Number of points of curves over finite fields in some relative situations from an euclidean point of view

Abstract

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper [HP19] from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.