Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915459741253632 |
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| author | Borodin, Matvey May, Liam |
| author_facet | Borodin, Matvey May, Liam |
| contents | We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of three-dimensional abelian varieties over finite fields up to $\mathbb{F}_{25}$ using the data in the L-functions and Modular Forms Database, we conjecture a collection of apparent obstructions. We provide a survey of known and conjectured results related to this problem, and a detailed statistical analysis of these findings. We conjecture that two of these obstructions classify all isogeny classes asymptotically as $q \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields Borodin, Matvey May, Liam Number Theory Algebraic Geometry We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of three-dimensional abelian varieties over finite fields up to $\mathbb{F}_{25}$ using the data in the L-functions and Modular Forms Database, we conjecture a collection of apparent obstructions. We provide a survey of known and conjectured results related to this problem, and a detailed statistical analysis of these findings. We conjecture that two of these obstructions classify all isogeny classes asymptotically as $q \to \infty$. |
| title | Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2508.16885 |