Inventiones mathematicae

, Volume 119, Issue 1, pp 37-66

First online:

Hardy-Littlewood varieties and semisimple groups

  • Mikhail BorovoiAffiliated withRaymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University
  • , Zeév RudnickAffiliated withDepartment of Mathematics, Princeton University

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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.