Potentially Semi-stable Deformation Rings

LetK/Qp be a finite extension and GK = Gal(K̄/K) the Galois group of an algebraic closure K̄. Let F be a finite field of characteristic p, and VF a finite dimensional F-vector space equipped with a continuous action of GK . The study of the deformation theory of Galois representations was initiated by Mazur [Ma], who showed that if VF has no non-trivial endomorphisms, then it admits a universal deformation ring RVF . After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand certain quotients of RVF corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of RVF was constructed by Ramakrishna [Ra]. Suppose that L/K is a finite extension and let a b be integers. It seems to be a kind of folklore conjecture that there should be a quotient of RVF whose points in finite extensions of W (F)[1/p] correspond to deformations of VF which become semi-stable over L and have Hodge-Tate weights in the interval [a, b]. This is closely related to the final conjecture of [Fo 3]. Special cases of this, when VF is 2-dimensional, are conjectured in the papers of Fontaine-Mazur [FM, p.191], BreuilConrad-Diamond-Taylor [BCDT, Conj. 1.1.1] and Breuil-Mézard [BM, 2.2.2.4]. The purpose of this paper is to prove such a result.