Class number and regulator computation in cubic function fields

TL;DR

This paper introduces an efficient computational method for determining the divisor class number and regulator of cubic function fields, providing the largest known examples and insights into zeta function zero distributions.

Contribution

It presents a new algorithm based on Euler product approximations for computing class numbers and regulators in cubic function fields, with implementation details for specific signatures.

Findings

Largest divisor class numbers and regulators computed for cubic function fields

Numerical evidence supporting the effectiveness of the algorithm

Data verifying heuristics on zero distribution of zeta functions

Abstract

We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such as purely cubic function fields defined by superelliptic curves, the examples provided are the largest divisor class numbers and regulators ever computed for a cubic function field over a large prime field. The ideas underlying the optimization of the class number algorithm can in turn be used to analyze the distribution of the zeros of the function field's zeta function. We provide a variety of data on a certain distribution of the divisor class number that verify heuristics by Katz and Sarnak on the distribution of the zeroes of the zeta function.