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April  2009, 3(2): 271-309. doi: 10.3934/jmd.2009.3.271

The variance of arithmetic measures associated to closed geodesics on the modular surface

Wenzhi Luo 1, , Zeév Rudnick 2, and  Peter Sarnak 3,

1. 

Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, United States
2. 

Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States
3. 

Department of Mathematics, Princeton University, Fine Hall,Washington Road, Princeton, NJ 08544 and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States

Received  October 2008 Published  May 2009

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.
Keywords: Arithmetic variance, modular surface, closed geodesics..
Mathematics Subject Classification: Primary: 11F72; Secondary: 37D4.
Citation: Wenzhi Luo, Zeév Rudnick, Peter Sarnak. The variance of arithmetic measures associated to closed geodesics on the modular surface. Journal of Modern Dynamics, 2009, 3 (2) : 271-309. doi: 10.3934/jmd.2009.3.271
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