The universal family of semistable p-adic Galois representations

Hartl, Urs; Hellmann, Eugen

doi:10.2140/ant.2020.14.1055

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The universal family of semistable $p$-adic Galois representations

Urs Hartl and Eugen Hellmann

Vol. 14 (2020), No. 5, 1055–1121

DOI: 10.2140/ant.2020.14.1055

Abstract

Let $K$ be a finite field extension of $ℚ_{p}$ and let $𝒢_{K}$ be its absolute Galois group. We construct the universal family of filtered $(ϕ, N)$ -modules, or (more generally) the universal family of $(ϕ, N)$ -modules with a Hodge–Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semistable $𝒢_{K}$ -representations in $ℚ_{p}$ -algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over $ℚ_{p}$ in the sense of Huber. This has conjectural applications to the $p$ -adic local Langlands program.

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Keywords

p-adic Galois representations, crystalline representations, semistable representations, moduli spaces, filtered modules

Mathematical Subject Classification 2010

Primary: 11S20

Secondary: 11F80, 13A35

Milestones

Received: 8 October 2015

Revised: 29 May 2019

Accepted: 24 November 2019

Published: 13 July 2020

Authors

Urs Hartl
Mathematisches Institut Universität Münster Münster Germany

Eugen Hellmann
Mathematisches Institut Universität Münster Münster Germany

Vol. 14, No. 5, 2020