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

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 $X \rightarrow Y$.