Abstract

Consider a smooth projective curve $\bar{C}$ over a finite field ${\mathbb{𝔽}}_q$, equipped with a simply branched morphism $\bar{C}\to {\mathbb{P}}^1$ 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 $\bar{C}$ to characteristic zero, such that it can be fed as input to Tuitman’s algorithm for computing the Hasse–Weil zeta function of $\bar{C}∕{\mathbb{𝔽}}_q$. Our method relies on the parametrizations of low rank rings due to Delone and Faddeev, and Bhargava.

Keywords

Mathematical Subject Classification 2010

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