Multivariable ({\phi},{\Gamma})-modules and products of Galois groups

TL;DR

This paper establishes an equivalence between categories of continuous Galois group representations and étale $(,g)$-modules over multivariable Laurent-series rings, extending the classical theory to products of Galois groups.

Contribution

It introduces a multivariable $(,g)$-module framework that generalizes existing one-variable theories to products of Galois groups.

Findings

Categories of Galois representations are equivalent to étale $(,g)$-modules over multivariable Laurent-series rings.

The equivalence holds over $_p$, $z_p$, and $q_p$ coefficient rings.

This generalizes the classical $(,g)$-module theory to higher-dimensional Galois groups.

Abstract

We show that the category of continuous representations of the $d$th direct power of the absolute Galois group of $\mathbb{Q}_p$ on finite dimensional $\mathbb{F}_p$-vector spaces (resp. finitely generated $\mathbb{Z}_p$-modules, resp. finite dimensional $\mathbb{Q}_p$-vector spaces) is equivalent to the category of \'etale $(\varphi,\Gamma)$-modules over a $d$-variable Laurent-series ring over $\mathbb{F}_p$ (resp. over $\mathbb{Z}_p$, resp. over $\mathbb{Q}_p$).