On reductions of families of crystalline Galois representations

TL;DR

This paper constructs analytic families of crystalline Galois representations over unramified extensions of Q_p, enabling the study of their reductions and providing new insights into their structure and classification.

Contribution

It introduces a method to generate families of crystalline Galois representations from given irreducible ones, expanding understanding of their reductions and deformations.

Findings

Constructed analytic families of étale (Φ,Gamma)-modules for Galois representations.

Generated infinite families of crystalline representations with the same Hodge-Tate weights.

Computed semisimplified mod p reductions of the constructed families.

Abstract

Let K_{f} be the finite unramified extension of Q_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of \'etale (Phi,Gamma)-modules which give rise to families of crystalline E-representations of the absolute Galois group G_{K_{f}} of K_{f}. For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0,-k_{i}}_{{\tau}_{i}} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod p reductions of the members of each such family.