Local constancy for the reduction mod p of 2-dimensional crystalline representations

Abstract

Irreducible crystalline representations of dimension 2 of Gal ( Q ¯ p / Q p ) are all twists of some representations V k , α p which depend on two parameters, the weight k and the trace of the Frobenius map ap. We show that the reduction modulo p of V k , α p is a locally constant function of ap (with an explicit radius), and a locally constant function of the weight k if ap≠0. We then give (for p≠2) an algorithm for computing the reduction modulo p of V k , α p . The main ingredient is Fontaine's theory of (ϕ, Γ)-modules, as well as the theory of Wach modules.