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arXiv · 2508.16885

Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields

Matvey Borodin·Liam May
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Abstract

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 $q \to \infty$.

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Matvey Borodin, Liam May. 2025-08-23. Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields. https://arxiv.org/abs/2508.16885

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