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

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.

The "local constancy" part is new; the algorithm is unchanged; the title of the article has been changed

References in corpus (2)

• Explicit reduction modulo $p$ of certain crystalline representations
• Towards a $p$-adic Langlands programme

Cited by in corpus (4)

• An algorithm for computing the reduction of $2$-dimensional crystalline representations of $\text{Gal}({\overline{\mathbf{Q}}}_p/{\mathbf{Q}}_p)$
• Semi-stable representations as limits of crystalline representations
• Reductions of $2$-dimensional semi-stable representations with large $\mathcal{L}$-invariant
• On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$