Points de hauteur born\'ee sur les hypersurfaces lisses des vari\'et\'es toriques. Cas g\'en\'eral

TL;DR

This paper proves the Batyrev-Manin Conjecture for the count of rational points of bounded height on hypersurfaces within certain toric varieties, employing a Hardy-Littlewood circle method inspired by Schindler's approach.

Contribution

It extends the verification of the Batyrev-Manin Conjecture to new classes of hypersurfaces in toric varieties using a novel adaptation of the Hardy-Littlewood circle method.

Findings

Confirmed the conjecture for specific toric hypersurfaces.

Derived the asymptotic formula with Peyre's constant.

Validated the method's effectiveness for these varieties.

Abstract

We demonstrate the Batyrev-Manin Conjecture for the number of points of bounded height on hypersurfaces of some toric varieties.. The method used is inspired by the one developed by Schindler for the study the case of hypersurfaces of biprojective spaces. This method is based on the Hardy-Littlewood circle method. The constant obtained in the final asymptotic formula is the one conjectured by Peyre. ----- Nous d\'emontrons la conjecture de Batyrev-Manin pour le nombre de points de hauteur born\'ee sur les hypersurfaces de certaines vari\'et\'es toriques. La m\'ethode utilis\'ee est inspir\'ee de celle d\'evelopp\'ee par Schindler pour l'\'etude du cas des hypersurfaces des espaces biprojectifs. Cette m\'ethode est bas\'ee sur la m\'ethode du cercle de Hardy-Littlewood. La constante obtenue dans la formule asymptotique finale est celle conjectur\'ee par Peyre.