Multivariable de Rham representations, Sen theory and $p$-adic differential equations

Abstract

Let $K$ be a complete valued field extension of ${\mathbf{Q}}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G_{K,\Delta }=G_K^{\Delta }$ of the absolute Galois group $G_K$ of $K$. This product appears as the fundamental group of a product of diamonds. We develop the corresponding $p$-adic Hodge theory by constructing analogues of the classical period rings ${\mathsf{B}}_{\mathrm{dR}}$ and ${\mathsf{B}}_{\mathrm{HT}}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G_{K,\Delta }$ an integrable $p$-adic differential system in several variables ${\mathsf{D}}_{\mathrm{dif}}(V)$. We prove that this system is trivial if and only if the representation $V$ is de Rham. Finally, we relate this differential system to the multivariable overconvergent $(\phi ,\Gamma )$-module of $V$ constructed by Pal and Zábrádi, along classical Berger's construction.

minor corrections, new format, to be published in Mathematical Research Letters Vol.31, no.1