Galois representations over pseudorigid spaces

TL;DR

This paper develops a framework for $p$-adic Hodge theory of Galois representations over pseudorigid spaces, introducing new period rings and using Tate--Sen methods to construct overconvergent modules, advancing understanding of eigenvarieties.

Contribution

It introduces perfect and imperfect overconvergent period rings and applies Tate--Sen techniques to construct $(, abla)$-modules over pseudorigid spaces, extending $p$-adic Hodge theory.

Findings

Construction of overconvergent $(, abla)$-modules over pseudorigid spaces.

Introduction of new overconvergent period rings.

Application to eigenvarieties at the boundary of weight space.

Abstract

We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We introduce perfect and imperfect overconvergent period rings, and we use the Tate--Sen method to construct overconvergent $(\varphi, \Gamma)$-modules for Galois representations over pseudorigid spaces.