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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.11048 |
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| _version_ | 1866912547697852416 |
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| author | Fu, Yu |
| author_facet | Fu, Yu |
| contents | Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the authors studied a special case involving specific one-parameter families of Jacobians of curves using analytic methods. We generalize their results, particularly Theorem B, to all families of abelian varieties with big geometric monodromy, employing an arithmetic approach. Our method applies Heath-Brown-type bounds on certain covers with level structures and optimizes the covers to derive the desired results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-simple abelian varieties in a family: arithmetic approaches Fu, Yu Number Theory 14G05, 11G05, 11G50 Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the authors studied a special case involving specific one-parameter families of Jacobians of curves using analytic methods. We generalize their results, particularly Theorem B, to all families of abelian varieties with big geometric monodromy, employing an arithmetic approach. Our method applies Heath-Brown-type bounds on certain covers with level structures and optimizes the covers to derive the desired results. |
| title | Non-simple abelian varieties in a family: arithmetic approaches |
| topic | Number Theory 14G05, 11G05, 11G50 |
| url | https://arxiv.org/abs/2412.11048 |