A closed formula for the Geil-Matsumoto bound on semigroups and curves with many points

TL;DR

This paper derives a closed-form formula for the Geil-Matsumoto bound on the number of rational points of algebraic curves over finite fields for general numerical semigroups, improving bounds in certain cases.

Contribution

It provides a new closed formula for the GM bound for any numerical semigroup, extending previous results limited to two generators.

Findings

Closed formula for GM bound for any numerical semigroup.

Simplified formula for semigroups generated by consecutive integers.

Improved upper bounds on rational points for some algebraic curves.

Abstract

The Geil-Matsumoto bound (GM bound) constrains the number of rational points on a curve over a finite field in terms of the Weierstrass semigroup of any of the points on the curve. For general numerical semigroups, the GM bound lacks a simple closed-form expression, making its computation a challenging problem. A closed formula has been obtained for the case when the semigroup is generated by two co-prime integers. In this work, for any numerical semigroup, we provide a closed formula for the GM bound in terms of the Ap\'ery set of a nonzero element of the semigroup. In the case where the numerical semigroup is generated by consecutive integers $n, n+1, \dots, n+t$ with $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, we obtain a simple closed formula for the bound. We apply these results to obtain upper bounds on the number of rational points for algebraic curves over finite fields. In some cases, the GM bound improves some well-known upper bounds on the number of rational points.