCICULE 0

12 Tong LIU 2.4. Lattices in D dr (V ) Let V be a de Rham representation of G. We denote D dr (V ) := (B dr Qp V ) G. Now let us summarize results from [12] and [13] to manipulate lattices in semi-stable representations. Let V be a semi-stable representation in Rep st,r Q p, Λ V a G-stable Z p -lattice. The main result of [12] is that there exists an anti-equivalence ˆT of categories between Rep st,r Z p of (ϕ, Ĝ)-modules of height r. Let ˆM = (M, ϕ M, and the category Ĝ) be the (ϕ, Ĝ)-module such that ˆT ( ˆM) Λ with (M, ϕ M ) the ambient finite free Kisin module of height r. In particular, this means that M is the unique finite free Kisin module such that T S (M) Λ G (see Theorem 2.3.1 in [12]). Let D = D st (V ) the filtered (ϕ, N)-module attached to V. Set D := D(D) the Breuil module associated to D and M := M S (M) the filtered ϕ- modules over S. There exists a natural isomorphism of ϕ-modules over S between Q p Zp M and D (see 3 in [11] for full details). In particular, Q p Zp M has the structure of monodromy N. We easily check that M/uM M/I + SM as ϕ-modules and f π (M) = M/Fil 1 SM is an O K - lattice in D/Fil 1 SD. By Proposition 2.4.1 in [13], we have N(M) M if p > 2 (1). Let M S (ϕ, N, Fil) the full subcategory of filtered ϕ-modules M over S such that There exists a finite free Kisin module M such that M M S (M). Q p Zp M has a structure of Breuil module and N(M) M. Hence we obtain a contravariant functor Tst 1 from Rep st,r Z p to M S (ϕ, N, Fil). If r < p 1 then Tst 1 is an anti-equivalence by the main result of [11]. But Tst 1 in general is not a full functor if r p 1. Now let us recall a little more details on the construction of M st (Λ) from 2.2 and 2.3 from [13], let D := D/I + SD, which is a ϕ-module over K 0. Then M := M/uM M/I + SM is a ϕ-stable W (k)-lattice in D. Note that D is canonically isomorphic to D = D st (V ) via the unique ϕ-compatible section s : D D (see Proposition 6.2.1.1 in [1]). Then M st (Λ) is just the image s(m). Now identify D/Fil 1 SD with D K = D dr (V ) via D S W (k) D. We obtain two O K -lattices: M st (Λ) K := O K W (k) s(m) and f π (M). Proposition 2.9. There exists a constant c 3 only depending on e and r such that p c3 (M st (Λ) K ) f π (M) and p c3 (f π (M)) M st (Λ) K. Proof. Note that M = S ϕ,s M with a Kisin module M of height r. If {ẽ 1,..., ẽ d } is an S-basis of M. Then {ê i := 1 ẽ i } forms an S-basis of M. (1) Here is the only place that we need p to be an odd prime. ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 13 Let e i be the image of ê i of the natural map M M/I + SM = M. Since M/I + SM has a unique ϕ-equivariant section s : M D, we just denote e i for s(e i ). Note that e 1,..., e d forms a basis of s(m). Let A M d (S) be the matrix such that ϕ(ê 1,..., ê d ) = (ê 1,..., ê d )A and A 0 := A mod u the matrix in M d (W (k)). Then we have ϕ(e 1,..., e d ) = (e 1,..., e d )A 0. Let X M d (S K0 ) be the matrix such that (ê 1,..., ê d ) = (e 1,..., e d )X inside D. It suffices to show that there exists a constant c 3 such that p c3 X and p c3 X 1 are in M d (S). To proceed the proof, note that we have the relation A 0 ϕ(x) = XA. Set X n := A 0 ϕ(a 0 ) ϕ n (A 0 )ϕ n (A 1 ) ϕ(a 1 )A 1. Following the same idea of the proof of Proposition 6.2.1.1 in [1], we show that X n converges to X and there exists a constant c 3 such that p c3 X n M d (S) and hence p c3 X M d (S). To prove this, we first claim that p r A 1 M d (S) and A 0 A 1 = I d + up p Y with Y M r d (S). We accept this claim and postpone the proof in the end. Set c 3 = max i0 (ri v p (q(p i )!)) where q(p i ) satisfies the relation p i = eq(p i ) + r(p i ) with 0 r(p i ) < e. Now X n = X 0 + n 1 (X i+1 X i ) = X 0 + n 1 i=0 i=0 u pi+2 p Z r i where Z i = A 0 ϕ(a 0 ) ϕ i (A 0 )ϕ i+1 (Y )ϕ i (A 1 ) ϕ(a 1 )A 1. Since p r A 1 i+2 M d (S), we see that upi+2 p Z r i = up p r(i+1) Z p r(i+2) i is in u pi+2 M p r(i+2) d (S). As any a S can be (uniquely) written as u a i i q(i)! with u pi+2 a i W (k), we conclude that p c3 is in S. Hence p c3 X p r(i+2) n and then p c3 X are in M d (S). Now let us prove the claim. Since M = S ϕ,s M with a Kisin module M of height r, we have A = ϕ(ã) with à M d(s) and there exists a matrix B M d (S) such that à B = Bà = E(u)r I d. Hence A 1 = ϕ(ã 1 ) = (ϕ(e(u) r ) 1 ϕ( B). Note that ϕ(e(u))/p is a unit in S. Hence p r A 1 is in M d (S). Now A 0 A 1 = A 0 ϕ(ã 1 B 1 )ϕ( B) = (ϕ(e(u) r )) 1 A 0 ϕ( B). Let pa 0 be the constant term of E(u). It is easy to see that A 0 ϕ( B) = (a 0 p) r I d + u p Y with Y M d (S). Now we have A 0 A 1 = I d +(b r 1)I d + up p (a r 0 b) r Y with b = (ϕ(e(u))/c 0 p) 1 S. We easily compute that b = 1 + b with b uep p S. Hence A 0A 1 = I d + up p Y with Y M r d (S). Finally, it remains to show that p c3 X 1 is in M d (S). In fact, we can use the same strategy to Xn 1. In this situation, we need to show that p r A 1 0 M d (W (k)) and AA 1 0 = I d + up p Y with Y M r d (S) and this is easy to show by a similar argument as the above. i=0 TOME 00 (XXXX), FASCICULE 0

14 Tong LIU The following example shows that c 3 0 in general. Example 2.10. 1 given by Let M be a finite free rank-2 Kisin module of height ( ) 1 u ϕ(e 1, e 2 ) = (e 1, e 2 ) 0 E(u) where e 1, e d forms an S-basis of M. By Theorem (0.4) in [8], M corresponds to a Z p -lattice of crystalline representation with Hodge-Tate weight ( ) 1 u p in {0, 1}. Using notations in the above proof we have A = 0 ϕ(e(u)) ( ) ( ) 1 0 1 x and A 0 =. The above proof showed we may write X =. 0 p 0 α Then the relation A 0 ϕ(x) = XA yields two equations: pϕ(α) = αϕ(e(u)) and ϕ(x) = u p + xϕ(e(u)). Since ϕ(e(u)) = pµ with µ a unit in S, we easily solve that α is a unit of S and x = up p µ 1 + higher degree term. If e > 1, we see that u p /p is not in S. And if e > p then x mod Fil 1 S is not in O K. So M st (Λ) K and f π (M) are different O K -lattices. By Corollary 3.2.3 in [11] and the construction of F i M S (M), we conclude that F i M S (M) = M Fil i D. So we may just denote F i M by Fil i M. Consider the natural projection f π : D D K. Write M K = f π (M) and define Fil i MK := M K Fil i D K. Obviously, f π (Fil i M) Fil i MK. Lemma 2.11. There exists a constant c 4 only depending on E(u) and r such that p c4 Fil i MK f π (Fil i M). Proof. Here we modify the idea used in the proof of Proposition 6.2.2.3 in [1]. We prove by induction on i that there exists a constant µ i depending on E(u) and i such that p µi Fil i MK f π (Fil i M). If i = 0 the case is trivial. Now assume the statement is valid for any i j. Without loss of generality, we may assume that µ i 1 µ i for any 1 i j. Now consider the case i = j + 1. Let x Fil j+1 MK then x Fil j MK. By induction, p µj x f π (Fil j M). That is, there exists a ˆx Fil j M such that f π (ˆx) = p µj x. Write N(E(u)) = R(u). Note that R(π) 0. So there exists Q(u) K 0 [u] such that Q(π)R(π) = 1. Let H(u) = Q(u)E(u). Note that 1 + N(H(u)) Fil 1 S K. Now set ŷ := ˆx + H(u)N(ˆx) + 1 2 H2 (u)n 2 (ˆx) + + 1 j! Hj (u)n j (ˆx). It is obvious that f π (ŷ) = f π (ˆx). Write m(u) = 1 + N(H(u)), we have j 1 1 N(ŷ) = m(u)n(ˆx) + i! m(u)(h(u))i N i+1 (ˆx) + 1 j! (H(u)j )N j+1 (ˆx). i=1 ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 15 Since N i (ˆx) Fil j i M for 1 i j, m(u) Fil 1 S K0 and H(u) Fil 1 S K0, we see that N(ŷ) Fil j D. Hence ŷ Fil j+1 D. Let λ be the minimal constant such that p λ Q(u) W (k)[u]. Apparently λ only depends on E(u). Now p λj (H(u)) j E(u) j W (k)[u] and then p λj ŷ M. Hence p λj ŷ Fil j+1 M. So set µ j+1 = µ j +λj. We see that f π (p λj ŷ) = p λj f π (ˆx) = p µj+1 x. Remark 2.12. The constant c 3 and c 4 are not optimal. In fact, assume that representations are crystalline, K 0 = K and 0 r p 2. One can choose that c 3 = c 4 = 0 by using Fontaine-Laffile theory in [7]. But in general, we do not expect c 3 or c 4 is zero. 2.5. A lemma that needs all constants The aim of this subsection is to prepare a lemma to prove Theorem 2.3. Let T S (h) : Λ Λ be a map in Rep st,r Z p and h : L L the corresponding map of Kisin modules. Assume that h is surjective. Then we have a surjective map of O K -modules h K : L K L K where L K = M st (Λ) K and L K = M st(λ ) K. Obviously we have that h K (Fil i L K ) Fil i L K. Lemma 2.13. There exists a constant c 5 only depending on E(u) and r such that p c5 Fil i L K h K(Fil i L K ). We need some preparations for the above lemma. Apply the functor M S to h, we obtain a surjection h S : L L where L := M S (L) and L = M S (L ) respectively. By the definition in Formula (2.3.1), it is easy to see that h S (F i L) F i L. Lemma 2.14. Notations as the above. There exists a constant α only depending on r such that p α F r L h S (F r L). Proof. Let K be the kernel of h. It is not hard to show that K is a ϕ-module of E(u)-height r (see Proposition 1.3.5 in [4]). In fact K can be shown to be S-free but we do not need this here. Now we have the following commutative diagram: 0 S ϕ,s K S ϕ,s L S ϕ,s h S ϕ,s L 0 1 ϕ 0 K L 1 ϕ h 1 ϕ L 0. TOME 00 (XXXX), FASCICULE 0

16 Tong LIU It is easy to see both rows are exact as L is finite S-free. Denote S ϕ,s L, S ϕ,s L and S ϕ,s h by L, L and h respectively. Set F r L = {x L (1 ϕ)(x) E(u) r L} and define F r L similarly. We first prove that h : F r L F r L is surjective. To see this, for any y = h (x) F r L with x L, we have (1 ϕ)(y) E(u) r L. So there exists a z K such that (1 ϕ)(x)+z is in E(u) r L. Then the fact that L has height r implies that (1 ϕ)(x) + z = (1 ϕ)(w ) for w L. So there exists w L such that (1 ϕ)(w) = z. As 1 ϕ in the last two columns are easily to see to be injective, w is in the kernel of h. So h (x w) = y and x w is in F r L. This proves that h : F r L F r L is surjective. Now set α = v p ((r 1)!). For any s S, note that s = i a i E(u)i i!. So s = s 0 +s 1 with s 1 Fil r S and p α s 0 W (k)[u]. Now pick any y = h S (x) F r L with x L. Then we can write y = y 0 +y 1 such that y 1 Fil r SL and p α y 0 L L. As h S is surjective, there exists x 1 Fil r SL F r L such that h S (x 1 ) = y 1. It is easy to see that p α y 0 F r L. Hence there exists x 0 F r L F r L such that h S (x 0 ) = p α y 0. This proves the lemma. Proof of Lemma 2.13. Note that both h S (F i L) and F i L satisfy Griffith Transversality. We denote F i (h S (L)) and F i (L ) for F i constructed from h S (F i L) and F i L above Proposition 2.7 respectively. Note that the construction of F i only depends on Fil r and N on L. So p α F i L F i (h S (L)) F i L by Lemma 2.14. Then by Lemma 2.7, we get p c1+α F i L p α F i L F i (h S (L)) h S (F i (L)). Applying functor f π to h S, we have a surjective map h K : LK L K where L K = f π (L) and L K = f π(l ). Hence p c1+α f π (F i L ) f π (h S (F i L)). By Lemma 2.11, we have p c3+c1+α Fil i L K p c1+α f π (F i L ) f π (h S (F i L)) h K (Fil i LK ). Finally, by Proposition 2.9 and set c 5 = 2c 4 + c 3 + c 1 + α, we have p c5 Fil i L K p c4+c3+c1+α Fil i L K h K (p c4 Fil i LK ) h K (Fil i L K ). ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 17 3. Filtration Attached to Torsion Semi-stable Representations 3.1. Construction of filtration to torsion representations Now let us first discuss more details on filtration associated to torsion semi-stable representations. Let T Rep st,r tor be a torsion semi-stable representation and j : Λ Λ is a lift of T. That is, j : Λ Λ are G-stable Z p -lattices inside a semi-stable representation with Hodge-Tate weights in {0,..., r} and we have the exact sequence of Z p [G]-modules 0 Λ j Λ q T 0. Recall that ˆT is an anti-equivalence between the category of (ϕ, Ĝ)-modules of height r and Repst,r Z p. Let L and L be the ambient Kisin modules of (ϕ, Ĝ)-modules correspond to Λ and Λ respectively, we obtain the injective morphism of Kisin modules j : L L. Write M := L/j(L ) which is a torsion Kisin module of height r. By Proposition 3.2.3 in [13], we have T S (M) T G. Now consider the exact sequence of Kisin modules to correspond the above exact sequence of Galois representations: (3.1.1) 0 L j L q M 0. Now modulo u, we have an exact sequence (3.1.2) 0 L j L q M 0, By the construction of M st, the exact sequence (3.1.2) is canonically isomorphic to the exact sequence 0 M st (Λ ) M st(j) M st (Λ) M st,j (T ) 0. By tensoring O K to the above exact sequence, we obtain an exact sequence of O K -modules (3.1.3) 0 L K jk L K q K M K 0, Recall that Fil i L K = L K Fil i D K where D K := D dr (Q p Zp Λ). For any i Z, by the construction in 2.1, we have Fil i M K := q K (Fil i L K ) and the following exact sequence (3.1.4) 0 Fil i L K jk Fil i L K q K Fil i M K 0. Using Snake Lemma, the above exact sequence induces the following exact sequence: TOME 00 (XXXX), FASCICULE 0

18 Tong LIU Corollary 3.1. The following sequence is exact 0 gr i L K jk gr i L K q K gr i M K 0. 3.2. The proof of Theorem 2.3 Now we need to recall a part of Theorem 3.1.1 in [13] and its proof to complete the proof of Theorem 2.3. Let M tor (ϕ, N) whose objects are finite length W (k)-modules with only ϕ and N-structures satisfying the properties required in the definition of filtered (ϕ, N)-modules. Theorem 3.2. There exists a constant c only depending on e and r such that the following statement holds: for any morphism f : T T in Rep st,r tor and any lift j, j of T, T respectively, there exists a morphism g : M st,j (T ) M st,j (T ) in M tor (ϕ, N) such that (1) if there exists a morphism of lifts ˆf : j j which lifts f then g = p c M st, ˆf (f). (2) let f : T T be a morphism in Rep st,r tor with j the lift of f and g : M st,j (T ) M st,j (T ) the morphism in M tor (ϕ, N) attached to f, j and j. If there exists a morphism of lifts ĥ : j j which lifts f f then g g = p 2c M st, ĥ (f f ). The above theorem is a part of Theorem 3.1.1 in [13]. To prove Theorem 2.3, it suffices to show that there exists a constant c 5 only depending on E(u) and r such that g := p c5 (O K W (k) g) and g := p c5 (O K W (k) g ) preserve filtration defined in the previous subsection. Now let us recall the construction of g. Let M be the torsion Kisin module obtained by the exact sequence (3.1.1). Proposition 3.2.3 in [13] explains that T S (M) T G. Similarly, the lift j : Λ Λ of T induces a torsion Kisin module M such that T S (M ) T G. By Theorem 2.4.2 in [10], there exists a unique morphism f : M M of Kisin modules such that T S (f) = p c f. Then g is constructed as g := f mod us. Define a map i L : L L L via i L (x) = (x, 0) and define i L : L L L via i L (y) = (0, y). Set q : L L M via (f q)(x) + q (y) for (x, y) L L and N := Ker q. By Corollary 2.3.8 in [10], N is a finite free Kisin module of height r. Then we have the following commutative ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 19 diagram of Kisin modules. 0 L L q M 0 (3.2.1) i L 0 N L L q 0 L i L L q M 0 It is easy to check that by functor T S or ˆT, the above commutative diagram corresponds the following commutative diagram of Galois representations f M 0 0 Λ Λ q T 0 (3.2.2) 0 Λ Λ j N q T 0 0 Λ Λ q p c f T 0 The above diagram can be constructed without using the previous diagram as the following: Taking dual of the first row and the last row of the above diagram, we obtain two exact sequences: 0 (Λ ) (Λ ) q (T ) 0 and 0 (Λ ) Λ q T 0. We can construct q : Λ (Λ ) (T ) by q ((x, y)) = p c f q (x) + q (y) and let N := Ker( q ). We embed Λ and Λ to Λ (Λ ) to the first factor and the second factor respectively, In this way, we obtain a commutative diagram as Diagram (3.2.1) 0 Λ Λ q (T ) 0 (3.2.3) 0 N Λ Λ q (T ) 0 0 (Λ ) Λ p c f q T 0 Then Diagram (3.2.2) is obtained by taking dual of the above diagram. In summary, we obtain another lift j of T. It is obviously that the map g : M st,j (T ) M st, j (T ) preserves filtration. We also have a map α : TOME 00 (XXXX), FASCICULE 0

20 Tong LIU M st,j (T ) M st, j (T ) by modulo u of the upper block of Diagram (3.2.1), which is a Frobenius, monodromy compatible isomorphism of W (k)-modules. It is clear that α K (Fil i M K,j ) Fili M K, j where α K := O K W (k) α, Fil i M K,j and Fili M K, j are filtration of O K W (k) M st,j (T ) and O K W (k) M st, j (T ) via the construction in the last subsection. Now to prove Theorem 2.3, it suffices to prove that there exists a constant c 5 only depending on E(u) and r such that p c5 Fil i M K, j α K(Fil i M K,j ). To prove this statement, consider the following commutative diagram 0 0 0 0 L L q M 0 (3.2.4) i L 0 N L L q M 0 0 L L 0 0 0 0 0 The upper block of the above diagram is the upper block of Diagram (3.2.1) and L := N/L. We easily check that all rows and columns are short exact and then the map L L is indeed an isomorphism. One easily checks that the above commutative diagram corresponds to the following commutative diagram of Galois representations 0 0 0 0 Λ Λ q T 0 (3.2.5) 0 Λ Λ j N q T 0 0 Λ Λ 0 0 0 0 0 ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 21 where Λ is the kernel of map N Λ. We easily see that all rows and all columns are exact and the map Λ Λ is an isomorphism of Z p [G]-modules. By the construction of filtration in the previous subsection, Diagram (3.2.4) yields a new commutative diagram 0 0 0 0 Fil i L K Fil i L K q K Fil i M K,j 0 (3.2.6) 0 Fil i NK Fil i L K Fil i L K q K α K Fil i M K, j 0 0 Fil i ˆLK Fil i L K Q i 0 0 0 0 where Fil i ˆLK := Fil i NK /Fil i L K and Q i := Fil i L K /Fil i ˆLK, which can easily be checked to be Fil i M K, j /Fili M K,j. We need to show that Q pc5 i = 0 for a constant c 5. Note that Fil i ˆLK can be regarded as h K (Fil i NK ) where h K : N K L K is a surjective map induced by the surjective map of Kisin modules N L L. By Lemma 2.13 there exists a constant c 5 such that p c5 kills Q i. This finishes the proof of Theorem 2.3 which is stated again in the following. Theorem 3.3 (Theorem 2.3). There exists a constant c only depending on E(u) and r such that the following statement holds: for any morphism f : T T in Rep st,r tor and any lift j, j of T, T respectively, there exists a morphism g : M st,j (T ) M st,j (T ) in M r tor(ϕ, N, Fil) such that (1) if there exists a morphism of lifts ˆf : j j which lifts f then g = p c M st, ˆf (f). (2) let f : T T be a morphism in Rep st,r tor with j the lift of T and g : M st,j (T ) M st,j (T ) the morphism in M r tor(ϕ, N, Fil) attached to f, j and j. If there exists a morphism of lifts ĥ : j j which lifts f f then g g = p 2c M st, ĥ (f f ). TOME 00 (XXXX), FASCICULE 0

22 Tong LIU 4. Application to Galois deformation ring The aim of this section is to reprove a part of Theorem (2.6.7) in [9] via a different approach. Throughout this section, we assume that K is a finite extension of Q p. 4.1. p-adic Hodge-Tate type We first recall the definition of p-adic Hodge-Tate type from [9] and prove several technical results on p-adic Hodge-Tate type. Let E be a finite extension of Q p. Suppose that we are given a finite dimensional E- vector space D E and a filtration (Fil i D E,K ) i Z of D E,K := K Qp D E by E Qp K-modules such that the associated graded is concentrated in degree in [0, r], namely the set {i gr i D E,K 0} {0, 1,..., r}. We set v = {D E, Fil i D E,K, i = 0, 1,..., r}. If B is a finite E-algebra and V B a finite free B-module with a continuous G-action, which makes V B a de Rham representation, then we say that V B has p-adic Hodge-Tate type v if V B has all its Hodge-Tate weights in {0,..., r} and there is an isomorphism of B Qp K-modules gr i (D dr (V B )) gr i (D E,K ) E B. Recall that D dr (V B ) := (B dr Qp V B )G. Lemma 4.1. Notations as the above. Assume that B is a B-algebra and finite over E. Then V B := B B V B has p-adic Hodge-Tate type v. The above lemma is an easy consequence of the following fact. For any E-algebra A, write A K := K Qp A. Lemma 4.2. (1) We have D dr (V B ) B B D dr (V B ) and gr i (D dr (V B )) B B gr i (D dr (V B )) for all i Z. (2) D dr (V B ) is a finite free B K -module. Proof. (1) We first show that D dr (V B ) B B D dr (V B ). Consider the canonical isomorphism V B Q p B dr D dr (V B ) K B dr. After tensoring B, we get an isomorphism V B Q p B dr B B D dr (V B ) K B dr. ANNALES DE L INSTITUT FOURIER

FILTRATION ASSOCIATED TO TORSION SEMI-STABLE REPRESENTATIONS 23 So dim K (B B D dr (V B )) = dim Qp (V B ). But it is obvious that B B D dr (V B ) (V B Q p B dr ) G. Therefore we conclude that B B D dr (V B ) = D dr (V B ) = (V B Q p B dr ) G. Similarly, we can show that B B D HT (V B ) = D HT (V B ), where D HT (V ) := (B HT Qp V ) G for a Hodge-Tate representation V. To show gr i (D dr (V B )) B B gr i (D dr (V B )) as B Qp K- modules, note that for a de Rham representation V of G, we have gr i (D dr (V )) (gr i B dr Qp V ) G for each i and i Z gr i (D dr (V )) D HT (V ). It is clear that B B gr i (D dr (V B )) gr i (D dr (V B )). Then B B D HT (V B ) B B gr i (D dr (V B )) i Z i Z D HT (V B ). gr i (D dr (V B )) Hence the fact that B B D HT (V B ) = D HT (V B ) implies that gr i (D dr (V B )) = B B gr i (D dr (V B )). (2) We have known that if B is a finite extension of E then D dr (V B ) is a finite free B K -module (see Lemma 2.1 in [16]). Write B red := B/N(B) where N(B) is the nilpotent ideal of B. Since B red is a reduced Artinian E-algebra, it is a direct product of finite extension E j over E for j = 1,..., m. Hence D dr (V Bred ) is a finite free K Qp B red -module where V Bred = B red B V B. By (1), we have D dr (V Bred ) = B red B D dr (V B ). Let e 1,..., e d be a K Qp B red -basis of D dr (V Bred ) with ê i D dr (V B ) a lift of e i and d = dim B (V B ). Then by Nakayama s lemma, ê i generates D dr (V ) as a B K -module. Hence there exists a finite free B K -module M with rank d and a surjection map f : M D dr (V B ). On the other hand, it is easy to compute that dim K (D dr (V B )) = d dim Qp B = rank K M. Hence f is an isomorphism and D dr (V B ) is finite B K -free. Remark 4.3. By Remarque 3.1.1.4 in [2], gr i D dr (V B ) is not necessarily B K -free even for B = E being a finite extension of Q p. Since E K := E Qp K is a reduced E-algebra, we have E K ι J F (ι) of E K -algebras with F (ι) a finite extension of E. Here ι : K F (ι) is an embedding of K to Q p such that F (ι) = E ι(k) in Q p, and J is a set of such embeddings ι. So F (ι) is an E K -algebra via ι : K F (ι) and E F (ι). Hence for any E K -module M, we get a decomposition M M (ι) with ι J M (ι) := F (ι) EK M. For a filtered E K -module D K, we also use Fil i (ι)d K TOME 00 (XXXX), FASCICULE 0

24 Tong LIU and gr i (ι) D K to denote (Fil i D K ) (ι) and (gr i D K ) (ι) respectively. It is easy to check gr i (ι) D K Fil i (ι)d K /Fil i+1 (ι) D K. Write B F(ι) := F (ι) E B. The following is a useful result: Lemma 4.4. V B has type v if and only if gr i (ι) (D dr(v B )) is B F(ι) -free and rank BF(ι) (gr i (ι) (D dr(v B ))) = dim F(ι) (gr i (ι) (D E,K)) for all ι J and i Z. Proof. One direction is clear by definition. Now suppose that gr i (ι) (D dr(v B )) is B F(ι) -free. Select a B F(ι) -basis e 1,..., e d of gr i (ι) (D dr(v B )) and set M (ι) be F (ι) -module generated by e i inside gr i (ι) (D dr(v B )). It is obvious that M (ι) is finite F (ι) -free and gr i (ι) (D dr(v B )) B F(ι) F(ι) M (ι). Then dim F(ι) M (ι) = dim F(ι) gr i (ι) D E,K. Set M = ι J M (ι). Then M gr i D E,K as E K -modules and B E M gr i D dr (V B ) as B K -modules. As the proof of Lemma 4.2, write B red := B/N(B) where N(B) is the nilradical of B. Since B red is a reduced Artinian E-algebra, it is a direct product of finite extension E j over E for j = 1,..., m. Write V Ej := V B B E j for j = 1,..., m. Proposition 4.5. V B has type v if and only if V Ej each j = 1,..., m. has type v for Proof. The only if" part is the consequence of Lemma 4.1. To prove if" part, note that the fact V Ej has type v for each j = 1,..., m implies that V Bred has type v where V Bred := B red B V B. So gr i D dr (V Bred ) B red E gr i D E,K as K Qp B red -modules. In particular, gr i (ι) D dr(v Bred ) is finite F (ι) E B red -free with the rank d i = dim F(ι) gr i (ι) D E,K. By Lemma 4.4, we have to show that gr i (ι) (D dr(v B )) is B F(ι) -free with rank dim F(ι) gr i (ι) D E,K. By Lemma 4.2 (1), it is easy to check that gr i (ι) D dr(v Bred ) = B red B gr i (ι) D dr(v B ). Select e 1,..., e di gr i (ι) (D dr(v B )) such that the image of {e l } in gr i (ι) (D dr(v Bred )) forms a F (ι) E B red -basis of gr i (ι) (D dr(v Bred )). By Nakayama s lemma, we know {e l } generates gr i (ι) (D dr(v B )). Hence we have a finite free B F(ι) -module with rank d i projects gr i (ι) (D dr(v B )). So dim F(ι) gr i (ι) (D dr(v B )) d i dim E B and gr i (ι) (D dr(v B )) is finite B F(ι) - free with rank d i = dim F(ι) gr i (ι) D E,K if only if the equality holds. On the other hand, by Lemma 4.2 (2), D dr (V B ) is a finite free B K -module with rank d = rank B (V B ) = rank Bred (V Bred ), we have dim F(ι) (D dr (V B ) (ι) ) = d dim E B. Note that dim F(ι) (D E,K,(ι) ) = d because V Bred has type v. Now ANNALES DE L INSTITUT FOURIER