Filtered Modules Corresponding to Potentially Semi-stable Representations

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable p-adic representations of the Galois groups of p-adic fields under the assumptions that p is odd and the coefficients are large enough. Introduction Let p be an odd prime number, and let K be a p-adic field. The absolute Galois group of K is denoted by GK . By the fundamental theorem of Colmez and Fontaine [CF], there exists a correspondence between potentially semi-stable p-adic representations and admissible filtered (φ,N)-modules with Galois action. The aim of this paper is the classification of the admissible filtered (φ,N)-modules with Galois action corresponding to two-dimensional potentially semi-stable p-adic representations of GK with coefficients in a p-adic field. IfK = Qp and a coefficient is alsoQp, the classification is given in [FM, Appendix A] under the assumption that p ≥ 5. If K = Qp and a coefficient is general, these filtered (φ,N)-modules are studied in [BM] and [Sav], and the classification is given by Ghate and Mézard in [GM] under some assumptions. In this paper, we generalize the results of [GM] to the case where K is a general p-adic field. After writing of this paper, the author has known that there is preceding research [Do] on this subject by Dousmanis. The author does not claim priority, but there are some differences. In [Do], a classification is given by Frobenius action, and in this paper, we give a classification by Galois action. Let F be a finite extension of K. A potentially semi-stable representation ρ is said to be F -semi-stable, if the restriction of ρ to the absolute Galois group of F is semi-stable. In [Do], a classification of F -semi-stable representations is given for a general finite Galois extension F of K. In this paper, we give a class of finite Galois extensions of K such that any potentially semi-stable representation is F -semi-stable for a field F in this class, and give a classification of F -semi-stable representations and a more explicit description of Galois action of Gal(F/K) for F in this class, assuming p 6= 2. Then, in this paper, we first fix a large enough coefficient field, and do not extend it in the classification. The study of the admissible filtered (φ,N)-modules is important in the p-adic Langlands program, and the classification in [GM] was used in [BE] to study the Galois representations attached to modular forms. So we could expect applications of our result to studies of the Galois representations attached to Hilbert modular forms. This paper is clearly influenced by the paper [GM], and we owe a lot of arguments to [GM]. We mention it here, and do not repeat it each times in the following. 1 2 NAOKI IMAI Acknowledgment. The author is supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. He would like to thank Gerasimos Dousmanis for permitting this paper. Notation. Throughout this paper, we use the following notation. Let p be an odd prime number, and Cp be the p-adic completion of the algebraic closure of Qp. Let K be a p-adic field. We consider K as a subfield of Cp. The residue field of K is denoted by k, whose cardinality is q. Let K0 be the maximal unramified extension of Qp contained in K. For any p-adic field L, the absolute Galois group of L is denoted by GL, the inertia subgroup of GL is denoted by IL, the Weil group of L is denoted by WL, the ring of integers of L is denoted by OL and the unique maximal ideal of OL is denoted by pL. For a Galois extension L of K, the inertia subgroup of Gal(L/K) is denoted by I(L/K). Let vp be the valuations of p-adic fields normalized by vp(p) = 1. 1. Filtered (φ,N)-modules Let E be a p-adic field. We consider a two-dimensional p-adic representation V of GK over E, which is denoted by ρ : GK → GL(V ). As in [Fon], we can construct K0-algebra Bst with a Frobenius endomorphism, a monodromy operator and Galois action. Further, we can define a decreasing filtration on K ⊗K0 Bst. Let F be a finite Galois extension of K, and F0 be the maximal unramified extension of Qp contained in F . Then we have BF st = F0. The p-adic representation ρ is called F -semi-stable if and only if the dimension of Dst,F (V ) = (Bst ⊗Qp V ) GF over F0 is equal to the dimension of V over Qp. If ρ is F -semi-stable for some finite Galois extension F of K, we say that ρ is potentially semi-stable representation. Potentially semi-stable representations are Hodge-Tate. To fix a convention, we recall the definition of the Hodge-Tate weights. For i ∈ Z, we put D HT(V ) = ( Cp(i)⊗Qp V )GK . Here and in the following, (i) means i times twists by the p-adic cyclotomic character of GK . Then there is a GK -equivariant isomorphism