Multivariable $p$-adic Hodge theory for products of Galois groups

TL;DR

This paper develops a multivariable $p$-adic Hodge theory framework for families of Galois representations, generalizing previous results and enabling the recovery of key properties from associated $(, abla)$-modules.

Contribution

It introduces a new multivariable $(, abla)$-module theory for Galois representations, extending prior work and connecting to existing multivariable $p$-adic Hodge structures.

Findings

Constructs overconvergent multivariable $(, abla)$-modules for Galois families

Defines rings of multivariable crystalline and semistable periods

Recovers main results of prior multivariable $p$-adic Galois theory

Abstract

In this paper we explain how to attach to a family of $p$-adic representations of a product of Galois groups an overconvergent family of multivariable $(\varphi,\Gamma)$-modules, generalizing results from Pal-Zabradi and Carter-Kedlaya-Zabradi, using Colmez-Sen-Tate descent. We also define rings of multivariable crystalline and semistable periods, and explain how to recover this multivariable $p$-adic theory attached to a family of representations from its multivariable $(\varphi,\Gamma)$-module. We also explain how our framework allows us to recover the main results of Brinon-Chiarellotto-Mazzari on multivariable $p$-adic Galois representations.