Inventiones mathematicae

, Volume 119, Issue 1, pp 37–66

Hardy-Littlewood varieties and semisimple groups


  • Mikhail Borovoi
    • Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv University
  • Zeév Rudnick
    • Department of MathematicsPrinceton University

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