Lattices in Tate modules
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866908589812088832 |
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| author | Poonen, Bjorn Rybakov, Sergey |
| author_facet | Poonen, Bjorn Rybakov, Sergey |
| contents | Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A \in \operatorname{M}_{2g}(\mathbb{Z})$ such that each Tate module $T_\ell X$ has a $\mathbb{Z}_\ell$-basis on which the action of $u$ is given by $A$, and similarly for the covariant Dieudonné module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_06363 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lattices in Tate modules Poonen, Bjorn Rybakov, Sergey Algebraic Geometry Number Theory 14K02 (Primary) 14K05 (Secondary) Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A \in \operatorname{M}_{2g}(\mathbb{Z})$ such that each Tate module $T_\ell X$ has a $\mathbb{Z}_\ell$-basis on which the action of $u$ is given by $A$, and similarly for the covariant Dieudonné module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$. |
| title | Lattices in Tate modules |
| topic | Algebraic Geometry Number Theory 14K02 (Primary) 14K05 (Secondary) |
| url | https://arxiv.org/abs/2107.06363 |