#### Vol. 4, No. 1, 2020

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

### Wouter Castryck and Floris Vermeulen

Vol. 4 (2020), No. 1, 109–125
##### Abstract

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

##### Keywords
point counting, Tuitman's algorithm, Delone–Faddeev correspondence, Bhargava correspondence
##### Mathematical Subject Classification 2010
Primary: 11G20, 11Y99, 14Q05