On the Distribution of the Eigenvalues of the Hyperbolic Laplacian for PSL(2, Z), II

✍ Scribed by C.J. Mozzochi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 164 KB
• Volume: 86
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2558

✦ Synopsis

Let H be the upper half plane and X=SL(2, Z)"H the corresponding modular surface. Theory and experiment suggest that the eigenvalues of the hyperbolic Laplacian, 2, on X, denoted by * j =1Â4+t 2 j , behave in many ways like a random sequence. In particular, for any A>0 the numbers A* j , j=1, 2, 3, ..., should be well distributed modulo 1 (that is to say, there should be square root cancellation in the corresponding Weyl sums). In this paper we show in sharp contrast to the above that the sequence 2t j log(t j Â?e) is not well distributed modulo 1. This reflects a certain structure that the closed geodesics on X carry, precisely that the norms of the hyperbolic conjugacy classes (which correspond to closed geodesics) of 1 are very close to being squares of integers. This phenomenon no doubt occurs for all arithmetic quotients X of H but not for the generic hyperbolic surface. 2001 Academic Press * j = 1 4 +t 2 j with t j >0, j=1, 2, 3, .... The Weyl Selberg formula states that for 1=PSL(2, Z), M(T )= :

0<t j T m(t j )= 1 12 T 2 +c 1 T log T+O(T )