Multivariable $(\varphi, \Gamma)$-modules and products of Galois groups

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 étale $(\varphi, \Gamma)$-modules over a $d$-variable Laurent-series ring over $\mathbb{F}_p$ (resp. over $\mathbb{Z}_p$, resp. over $\mathbb{Q}_p$).

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This research was supported by a Hungarian OTKA Research grant K-100291 and by the János Bolyai Scholarship of the Hungarian Academy of Sciences. I would like to thank the Arithmetic Geometry and Number Theory group of the University of Duisburg-Essen, campus Essen, for its hospitality and for financial support from SFB TR45 where parts of this paper was written. I am grateful to Christophe Breuil, Elmar Große-Klönne, Kiran Kedlaya, and Vytas Paškūnas for useful discussions on the topic. I would like to thank Peter Scholze for clarifying the relation of this work to his theory of realizing $G_{\mathbb{Q}_p , \triangle}$ as the étale fundamental group of a diamond.

Received 14 March 2016

Accepted 18 December 2016

Published 5 July 2018