Multivariable de Rham representations, Sen theory and $p$-adic differential equations
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| Format: | Preprint |
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2021
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| _version_ | 1866929499085471744 |
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| author | Brinon, Olivier Chiarellotto, Bruno Mazzari, Nicola |
| author_facet | Brinon, Olivier Chiarellotto, Bruno Mazzari, Nicola |
| contents | Let $K$ be a complete valued field extension of $\mathbf{Q}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G_{K,Δ}=G_K^Δ$ 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 $\mathsf{B}_{\rm dR}$ and $\mathsf{B}_{\rm HT}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G_{K,Δ}$ an integrable $p$-adic differential system in several variables $\mathsf{D}_{\rm dif}(V)$. 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 $(φ,Γ)$-module of $V$ constructed by Pal and Zábrádi, along classical Berger's construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_11563 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Multivariable de Rham representations, Sen theory and $p$-adic differential equations Brinon, Olivier Chiarellotto, Bruno Mazzari, Nicola Number Theory Let $K$ be a complete valued field extension of $\mathbf{Q}_p$ with perfect residue field. We consider $p$-adic representations of a finite product $G_{K,Δ}=G_K^Δ$ 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 $\mathsf{B}_{\rm dR}$ and $\mathsf{B}_{\rm HT}$, and multivariable Sen theory. In particular, we associate to any $p$-adic representation $V$ of $G_{K,Δ}$ an integrable $p$-adic differential system in several variables $\mathsf{D}_{\rm dif}(V)$. 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 $(φ,Γ)$-module of $V$ constructed by Pal and Zábrádi, along classical Berger's construction. |
| title | Multivariable de Rham representations, Sen theory and $p$-adic differential equations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2111.11563 |