Explicit bounds and heuristics on class numbers in hyperelliptic function fields (Q2781230)

Let \(X\) be a hyperelliptic curve over the finite field \({\mathbb F}_q\). Let \(h\) denote the divisor class number, the number of rational points on the Jacobian of the curve. The values of \(\#X({\mathbb F}_{q^i})\) for all positive integers \(i\) up to \(g\) together determine \(h\). If one knows only the first few of these values, or equivalently one knows the values of certain character sums, then one can obtain partial information about \(h\). More precisely, one can obtain upper and lower bounds for \(h\) that are better than the general Weil bounds \((\sqrt{q}-1)^{2g} \leq h \leq (\sqrt{q}+1)^{2g}\). NEWLINENEWLINENEWLINEThe present paper uses an analysis of character sums to prove two theorems, each of which gives such improved upper and lower bounds for \(h\). These improve upon the bounds given in \textit{A. Stein} and \textit{H. C. Williams}, Exp. Math. 8, 119-133 (1999; Zbl 0987.11071)]. These theoretical estimates are compared against experimental data, and the expected size of the error term is understood in terms of equidistribution results of Katz and Sarnak. NEWLINENEWLINENEWLINEBounds such as those obtained in this paper are important because they can be combined with either the baby-step--giant-step method or the Pollard kangaroo method to give fast algorithms for computing \(h\) precisely.