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

TL;DR

This work develops a comprehensive multivariable extension of p-adic Hodge theory for representations of products of Galois groups. It constructs overconvergent multivariable $(\varphi,\Gamma)$-modules attached to families of representations, builds multivariable rings of crystalline, semistable, and de Rham periods, and proves descent theorems via a multivariable Colmez–Sen–Tate framework. The authors show that multivariable de Rham, crystalline, and semistable structures recover their classical single-variable counterparts on each factor, and they establish equivalences and finiteness results for these multivariable objects, including how to recover multivariable invariants from their $(\varphi,\Gamma)$-modules. Applications include a unified treatment of Brinon–Chiarellotto–Mazzari-type results and a robust framework for studying p-adic Galois representations in geometric Langlands-inspired multivariable settings.

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 $(\varphi,Γ)$-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 $(\varphi,Γ)$-module. We also explain how our framework allows us to recover the main results of Brinon-Chiarellotto-Mazzari on multivariable $p$-adic Galois representations.

Theorems & Definitions (62)

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