RESEARCH ARTICLE

Manin’s conjecture for a class of singular cubic hypersurfaces

  • Wenguang ZHAI
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  • Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

Received date: 05 Nov 2020

Accepted date: 31 May 2021

Copyright

2022 Higher Education Press

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.

Cite this article

Wenguang ZHAI . Manin’s conjecture for a class of singular cubic hypersurfaces[J]. Frontiers of Mathematics in China, 2022 , 17(6) : 1089 -1132 . DOI: 10.1007/s11464-021-0945-2

1
Batyrev V , Tschinkel Y . Manin’s conjecture for toric varieties. J Algebraic Geom, 1998, 7: 15- 53

2
Batyrev V , Tschinkel Y . Tamagawa numbers of polarized algebraic varieties. Astérisque, 1998, 251: 299- 340

3
Bhowmik G , Wu J . On the asymptotic behaviour of the number of subgroups of finite abelian groups. Arch Math (Basel), 1997, 69: 95- 104

4
de La Bretèche R . Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Astérisque, 1998, 251: 51- 77

5
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

6
Conrey J B . The fourth moment of derivatives of the Riemann zeta-function. Quart J Math Oxford (2), 1988, 39 (1): 21- 36

7
Davenport H . Cubic forms in sixteen variables. Proc R Soc Lond A, 1963, 272: 285- 303

8
Deligne P . La Conjecture de Weil. I. Publ Math Inst Hautes Études Sci, 1974, 43: 29- 39

9
Fouvry É . Sur la hauteur des points d’une certaine surface cubique singulière. Astérisque, 1998, 251: 31- 49

10
Franke J , Manin Y I , Tschinkel Y . Rational points of bounded height on Fano varieties. Invent Math, 1989, 95: 421- 435

11
Heath-Brown D R . Cubic forms in 14 variables. Invent Math, 2007, 170: 199- 230

12
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

13
Ivić A . The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications. New York: Wiley, 1985

14
Iwaniec H . Topics in Classical Automorphic Forms. Grad Stud Math, Vol 17. Providence: Amer Math Soc, 1997

15
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

16
Robert O , Sargos P . Three-dimensional exponential sums with monomials. J Reine Angew Math, 2006, 591: 1- 20

17
Salberger P . Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque, 1998, 351: 91- 258

18
Tenenbaum G . Introduction to Analytic and Probabilistic Number Theory. 3rd ed. Grad Stud Math, Vol 163. Providence: Amer Math Soc, 2015

19
Tóth L , Zhai W . On the error term concerning the number of subgroups of the groups Zm × Zn with m, nx. Acta Arith, 2018, 183: 285- 299

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