Ray class fields of global function fields with many rational places

Abstract

0. Introduction. Algebraic curves over finite fields with many rational points have been of increasing interest in the last two decades. The question of explicitly determining the maximal number of points on a curve of given genus was initiated and in some special cases solved by Serre [34, 35, 36] around 1982. Since then there have been attempts to attack the problem by means of algebraic geometry as well as field arithmetic. Constructions by explicit equations have been carried out by van der Geer and van der Vlugt [7, 8]. The present paper, which makes use of class field theory, has its immediate predecessors in work by Lauter [13, 14, 15] and Niederreiter and Xing [19, 20, 21, 25, 26, 38]. The numerical results obtained improve several entries of the tables given in [9], [17] and [27]. As we are looking from the field theoretic point of view, with an algebraic curveX (smooth, projective, absolutely irreducible) defined over a finite field Fq we associate its field K = Fq(X) of algebraic functions, a global function field with full constant field Fq. Its genus is that of X, and coverings of X correspond to field extensions of K, the degree of the covering being the degree of the extension. A place of K, by which we mean the maximal ideal p in some discrete valuation ring of K, with (residue field) degree d = deg p, corresponds to (a Galois conjugacy class of) d points on X(Fqd), and each point on X having Fqd as its minimal field of definition over Fq lies in such a conjugacy class. In particular the rational places, i.e. the places of degree 1, of K|Fq are in 1-1 correspondence with the Fq-rational points on X. The (normalized) discrete valuation associated with a place p of K will be denoted by vp.