  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.
