Integral points on affine quadric surfaces
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 141-161.

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely accounted for by the Brauer–Manin obstruction. We investigate how often the family of quadric hypersurfaces ax 2 +by 2 +cz 2 =n has a Brauer–Manin obstruction where a,b,c,n are integers. We improve previous bounds of Mitankin [7].

Il est bien connu que le principe de Hasse est valable pour les hypersurfaces quadratiques. Le principe de Hasse échoue pour les points entiers sur les hypersurfaces quadratiques lisses de dimension 2, mais cet échec peut être complètement expliqué par l’obstruction de Brauer–Manin. Nous étudions à quelle fréquence la famille d’hypersurfaces quadratiques ax 2 +by 2 +cz 2 =n a une obstruction de Brauer–Manin, où a,b,c,n sont des entiers. Nous améliorons les éstimations précédentes de Mitankin [7].

Published online:
DOI: 10.5802/jtnb.1196
Classification: 14G05, 14G12
Keywords: Brauer–Manin obstruction, integral point, quadric surface
Tim Santens 1

1 Departement Wiskunde KU Leuven Celestijnenlaan 200B 3001 Heverlee, Belgium
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Tim Santens. Integral points on affine quadric surfaces. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 141-161. doi : 10.5802/jtnb.1196.

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