Explicit bounds and heuristics on class numbers in hyperelliptic function fields

In this paper, we provide tight estimates for the divisor class number of hyperelliptic function fields. We extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results. We thus obtain a faster method of computing regulators and class numbers. Furthermore, we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest.