Crystalline representations of G_Qp^a with coefficients

Abstract

This paper studies crystalline representations of G_K with coefficients of any dimension, where K is the unramified extension of Q_p of degree a. We prove a theorem of Fontaine-Laffaille type when σ-invariant Hodge-Tate weight less than p-1, which establishes the bijection between Galois stable lattices in crystalline representations and strongly divisible ϕ-lattice. In generalizing Breuil's work, we classify all reducible and irreducible crystalline representations of G_K of dimensional 2, then describe their mod p reductions. We generalize some results (of Deligne, Fontaine-Serre, and Edixhoven) to representations arising from Hilbert modular forms when σ-invariant Hodge-Tate weight less than p-1.