Vol. 8, No. 4, 2014

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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
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