Tabulation of cubic function fields via polynomial binary cubic forms

TL;DR

This paper introduces a method to systematically list all cubic function fields over finite fields with certain discriminant properties, generalizing existing number field techniques to function fields.

Contribution

It extends Belabas' method and a theorem of Davenport and Heilbronn to cubic function fields, providing an efficient algorithm for tabulation.

Findings

Algorithm requires $O(B^4 q^B)$ operations as $B$ grows

Includes examples and data for q=5,7,11,13

Generalizes number field methods to function fields

Abstract

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B \rightarrow \infty$. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.