Lifting low-gonal curves for use in Tuitman's algorithm

Abstract

Consider a smooth projective curve $\overline{C}$ over a finite field ${\mathbb{F}}_q$, equipped with a simply branched morphism $\overline{C}\to {\mathbb{P}}^1$ of degree $d\le 5$. Assume char${\mathbb{F}}_q>2$ if $d\le 4$, and char${\mathbb{F}}_q>3$ if $d=5$. In this paper we describe how to efficiently compute a lift of $\overline{C}$ to characteristic zero, such that it can be fed as input to Tuitman's algorithm for computing the Hasse-Weil zeta function of $\overline{C}/{\mathbb{F}}_q$. Our method relies on the parametrizations of low rank rings due to Delone-Faddeev and Bhargava.

18 pages