Séminaires et Congrès - Parutions - 21 (2009) 125-159 < Parutions < 21

Arithmetics, Geometry and Code theory
François Rodier, Serge Vladut
Séminaires et Congrès 21 (2009)
Présentation, Sommaire

Bounding Picard numbers of surfaces using -adic cohomology
Timothy G. Abbott, Kiran S. Kedlaya, David Roe
Séminaires et Congrès 21 (2009), 125-159
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Résumé :
Estimation du nombre de Picard des surfaces par la cohomologie -adique
Nous décrivons un algorithme qui permet de calculer une borne supérieure sur le nombre de Picard (arithmétique ou géométrique) d'une surface lisse projective sur un corps fini, en utilisant un calcul de l'action de Frobenius sur la cohomologie -adique en petite précision; cette question est suggérée par une application à la théorie des codes algébro-géométriques de type LDPC (low density parity check), décrits par Voloch et Zarzar. Grâce à une implémentation de cet algorithme en Magma, nous présentons divers exemples: des surfaces quartiques sur et dont le nombre de Picard arithmétique est égal à , et une surface quintique sur dont le nombre de Picard géométrique est égal à . De plus, nous présentons des exemples d'une construction de van Luijk, de certaines surfaces K3 sur un corps fini avec groupe d'automorphismes géométriques trivial; ceci nécessite de vérifier que le nombre de Picard géométrique de ces surfaces est égal à .

Mots-clefs : Nombre de Picard, surfaces sur les corps finis, cohomologie -adique, cohomologie de de Rham, théorie des codes.

Abstract:
Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on -adic cohomology to a small degree of -adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over and with arithmetic Picard number , and a smooth quintic over with geometric Picard number . We also produce some examples of a construction of van Luijk of K3 surfaces over finite fields with trivial geometric automorphism group; this requires verifying that these surfaces have geometric Picard number .

Keywords: Picard number, surfaces over finite fields, -adic cohomology, de Rham cohomology, coding theory.

Class. math. : Primary 14C22, 14F30; secondary 14F40, 14G10, 14G15, 14G50, 14J20.

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