Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T20:13:19.715Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONAL POINTS ON THE INNER PRODUCT CONE VIA THE HYPERBOLA METHOD

Part of: Arithmetic problems. Diophantine geometry Elementary classical functions Additive number theory; partitions

Published online by Cambridge University Press:  29 November 2017

V. Blomer  and
J. Brüdern
V. Blomer
Affiliation:
Mathematisches Institut, Bunsenstr. 3–5, 37073 Göttingen, Germany email vblomer@math.uni-goettingen.de
J. Brüdern
Affiliation:
Mathematisches Institut, Bunsenstr. 3–5, 37073 Göttingen, Germany email jbruede@mathematik.uni-goettingen.de
Get access

Abstract

A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.

MSC classification

Primary: 11P55: Applications of the Hardy-Littlewood method 14G05: Rational points 33B99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 780 - 796
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blomer, V. and Brüdern, J., Counting in hyperbolic spikes: the diophantine analysis of multihomogeneous diagonal equations. J. reine angew. Math., doi:10.1515/crelle-2015-0037.Google Scholar
Brüdern, J., Einführung in die analytische Zahlentheorie, Springer (Berlin, 1995).CrossRefGoogle Scholar
Bump, D., Automorphic Forms on GL(3, R) (Lecture Notes in Mathematics 1083 ), Springer (Berlin, 1984).Google Scholar
Buttcane, J. and Miller, S. D., Weights, raising and lowering operators, and $K$ -types for automorphic forms on $\text{SL}(3,\mathbb{R})$ . Preprint, 2017, arXiv:1702.08851.Google Scholar
Franke, J., Manin, Y. I. and Tschinkel, Y., Rational points of bounded height on Fano varieties. Invent. Math. 95 1989, 421–435.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, 7th edn., Academic Press (New York, 2007).Google Scholar
Spencer, C. V., The Manin conjecture for x 0 y 0 + ⋯ + x s y s = 0. J. Number Theory 129 2009, 1505–1521.CrossRefGoogle Scholar
Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn. (Cambridge Tracts in Mathematics 125 ), Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar
Vaughan, R. C., A variance for k-free numbers in arithmetic progressions. Proc. Lond. Math. Soc. (3) 91 2005, 573–597.CrossRefGoogle Scholar