Approximating Euler Products and Class Number Computation in Algebraic Function Fields

We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the class number algorithms in turn can be used to analyze the distribution of the zeros of its zeta function. 1. Background and Motivation The zeta function of an algebraic object incorporates a large amount of information about its associated object. For computational purposes, it is often necessary to compute a large but finite number of terms in the Euler product representation of the zeta function. This idea has been primarily used for regulator and class number computation of global fields, as well as in other applications. Analytic class number formulas are a powerful number theoretic tool, since they relate the class number of a global field to its zeta function. In the 1970’s and 80’s, a number of algorithms for computing the class number and, where applicable, the regulator, of an algebraic number field by way of truncated Euler products were proposed. Quadratic number fields were investigated by Shanks [47] and Lenstra [38], and cubic extensions by Williams et al. [4, 57, 17]. Hafner and McCurley’s seminal subexponential algorithm for imaginary quadratic fields [22] was subsequently generalized to arbitrary number fields by Buchmann et al. [9, 8] and has since undergone many improvements, especially in for quadratic number fields. Newer methods use Bach’s improved method for approximating Euler products [3]. In addition, the analytic class number and truncated Euler products have been employed in numerous other applications by Buchmann, Williams, and others; these applications include proving regulator and class number computation NP-complete [10], testing whether a real quadratic field has class number one [51], investigating bounds on the regulator of a real quadratic field [25], finding polynomials with high prime value density and testing the Hardy-Littlewood Conjecture F [39, 28], as well as numerically verifying the Cohen-Lenstra heuristics [53] and the Ankeny-ArtinChowla conjecture [54, 55]. The function field analog of the class number question is the problem of finding the divisor class number of an algebraic function field over a finite field, or Date: May 27, 2008. 1991 Mathematics Subject Classification. Primary 11R58, 11Y16. Secondary 11M38, 11R65.