$(\phi,\tau)$-modules diff\'erentiels et repr\'esentations potentiellement semi-stables

TL;DR

This paper develops a method to recover key invariants of $p$-adic Galois representations from differential modules associated with $(, au)$-modules, characterizing potentially semi-stable and finite $E$-height representations via connections.

Contribution

It introduces a way to extract $D_{cris}$ and $D_{st}$ invariants from differential $(, au)$-modules, providing new characterizations of potentially semi-stable and finite $E$-height representations.

Findings

Recovered $D_{cris}(V)$ and $D_{st}(V)$ from $D_{ au,rig}^ullet(V)$.

Characterized potentially semi-stable representations via the connection.

Identified finite $E$-height representations through the connection operator.

Abstract

Soit $K$ un corps $p$-adique et soit $V$ une repr\'esentation $p$-adique de $\mathcal{G}_K = \mathrm{Gal}(\bar{K}/K)$. La surconvergence des $(\phi,\tau)$-modules nous permet d'attacher \`a $V$ un $\phi$-module diff\'erentiel \`a connexion $D_{\tau,\mathrm{rig}}^\dagger(V)$ sur l'anneau de Robba $\mathbf{B}_{\tau,\mathrm{rig},K}^\dagger$. On montre dans cet article comment retrouver les invariants $D_{\mathrm{cris}}(V)$ et $D_{\mathrm{st}}(V)$ \`a partir de $D_{\tau,\mathrm{rig}}^\dagger(V)$, et comment caract\'eriser les repr\'esentations potentiellement semi-stables, ainsi que celles de $E$-hauteur finie, \`a partir de la connexion. Let $K$ be a $p$-adic field and let $V$ be a $p$-adic representation of $\mathcal{G}_K=\mathrm{Gal}(\bar{K}/K)$. The overconvergence of $(\phi,\tau)$-modules allows us to attach to $V$ a differential $\phi$-module $D_{\tau,\mathrm{rig}}^\dagger(V)$ on the Robba ring $\mathbf{B}_{\tau,\mathrm{rig},K}^\dagger$ that comes equipped with a connection. We show in this paper how to recover the invariants $D_{\mathrm{cris}}(V)$ and $D_{\mathrm{st}}(V)$ from $D_{\tau,\mathrm{rig}}^\dagger(V)$, and give a characterization of both potentially semi-stable representations of $\mathcal{G}_K$ and finite $E$-height representations in terms of the connection operator.