Bounding the number of rational places using Weierstrass semigroups

TL;DR

This paper establishes a new upper bound on the number of rational places in algebraic function fields over finite fields, based on Weierstrass semigroups, improving previous bounds especially for small q.

Contribution

It introduces an improved upper bound for rational places using Weierstrass semigroup generators, surpassing Lewittes' bound and impacting tower theory.

Findings

Derived a new upper bound depending on semigroup generators and q

Improved Serre's bound for q=2, 3, 4

Linked bounds to tower of function fields theory

Abstract

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields.