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

TL;DR

This paper demonstrates that the mod p reduction of 2-dimensional crystalline Galois representations varies locally constantly with respect to key parameters, and provides an algorithm for computing these reductions.

Contribution

It establishes the local constancy of reductions modulo p in terms of parameters and introduces an explicit algorithm for their computation.

Findings

Reduction modulo p is locally constant in a_p and weight k (when a_p ≠ 0)

Explicit radius of local constancy is determined

An algorithm for computing reductions modulo p is developed

Abstract

Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant function of a_p (with an explicit radius) and a locally constant function of the weight k if a_p <> 0. We then give an algorithm for computing the reductions modulo p of these representations. The main ingredient is Fontaine's theory of (phi,Gamma)-modules as well as the theory of Wach modules.