Tabulation of cubic function fields via polynomial binary cubic forms

Abstract

We present a method for tabulating all cubic function fields overFq(t)\mathbb {F}_q(t)whose discriminantDDhas either odd degree or even degree and the leading coefficient of−3D-3Dis a non-square inFq∗\mathbb {F}_{q}^*, up to a given boundBBondeg⁡(D)\deg (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 requiresO(B4qB)O(B^4 q^B)field operations asB→∞B \rightarrow \infty. The algorithm, examples and numerical data forq=5,7,11,13q=5,7,11,13are included.