p-adic Hodge theory in rigid analytic families

Bellovin, Rebecca

doi:10.2140/ant.2015.9.371

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$p$-adic Hodge theory in rigid analytic families

Rebecca Bellovin

Vol. 9 (2015), No. 2, 371–433

DOI: 10.2140/ant.2015.9.371

Abstract

We study the functors $D_{B_{*}} (V)$ , where $B_{*}$ is one of Fontaine’s period rings and $V$ is a family of Galois representations with coefficients in an affinoid algebra $A$ . We first relate them to $(φ, Γ)$ -modules, showing 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 $D_{cris} (V) = D_{rig} (V) {[1 ∕ t]}^{Γ_{K}}$ ; this generalizes results of Sen, Fontaine, and Berger. We then deduce that 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 $B_{*}$ -admissible loci in $Sp (A)$ , generalizing a result of Berger and Colmez to the case where $A$ is not necessarily reduced.

Keywords

$p$-adic Hodge theory, rigid analytic geometry

Mathematical Subject Classification 2010

Primary: 11S20

Secondary: 14G22

Milestones

Received: 23 January 2014

Revised: 21 November 2014

Accepted: 25 December 2014

Published: 5 March 2015

Authors

Rebecca Bellovin
Department of Mathematics University of California, Berkeley Evans Hall Berkeley, CA 94720 United States

Vol. 9, No. 2, 2015