Closed Geodesics and Periods of Automorphic Forms

✍ Scribed by Richard Sharp

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 146 KB
• Volume: 160
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.1987

✦ Synopsis

We study the detailed structure of the distribution of Eichler Shimura periods of an automorphic form on a compact hyperbolic surface. We show that these periods do not cluster around the asymptotic period over a homology class discovered by Zelditch.

2001 Academic Press

0. INTRODUCTION AND RESULTS

Let M=H 2 Â1 be a compact hyperbolic surface, where 1 is a discrete subgroup of PSL(2, R)=SL(2, R)Â [ \I]. Such a surface has a countable infinity of closed geodesics, one corresponding to each non-zero conjugacy class in 1$? 1 M. We shall denote a typical prime closed geodesic by #, its length by l(#), and its homology class by [#] # H 1 (M, Z). We shall say that 1 is symmetric if it is normalized by ==( &1 0 0 1 ), i.e., if =1==1. An interesting and much studied problem is to understand the distribution of the closed geodesics on M. For example, one may study the asymptotic behaviour of the prime geodesic counting function ?(T ) := *[# : l(#) T]. In the 1940s, Delsarte [4] showed that ?(T )te T ÂT, i.e., the ratio of the two sides converges to 1, as T Ä ; since then more precise results have been obtained. A more refined problem is to fix a homology class : # H 1 (M, Z) and to study ?(T, :) :=*[# : l(#) T, [#]=:]. It is known that ?(T, :)tC 0 e T ÂT g+1 , where g 2 denotes the genus of M, with an explicit formula for C 0 [11,16]. A related problem is to count closed geodesics subject to a constraint on the periods # |, where | is a harmonic 1-form. This is an example of the type of problem considered in [2,13,20]. Here there are additional features depending on whether or not the periods lie in a discrete subgroup of R.

A natural generalization is to consider (holomorphic) m-forms. These correspond exactly to automorphic forms f: H 2 Ä C of weight 2m with