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arXiv · 2502.12341

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

Léo Poyeton·Pietro Vanni
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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.

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Léo Poyeton, Pietro Vanni. 2025-02-17. Multivariable $p$-adic Hodge theory for products of Galois groups. https://arxiv.org/abs/2502.12341

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