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 fiel
d. 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 fields 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.
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 Pollards Kangaroo method in infrastructures, usin
g 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.
