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Counting and equidistribution in Heisenberg groups

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Abstract

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.

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Notes

  1. It is called the Korányi distance by many people working in sub-Riemannian geometry, though Korányi [27] does attribute it to Cygan [10].

  2. called a hyperconic by Segre.

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Acknowledgments

The first author thanks the Université de Paris-Sud (Orsay) for a visit of a month and a half which allowed an important part of the writing of this paper, under the financial support of the ERC grant GADA 208091. We thank Y. Benoist and L. Clozel for their help with Proposition 20.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014, University of Jyväskylä, Finland

    Jouni Parkkonen

  2. Laboratoire de mathématique d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Frédéric Paulin

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  1. Jouni Parkkonen
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Correspondence to Jouni Parkkonen.

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Parkkonen, J., Paulin, F. Counting and equidistribution in Heisenberg groups. Math. Ann. 367, 81–119 (2017). https://doi.org/10.1007/s00208-015-1350-5

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  • DOI: https://doi.org/10.1007/s00208-015-1350-5

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