Châtelet surface

In algebraic geometry, a Châtelet surface is a rational surface studied by Châtelet (1959) given by an equation

Some snapshots showing the real points of the Châtelet surface with P(x)=x^3-5*x^2-6*x. Axis: x=red, y=yellow, z=blue

$y^{2}-az^{2}=P(x),\,$

where P has degree 3 or 4. They are conic bundles.

References

• Châtelet, F. (1959), "Points rationnels sur certaines courbes et surfaces cubiques", L'Enseignement mathématique, 2nd ser. (in French), 5: 153–170, MR 0130218, archived from the original on 2014-04-19, retrieved 2009-06-23
• Viray, Bianca (2012), "Failure of the Hasse principle for Châtelet surfaces in characteristic 2", Journal de Théorie des Nombres de Bordeaux , 24 (1): 231–236, arXiv:0902.3644, doi:10.5802/jtnb.794, MR 2914907, S2CID 13878632