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THE POWER-SAVING MANIN–PEYRE CONJECTURE FOR A SENARY CUBIC

Part of: Zeta and $L$-functions: analytic theory Diophantine equations Multiplicative number theory

Published online by Cambridge University Press:  22 May 2019

Sandro Bettin  and
Kevin Destagnol
Sandro Bettin
Affiliation:
DIMA – Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy email [email protected]
Kevin Destagnol
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email [email protected]
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Abstract

Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory 47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^{2}\times \mathbb{P}^{2}$ with bihomogeneous coordinates $[x_{1}:x_{2}:x_{3}],[y_{1}:y_{2},y_{3}]$ and in $\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$ with multihomogeneous coordinates $[x_{1}:y_{1}],[x_{2}:y_{2}],[x_{3}:y_{3}]$ defined by the same equation $x_{1}y_{2}y_{3}+x_{2}y_{1}y_{3}+x_{3}y_{1}y_{2}=0$. We thus improve on recent work of Blomer et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann. 370 (2018), 491–553] and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type $\mathbf{A}_{1}$ and three lines (the other existing proof relying on harmonic analysis by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc. 108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc. 166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.

MSC classification

Primary: 11D45: Counting solutions of Diophantine equations 11N37: Asymptotic results on arithmetic functions 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 789 - 830
Copyright
Copyright © University College London 2019 

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