Counting and equidistribution in Heisenberg groups

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• 作者：Jouni Parkkonen ; Frédéric Paulin
• 关键词：Mathematics Subject Classification11E39 ; 11F06 ; 11N45 ; 20G20 ; 53C17 ; 53C22 ; 53C55
• 刊名：Mathematische Annalen
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：367
• 期：1-2
• 页码：81-119
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1807
• 卷排序：367

文摘

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension 2. We prove a Mertens formula for the integer points over a quadratic imaginary number fields K in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over K in Heisenberg groups. We give a counting formula for the cubic points over K in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over K, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to 0.