Class number approximation in cubic function fields

We develop explicitly computable bounds for the order of the Jacobian of a cubic function field. We use approximations via truncated Euler products and thus derive effective methods of computing the order of the Jacobian of a cubic function field. Also, a detailed discussion of the zeta function of a cubic function field extension is included

Computing zeta functions of Artin-Schreier curves over finite fields II

We describe a method which may be used to compute the zeta function of an arbitrary Artin-Schreier cover of the projective line over a finite field. Specifically, for covers defined by equations of the form Zp - Z = f (X) we present, and give the complexity analysis of, an algorithm for the case in which f (X) is a rational function whose poles all have order 1. However, we only prove the correctness of this algorithm when the field characteristic is at least 5. The algorithm is based upon a cohomological formula for the L-function of an additive character sum. One consequence is a practical method of finding the order of the group of rational points on the Jacobian of a hyperelliptic curve in characteristic 2. © 2003 Elsevier Inc. All rights reserved

Computing zeta functions of sparse nondegenerate hypersurfaces

Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well-suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the L-function of an exponential sum.Comment: 37 pages; minor revisio