Interpolating Hodge-Tate and de Rham Periods

TL;DR

This paper develops a method to interpolate Hodge-Tate and de Rham periods across families of Galois representations, providing a detailed stratification and cohomology vanishing results that unify and extend previous foundational work.

Contribution

It introduces a new stratification technique for Galois representations over rigid spaces, generalizing prior results of Sen, Kisin, and Berger-Colmez in the context of period interpolation.

Findings

Stratification of rigid spaces by period conditions.

Vanishing of higher cohomology groups.

Application to geometricity of Galois representations.

Abstract

We study the interpolation of Hodge-Tate and de Rham periods over rigid analytic families of Galois representations. Given a Galois representation on a coherent locally free sheaf over a reduced rigid space and a bounded range of weights, we obtain a stratification of this space by locally closed subvarieties where the Hodge-Tate and bounded de Rham periods (within this range) as well as 1-cocycles form locally free sheaves. We also prove strong vanishing results for higher cohomology. Together, these results give a simultaneous generalization of results of Sen, Kisin, and Berger-Colmez. The main result has been applied by Varma in her proof of geometricity of Harris-Lan-Taylor-Thorne Galois representations as well as in several works of Ding.