Tabulation of cubic function fields via polynomial binary cubic forms

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.

Computing quadratic function fields with high 3-rank via cubic field tabulation

This paper presents an algorithm for generating all imaginary and unusual discriminants up to a fixed degree bound that define a quadratic function field of positive 3-rank. Our method makes use of function field adaptations of a method due to Belabas for finding quadratic number fields of high 3-rank and of a refined function field version of a theorem due to Hasse. We provide numerical data for discriminant degree up to 11 over the finite fields $mathbb{F}_{5}, mathbb{F}_{7}, mathbb{F}_{11}$ and $mathbb{F}_{13}$. A special feature of our technique is that it produces quadratic function fields of minimal genus for any given 3-rank. Taking advantage of certain $mathbb{F}_{q}(t)$-automorphisms in conjunction with Horners rule for evaluating polynomials significantly speeds up our algorithm in the imaginary case; this improvement is unique to function fields and does not apply to number field tabulation. These automorphisms also account for certain divisibility properties in the number of fields found with positive 3-rank. Our numerical data mostly agrees with the predicted heuristics of Friedman-Washington and partial results on the distribution of such values due to Ellenberg-Venkatesh-Westerland for quadratic function fields over the finite field $finfldq{q}$ where $q equiv -1 pmod{3}$. The corresponding data for $q equiv 1 pmod{3}$ does not agree closely with the previously mentioned heuristics and results, but does agree more closely with some recent number field conjectures of Malle and some work in progress on proving such conjectures for function fields due to Garton.