Explicit uniform bounds for Brauer groups of singular K3 surfaces
Annales de l'Institut Fourier, Online first, 41 p.

Let k be a number field. We give an explicit bound, depending only on [k:] 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:] and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on [k:], 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.

Soit k un corps de nombres. On donne une borne explicite, dépendant uniquement de [k:] 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:] et supprimer la contrainte d’isogénéité des courbes elliptiques. En outre, on montre comment obtenir une borne, dépendant uniquement de [k:], 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.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3526
Classification: 11G35,  14J28,  14F22
Keywords: Brauer groups, K3 surfaces, uniform bounds, Brauer–Manin obstructions, effective strong Shafarevich conjecture.
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)
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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, Online first, 41 p.

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