Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866916424776155136 |
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| author | Kisin, Mark Zhou, Rong |
| author_facet | Kisin, Mark Zhou, Rong |
| contents | Let $A$ be an abelian variety over a number field $\mathrm E\subset \mathbb C$ and let $\mathbf G$ denote the Mumford--Tate group of $A$. After replacing $\mathrm E$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ factors through $\mathbf G(\mathbb Q_\ell).$ We show that for $v$ an odd prime of $\mathrm E$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf G(\mathbb Q_\ell)$ is independent of $\ell$. Along the way we prove that every point in the $μ$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_09945 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties Kisin, Mark Zhou, Rong Number Theory Let $A$ be an abelian variety over a number field $\mathrm E\subset \mathbb C$ and let $\mathbf G$ denote the Mumford--Tate group of $A$. After replacing $\mathrm E$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ factors through $\mathbf G(\mathbb Q_\ell).$ We show that for $v$ an odd prime of $\mathrm E$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf G(\mathbb Q_\ell)$ is independent of $\ell$. Along the way we prove that every point in the $μ$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it. |
| title | Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties |
| topic | Number Theory |
| url | https://arxiv.org/abs/2103.09945 |