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Computing class groups and gonalities of algebraic curves over finite fields

Maarten Derickx, Kenji Terao

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.

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.
Paper Structure (11 sections, 3 theorems, 25 equations, 6 tables, 7 algorithms)

This paper contains 11 sections, 3 theorems, 25 equations, 6 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Computing Riemann-Roch spaces
  3. Data availability and reproducibility.
  4. Acknowledgements
  5. Notation
  6. Computing Riemann-Roch spaces
  7. Computing gonalities
  8. Computing class groups
  9. Timings
  10. Gonalities
  11. Class groups

Key Result

Theorem 3.1

Let $D_0 \in \mathop{\mathrm{Div}}\nolimits X$ be a divisor on X, and let $\{f_1, \dots, f_{\ell}\}$ be a $k$-basis of $\mathcal{L}(D_0)$, where $\ell = \ell(D_0)$. Note that this choice of basis defines an isomorphism $\mathcal{L}(D_0) \cong k^{\ell}$ of $k$-vector spaces. For any closed point $x \ where all coefficients $a_{x, i, j, l}$ lie in $k$. For all $i, j$, we let $\mathbf{a}_{x, i, j} \i

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Lemma 4.1: najman2024gonality
  • Lemma 5.1
  • proof