Class Number and Regulator Computation in Purely Cubic Function Fields of Unit Rank Two

TL;DR

This paper introduces an improved computational method for determining the divisor class number and regulator of purely cubic function fields with unit rank 2, achieving the largest known values for genus 3 fields.

Contribution

It presents the first efficient square-root algorithm applied to the infrastructure of a global field of unit rank 2, enhancing previous methods with distribution-based optimizations.

Findings

Largest known divisor class numbers and regulators for genus 3 fields.

Successful implementation of a square-root algorithm in this context.

Enhanced computational efficiency over previous approaches.

Abstract

We describe and give computational results of a procedure to compute the divisor class number and regulator of most purely cubic function fields of unit rank 2. Our implementation is an improvement to Pollard's Kangaroo method in infrastructures, using distribution results of class numbers as well as information on the congruence class of the divisor class number, and an adaptation that efficiently navigates these torus-shaped infrastructures. Moreover, this is the first time that an efficient "square-root" algorithm has been applied to the infrastructure of a global field of unit rank 2. With the exception of certain function fields defined by Picard curves, our examples are the largest known divisor class numbers and regulators ever computed for a function field of genus 3.