Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

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 $g\ge 1$, 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.

15 pages, 1 figure. To appear in the Simons Symposia proceedings volume for the Simons Collaboration "Arithmetic Geometry, Number Theory and Computation"