Inventiones mathematicae

, Volume 119, Issue 1, pp 37–66

Hardy-Littlewood varieties and semisimple groups

  • Mikhail Borovoi
  • Zeév Rudnick

DOI: 10.1007/BF01245174

Cite this article as:
Borovoi, M. & Rudnick, Z. Invent Math (1995) 119: 37. doi:10.1007/BF01245174


We are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for which the Hardy-Littlewood circle method is applicable are strongly Hardy-Littlewood.

We prove that certain affine homogeneous spaces of semisimple groups are strongly Hardy-Littlewood varieties. Moreover, we prove that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties for with the asymptotic density of integer points can be computed in terms of a product of local densities.

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Mikhail Borovoi
    • 1
  • Zeév Rudnick
    • 2
  1. 1.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA

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