Let be a number field. We give an explicit bound, depending only on and the discriminant of the Néron–Severi lattice, on the size of the Brauer group of a K3 surface 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 and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on , on the number of -isomorphism classes of singular K3 surfaces defined over , thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.
Soit un corps de nombres. On donne une borne explicite, dépendant uniquement de et du discriminant du réseau de Néron–Severi, pour la taille du groupe de Brauer de toute surface K3 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 et supprimer la contrainte d’isogénéité des courbes elliptiques. En outre, on montre comment obtenir une borne, dépendant uniquement de , pour le nombre de classes d’isomorphismes sur de surfaces K3 singulières définies sur , prouvant ainsi une version effective de la conjecture forte de Shafarevich pour les surfaces K3 singulières.
Revised:
Accepted:
Published online:
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.
Balestrieri, Francesca 1; Johnson, Alexis 2; Newton, Rachel 3
@article{AIF_2023__73_2_567_0, author = {Balestrieri, Francesca and Johnson, Alexis and Newton, Rachel}, title = {Explicit uniform bounds for {Brauer} groups of singular {K3} surfaces}, journal = {Annales de l'Institut Fourier}, pages = {567--607}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {2}, year = {2023}, doi = {10.5802/aif.3526}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3526/} }
TY - JOUR AU - Balestrieri, Francesca AU - Johnson, Alexis AU - Newton, Rachel TI - Explicit uniform bounds for Brauer groups of singular K3 surfaces JO - Annales de l'Institut Fourier PY - 2023 SP - 567 EP - 607 VL - 73 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3526/ DO - 10.5802/aif.3526 LA - en ID - AIF_2023__73_2_567_0 ER -
%0 Journal Article %A Balestrieri, Francesca %A Johnson, Alexis %A Newton, Rachel %T Explicit uniform bounds for Brauer groups of singular K3 surfaces %J Annales de l'Institut Fourier %D 2023 %P 567-607 %V 73 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3526/ %R 10.5802/aif.3526 %G en %F AIF_2023__73_2_567_0
Balestrieri, Francesca; Johnson, Alexis; Newton, Rachel. Explicit uniform bounds for Brauer groups of singular K3 surfaces. Annales de l'Institut Fourier, Volume 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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