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

Abstract: Let $K$ be a complete valued field extension of ${\mathbb Q}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G^{\Delta}_K$ 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 ${\mathbb B}_{\rm dR}$ and ${\mathbb B}_{\rm HT}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G^{\Delta}_K$ an integrable $p$-adic differential system in several variables ${\mathbb D}_{\rm dif }(V)$. 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 $(\varphi,\Gamma)$-module of $V$ constructed by Pal and Zabradi along classical Berger's construction. We will also deal with some new ideas on locally analytic vectors in this framework. Joint work with O. Brinon and N. Mazzari.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)