Strongly compatible systems associated to semistable abelian varieties
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912360862580736 |
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| author | Kisin, Mark Zhou, Rong |
| author_facet | Kisin, Mark Zhou, Rong |
| contents | We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given
an abelian variety $A$ over a number field $\mathrm{E}\subset \mathbb C$, we prove that after replacing $\mathbb E$ by a finite extension, the action of $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm H^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\mathbf G$ of $A$. This involves an independence of $\ell$-statement for the Weil--Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly compatible systems associated to semistable abelian varieties Kisin, Mark Zhou, Rong Number Theory Algebraic Geometry We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety $A$ over a number field $\mathrm{E}\subset \mathbb C$, we prove that after replacing $\mathbb E$ by a finite extension, the action of $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm H^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\mathbf G$ of $A$. This involves an independence of $\ell$-statement for the Weil--Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction. |
| title | Strongly compatible systems associated to semistable abelian varieties |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2505.02165 |