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

Abstract

Let $K$ be a finite field extension of ${\mathbb{Q}}_p$ and let ${\mathcal{𝒢}}_K$ 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 ${\mathcal{𝒢}}_K$-representations in ${\mathbb{Q}}_p$-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 ${\mathbb{Q}}_p$ in the sense of Huber. This has conjectural applications to the $p$-adic local Langlands program.