### References & Citations

# Mathematics > Number Theory

# Title: Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

(Submitted on 6 Feb 2020 (v1), last revised 25 Nov 2020 (this version, v2))

Abstract: Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

## Submission history

From: Caleb Springer [view email]**[v1]**Thu, 6 Feb 2020 02:02:13 GMT (54kb,D)

**[v2]**Wed, 25 Nov 2020 18:08:46 GMT (56kb,D)

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