From Hodge Index Theorem to the number of points of curves over finite fields

TL;DR

This paper extends classical bounds on the number of rational points of curves over finite fields by developing higher order Weil bounds through geometric and algebraic methods, with computational support.

Contribution

It introduces a new third order Weil upper bound and provides a framework for higher order bounds using Euclidean relationships of Neron Severi classes.

Findings

Recovered Ihara's bound as a second order case

Established a new third order Weil upper bound

Produced numerical tables for higher order bounds using magma

Abstract

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect of exact recursive towers, and give several new bounds for points of curves in relative situation $X \rightarrow Y$.