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

TL;DR

This paper introduces an efficient algorithm for generating quadratic function fields with high 3-rank over finite fields, providing numerical data that supports existing heuristics and conjectures in the field.

Contribution

It develops a novel function field adaptation of a method for high 3-rank quadratic fields, including automorphism-based speed-ups and minimal genus field generation.

Findings

Numerical data aligns with Friedman-Washington heuristics for certain finite fields.

Algorithm efficiently produces quadratic function fields of minimal genus for given 3-rank.

Data supports recent conjectures by Malle and ongoing work by Garton.

Abstract

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 Horner's 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.