Bounding the number of points on a curve using a generalization of Weierstrass semigroups

TL;DR

This paper develops new upper bounds for the number of rational points on algebraic curves over finite fields by generalizing Weierstrass semigroups and applying coding theory techniques, even without explicit curve equations.

Contribution

It introduces a method to bound rational points using generalized Weierstrass semigroups, extending previous bounds without requiring explicit curve equations.

Findings

Provides upper bounds for rational points on algebraic curves

Extends previous bounds using generalized Weierstrass semigroups

Applicable to families of function fields

Abstract

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup [P. Beelen, N. Tuta\c{s}: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in [O. Geil, R. Matsumoto: Bounding the number of $\mathbb{F}_q$-rational places in algebraic function fields using Weierstrass semigroups. Pure Appl. Algebra, 213(6), 2009].