Enumerating hyperelliptic curves over finite fields in quasilinear time

Abstract

We present an algorithm that, for every fixed genus $g$, will enumerate all hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic in quasilinear time; that is, the time required for the algorithm is $\tilde{O}(q^{2g-1})$, where $q=\#k$. Such an algorithm already exists in the case $g=2$, thanks to work of Mestre and Cardona and Quer, and in the case $g=3$, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.

23 pages. Corrected small typos, added a reference to a paper of Nart, and added links to Internet Archive library copies of two books