Explicit uniform bounds for Brauer groups of singular K3 surfaces

Balestrieri, Francesca; Johnson, Alexis; Newton, Rachel

doi:10.5802/aif.3526

Explicit uniform bounds for Brauer groups of singular K3 surfaces
[Bornes uniformes explicites pour les groupes de Brauer des surfaces K3 singulières]

Balestrieri, Francesca ¹ ; Johnson, Alexis ² ; Newton, Rachel ³

¹ The American University of Paris 5 Boulevard de La Tour-Maubourg 75007 Paris (France)
² Department of Mathematics University of Minnesota 206 Church St SE Minneapolis, MN 55455 (USA)
³ Department of Mathematics King’s College London Strand, London WC2R 2LS (UK)

Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 567-607.

Résumé
Abstract

Soit $k$ un corps de nombres. On donne une borne explicite, dépendant uniquement de $[k : ℚ]$ et du discriminant du réseau de Néron–Severi, pour la taille du groupe de Brauer de toute surface K3 $X / k$ qui est géométriquement isomorphe à la surface Kummer attachée à un produit de courbes elliptiques de type CM isogènes. Comme application, on montre que l’ensemble de Brauer–Manin pour une telle variété est effectivement calculable. Sous l’hypothèse de Riemann généralisée, on peut de plus faire dépendre la borne explicite uniquement de $[k : ℚ]$ et supprimer la contrainte d’isogénéité des courbes elliptiques. En outre, on montre comment obtenir une borne, dépendant uniquement de $[k : ℚ]$ , pour le nombre de classes d’isomorphismes sur $C$ de surfaces K3 singulières définies sur $k$ , prouvant ainsi une version effective de la conjecture forte de Shafarevich pour les surfaces K3 singulières.

Let $k$ be a number field. We give an explicit bound, depending only on $[k : ℚ]$ and the discriminant of the Néron–Severi lattice, on the size of the Brauer group of a K3 surface $X / k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on $[k : ℚ]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on $[k : ℚ]$ , on the number of $C$ -isomorphism classes of singular K3 surfaces defined over $k$ , thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.

Reçu le : 2020-11-03
Révisé le : 2021-03-11
Accepté le : 2021-06-23
Publié le : 2023-05-12

DOI : 10.5802/aif.3526

Classification : 11G35, 14J28, 14F22
Keywords: Brauer groups, K3 surfaces, uniform bounds, Brauer–Manin obstructions, effective strong Shafarevich conjecture.
Mot clés : Groupes de Brauer, surfaces K3, bornes uniformes, obstructions de Brauer–Manin, conjecture forte de Sharafevich effective.

Affiliations des auteurs :

Balestrieri, Francesca ¹ ; Johnson, Alexis ² ; Newton, Rachel ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Explicit uniform bounds for {Brauer} groups of singular {K3} surfaces},
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Balestrieri, Francesca; Johnson, Alexis; Newton, Rachel. Explicit uniform bounds for Brauer groups of singular K3 surfaces. Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 567-607. doi : 10.5802/aif.3526. https://aif.centre-mersenne.org/articles/10.5802/aif.3526/

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