Number of points of curves over finite fields in some relative situations from an euclidean point of view
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 125-138.

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 [4] from an euclidean point of view. 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 product of the curve with itself endowed with the opposite of the intersection product.

Nous étudions le nombre de points rationnels d’une courbe projective lisse sur un corps fini dans certaines situations relatives et dans l’esprit d’un précédent article [4], où nous adoptions un point de vue euclidien. Nous prouvons une borne de Weil relative, conséquence de l’application de l’inégalité de Cauchy–Schwarz à des parties relatives de la diagonale et du graphe du Frobenius dans un sous-espace euclidien du groupe des diviseurs de la surface produit de la courbe avec elle-même, à équivalence numérique près, muni de l’opposé de la forme d’intersection.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1155
Classification: 11G20, 14G05, 14G15, 14H99
Keywords: Curves over a finite field, rational point, Weil bound, Intersection Theory
Hallouin, Emmanuel 1; Perret, Marc 1

1 Institut de Mathématiques de Toulouse ; UMR 5219, Université de Toulouse ; CNRS, UT2J F-31058 Toulouse, France
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Hallouin, Emmanuel; Perret, Marc. Number of points of curves over finite fields in some relative situations from an euclidean point of view. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 125-138. doi : 10.5802/jtnb.1155. http://www.numdam.org/articles/10.5802/jtnb.1155/

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