HARDY-LITTLEWOOD VARIETIES AND SEMISIMPLE GROUPS

• 成果类型: Article
• 署名作者: BOROVOI, M; RUDNICK, Z
• 署名单位: Princeton University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/BF01245174
• 发表日期: 1995
• 页码: 37-66
• 关键词: stable trace formula; homogeneous varieties; Strong Approximation; integer points; density; numbers; SPACES; terms; field

摘要

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.