Tabulation of cubic function fields via polynomial binary cubic forms

Details Tabulation of cubic function fields via polynomial binary cubic forms Tabulation of cubic function fields via polynomial binary cubic forms

2010-04-27

arxiv.org/abs/1004.4785v2

Mathematics

Number Theory

30 pages, minor typos corrected, extra table entries added, revamped complexity analysis of the algorithm. To appear in Mathem

Scientific paper

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.

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