#### Vol. 14, No. 5, 2020

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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
##### Abstract

Let $K$ be a finite field extension of ${ℚ}_{p}$ and let ${\mathsc{𝒢}}_{K}$ be its absolute Galois group. We construct the universal family of filtered $\left(\varphi ,N\right)$-modules, or (more generally) the universal family of $\left(\varphi ,N\right)$-modules with a Hodge–Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semistable ${\mathsc{𝒢}}_{K}$-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.

##### 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