Point counting on curves using a gonality preserving lift

TL;DR

This paper presents an efficient method for lifting algebraic curves from finite fields to number fields while preserving genus and gonality, enabling improved computation of zeta functions via p-adic cohomology.

Contribution

It introduces a gonality-preserving lift technique for curves of gonality up to four and genus up to five, with an implementation in Magma for practical use.

Findings

Efficient lifting method for curves of gonality ≤ 4

Implementation of the lift for genus ≤ 5 curves over finite fields of odd characteristic

Enhanced computation of zeta functions using the lift in p-adic cohomology

Abstract

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology.