Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields

TL;DR

This paper develops criteria to determine when an isogeny class of abelian threefolds over finite fields contains a hyperelliptic Jacobian, supported by data analysis and conjectures on classification.

Contribution

It introduces new obstructions for hyperelliptic Jacobians in isogeny classes and conjectures their completeness in classifying such classes asymptotically.

Findings

New criteria for hyperelliptic Jacobian existence in isogeny classes

Data-driven conjectures on obstructions based on Weil polynomials

Asymptotic classification conjecture as field size grows

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 $q \to \infty$.