Pair correlation of angles between reciprocal geodesics on the modular surface

Boca, Florin; Paşol, Vicenţiu; Popa, Alexandru; Zaharescu, Alexandru

doi:10.2140/ant.2014.8.999

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Pair correlation of angles between reciprocal geodesics on the modular surface

Florin P. Boca, Vicenţiu Paşol, Alexandru A. Popa and Alexandru Zaharescu

Vol. 8 (2014), No. 4, 999–1035

DOI: 10.2140/ant.2014.8.999

Abstract

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

Keywords

modular surface, reciprocal geodesics, pair correlation, hyperbolic lattice points, hyperbolic lattice angles

Mathematical Subject Classification 2010

Primary: 11K36

Secondary: 11J71, 11M36, 11P21, 37D40

Milestones

Received: 3 September 2013

Revised: 21 October 2013

Accepted: 22 November 2013

Published: 10 August 2014

Authors

Florin P. Boca
Department of Mathematics University of Illinois 1409 W. Green Street Urbana, IL 61801 United States

Vicenţiu Paşol
Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764 014700 Bucharest Romania

Alexandru A. Popa
Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764 014700 Bucharest Romania

Alexandru Zaharescu
Department of Mathematics University of Illinois 1409 W. Green Street Urbana, IL 61801 United States

Vol. 8, No. 4, 2014