Computing class groups and gonalities of algebraic curves over finite fields

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.

22 pages. Revised according to referee feedback. Comments welcome!