Integral points on affine quadric surfaces

Santens, Tim

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 141-161.

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 $a{x}^2+b{y}^2+c{z}^2=n$ a une obstruction de Brauer–Manin, où $a,b,c,n$ sont des entiers. Nous améliorons les éstimations précédentes de Mitankin [7].

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 $a{x}^2+b{y}^2+c{z}^2=n$ has a Brauer–Manin obstruction where $a,b,c,n$ are integers. We improve previous bounds of Mitankin [7].

Classification : 14G05, 14G12
Mots clés : Brauer–Manin obstruction, integral point, quadric surface

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