Counting hyperelliptic curves

Abstract

We find a closed formula for the number $\mathrm{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k={\mathbb{F}}_q$ of odd characteristic. These numbers $\mathrm{hyp}(g)$ are expressed as a polynomial in $q$ with integer coefficients that depend on the set of divisors of $q-1$ and $q+1$. As a by-product we obtain a closed formula for the number of self-dual curves of genus $g$. A hyperelliptic curve is self-dual if it is $k$-isomorphic to its own hyperelliptic twist.