Improvements of the Hasse-Weil-Serre bound over global function fields

Abstract

We improve the Hasse-Weil-Serre bound over a global function field K with relatively large genus in terms of the ramification behavior of the finite places and the infinite places for $K/k$, where k is the rational function field $F_q\left(T\right)$. Furthermore, we improve the Hasse-Weil-Serre bound over a global function field K in terms of the defining equation of K. As an application of our main result, we apply our bound to some well-known extensions: Kummer extensions and elementary abelian p-extensions, where p is the characteristic of k. In fact, elementary abelian p-extensions include Artin-Schreier type extensions, Artin-Schreier extensions, and Suzuki function fields. Moreover, we present infinite families of global function fields for Kummer extensions, Artin-Schreier type extensions, and elementary abelian p-extensions but not Artin-Schreier type extensions, which meet our improved bound: our bound is a sharp bound in these families. We also compare our new bound with some known data given in manypoints.org, which is the database on the rational points of algebraic curves. This comparison shows a meaningful improvement of our results on the bound of the number of the rational places of K.