Year: 2016
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2234832767