$p$-adic Hodge theory in rigid analytic families

TL;DR

This paper extends $p$-adic Hodge theory to families of Galois representations over affinoid algebras, establishing new structural results and stratifications of period modules, and constructing admissible loci in rigid analytic geometry.

Contribution

It generalizes classical $p$-adic Hodge theory results to families over non-reduced affinoid algebras, including stratifications and functorial admissible loci.

Findings

$ ext{D}_{ ext{HT}}(V)$ and $ ext{D}_{ ext{dR}}(V)$ are coherent sheaves on $ ext{Sp}(A)$.

$ ext{Sp}(A)$ is stratified by ranks of period submodules.

Constructs functorial $ ext{B}_ullet$-admissible loci in $ ext{Sp}(A)$.

Abstract

We study the functors $\D_{\B_\ast}(V)$, where $\B_\ast$ is one of Fontaine's period rings and $V$ is a family of Galois representations with coefficients in an affinoid algebra $A$. We show that $\D_{\HT}(V)=\oplus_{i\in\Z}(\D_{\Sen}(V)\cdot t^i)^{\Gamma_K}$, $\D_{\dR}(V)=\D_{\dif}(V)^{\Gamma_K}$, and $\D_{\cris}(V)=\D_{\rig}(V)[1/t]^{\Gamma_K}$, generalizing results of Sen, Fontaine, and Berger. The modules $\D_{\HT}(V)$ and $\D_{\dR}(V)$ are coherent sheaves on $\Sp(A)$, and $\Sp(A)$ is stratified by the ranks of submodules $\D_{\HT}^{[a,b]}(V)$ and $\D_{\dR}^{[a,b]}(V)$ of "periods with Hodge-Tate weights in the interval $[a,b]$". Finally, we construct functorial $\B_\ast$-admissible loci in $\Sp(A)$, generalizing a result of Berger-Colmez to the case where $A$ is not necessarily reduced.