Bounds for the number of points on curves over finite fields

TL;DR

This paper introduces new bounds for the number of rational points on algebraic curves over finite fields, improving upon classical bounds like Weil's in specific cases.

Contribution

It develops a variation of the Störh-Voloch theory to establish tighter bounds on rational points of algebraic curves over finite fields.

Findings

New bounds improve existing estimates in certain cases

Results outperform Weil's, Störh-Voloch's, and Ihara's bounds under specific conditions

Enhanced understanding of rational points distribution on algebraic curves

Abstract

Let $\mathcal{X}$ be a projective irreducible nonsingular algebraic curve defined over a finite field $\mathbb{F}_q$. This paper presents a variation of the St\"orh-Voloch theory and sets new bounds to the number of $\mathbb{F}_{q^r}$-rational points on $\mathcal{X}$. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, St\"orh-Voloch's and Ihara's.