Points de hauteur born\'ee sur les hypersurfaces lisses des vari\'et\'es toriques

TL;DR

This paper proves the Batyrev-Manin Conjecture for counting rational points of bounded height on certain hypersurfaces in toric varieties with Picard rank 2, using Hardy-Littlewood circle method and confirming Peyre's predicted constant.

Contribution

It extends the Hardy-Littlewood circle method approach to new classes of hypersurfaces in toric varieties with Picard rank 2, confirming Peyre's constant in the asymptotic formula.

Findings

Confirmed Batyrev-Manin Conjecture for specific toric hypersurfaces

Derived the asymptotic formula matching Peyre's constant

Extended circle method techniques to higher Picard rank cases

Abstract

We demonstrate the Batyrev-Manin Conjecture for the number of points of bounded height on hypersurfaces of some toric varieties whose rank of the Picard group is 2. 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 dont le rang du groupe de Picard est 2. 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.