Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places

Abstract

where the maximum is extended over all K with g(K) = g. Equivalently, Nq(g) is the maximum number of Fq-rational points that a smooth, projective, absolutely irreducible algebraic curve over Fq of given genus g can have. Global function fields K with many rational places, that is, with N(K) close to Nq(g(K)), have received a lot of attention in the literature. Quite a number of papers on the subject have also been written in the language of algebraic curves over finite fields. We refer e.g. to the work of Ihara [7] and Serre [16]–[19] in the 1980s and to the more recent papers of Garcia and Stichtenoth [2], [3], Niederreiter and Xing [10], [11], Perret [12], Schoof [15], van der Geer and van der Vlugt [22], [23], Xing [25], and Xing and Niederreiter [27], [28]. The construction of global function fields with many rational places, or equivalently of algebraic curves over Fq with many Fqrational points, is an interesting problem per se, but it is also important for applications in the theory of algebraic-geometry codes (see [20], [21]) and in the recent constructions of low-discrepancy sequences introduced by the authors [9], [11], [26].