Computing class groups and gonalities of algebraic curves over finite fields

TL;DR

This paper presents practical algorithms that significantly speed up the computation of divisor class groups and gonalities of algebraic curves over finite fields, especially for large genus or residue fields, using a novel precomputation approach.

Contribution

It introduces a new precomputation-based method that enhances the efficiency of computing class groups and gonalities of curves over finite fields.

Findings

Achieves several orders of magnitude speedup over existing methods.

Effective for curves with large genus or residue field.

Utilizes power series-expansions for efficient Riemann-Roch space computation.

Abstract

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The approach relies on introducing a precomputation step involving power series-expansions, which allows for an efficient amortized computation of large numbers of Riemann-Roch spaces.