BRINON, O; CHIARELLOTTO, B; MAZZARI, N

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BRINON, O
Institut de Mathématiques de Bordeaux [IMB]

CHIARELLOTTO, B
Dipartimento di Matematica [Padova]

MAZZARI, N
Dipartimento di Matematica [Padova]

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Document de travail - Pré-publication

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Let $K$ be a complete discretely valued field extension of $\mathbf{Q}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G_{K,\Delta}=G_K^{\Delta}$ of the absolute Galois group $G_K$ of $K$. This product appears as the fundamental group of a product of diamonds. We develop the corresponding $p$-adic Hodge theory by constructing analogues of the classical period rings $\mathsf{B}_{\text{dR}}$ and $\textsf{B}_{\text{HT}}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G_{K,\Delta}$ an integrable $p$-adic differential system in several variables $\text{D}_{\text{dif}}(V)$. We prove that this system is trivial if and only if the representation $V$ is de Rham. Finally, we relate this differential system to the multivariable overconvergent $(\varphi,\Gamma)$-module of $V$ constructed by Pal and Zábrádi, along classical Berger's construction.< Réduire

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Multivariable de Rham representations, Sen theory, an $p$-adic differential operators

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Résumé en anglais

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