$p$-adic Hodge theory in rigid analytic families

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 $\D_{\HT}(V)=\oplus_{i\in\Z}(\D_{\Sen}(V)\cdot t^i)^{Γ_K}$, $\D_{\dR}(V)=\D_{\dif}(V)^{Γ_K}$, and $\D_{\cris}(V)=\D_{\rig}(V)[1/t]^{Γ_K}$, 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.