Abstract
Let ℓ ⩾ 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in ℓ variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u3 = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batyrev V, Tschinkel Y. Manin’s conjecture for toric varieties. J Algebraic Geom, 1998, 7: 15–53
Batyrev V, Tschinkel Y. Tamagawa numbers of polarized algebraic varieties. Astérisque, 1998, 251: 299–340
Bhowmik G, Wu J. On the asymptotic behaviour of the number of subgroups of finite abelian groups. Arch Math (Basel), 1997, 69: 95–104
de La Bretèche R. Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Astérisque, 1998, 251: 51–77
de La Bretèche R, Destagnol K, Liu J, Wu J, Zhao Y. On a certain non-split cubic surface. Sci China Math, 2019, 62(12): 2435–2446
Conrey J B. The fourth moment of derivatives of the Riemann zeta-function. Quart J Math Oxford (2), 1988, 39(1): 21–36
Davenport H. Cubic forms in sixteen variables. Proc R Soc Lond A, 1963, 272: 285–303
Deligne P. La Conjecture de Weil. I. Publ Math Inst Hautes Études Sci, 1974, 43: 29–39
Fouvry É. Sur la hauteur des points d’une certaine surface cubique singulière. Astérisque, 1998, 251: 31–49
Franke J, Manin Y I, Tschinkel Y. Rational points of bounded height on Fano varieties. Invent Math, 1989, 95: 421–435
Heath-Brown D R. Cubic forms in 14 variables. Invent Math, 2007, 170: 199–230
Heath-Brown D R, Moroz B Z. The density of rational points on the cubic surface X3 = X1X2X3. Math Proc Cambridge Philos Soc, 1999, 125: 385–395
Ivić A. The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications. New York: Wiley, 1985
Iwaniec H. Topics in Classical Automorphic Forms. Grad Stud Math, Vol 17. Providence: Amer Math Soc, 1997
Liu J, Wu J, Zhao Y. Manin’s conjecture for a class of singular cubic hypersurfaces. Int Math Res Not IMRN, 2019, 2019(7): 2008–2043
Robert O, Sargos P. Three-dimensional exponential sums with monomials. J Reine Angew Math, 2006, 591: 1–20
Salberger P. Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque, 1998, 351: 91–258
Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory. 3rd ed. Grad Stud Math, Vol 163. Providence: Amer Math Soc, 2015
Tóth L, Zhai W. On the error term concerning the number of subgroups of the groups ℤm × ℤn with m,n ⩽ x. Acta Arith, 2018, 183: 285–299
Acknowledgements
The author deeply thanks the referees for valuable suggestions. The first version of this paper was finished while the author was visiting Three Gorges Math. Research Center of Sanxia University in the beginning of June, 2019. The author thanks TGMRC for hospitality. This work was supported by the National Natural Science Foundation of China (Grant No. 11971476).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhai, W. Manin’s conjecture for a class of singular cubic hypersurfaces. Front. Math 17, 1089–1132 (2022). https://doi.org/10.1007/s11464-021-0945-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-021-0945-2