Closed Geodesics and Periods of Automorphic Forms

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. 0. Introduction and Results Let M = H/Γ be a compact hyperbolic surface, where Γ is a discrete subgroup of PSL(2,R) = SL(2,R)/{±I}. Such a surface has a countable infinity of closed geodesics, one corresponding to each non-zero conjugacy class in Γ ∼= π1M . We shall denote a typical prime closed geodesic by γ, its length by l(γ), and its homology class by [γ] ∈ H1(M,Z). We shall say that Γ is symmetric if it is normalized by = ( −1 0 0 1 ) , i.e., if Γ = Γ. 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 π(T ) := #{γ : l(γ) ≤ T}. In the 1940’s Delsarte [4] showed that π(T ) ∼ e /T , i.e., the ratio of the two sides converge to 1, as T → ∞; since then more precise results have been obtained. A more refined problem is to fix a homology class α ∈ H1(M,Z) and to study π(T, α) := #{γ : l(γ) ≤ T, [γ] = α}. It is known that π(T, α) ∼ C0e /T , where g ≥ 2 denotes the genus of M , with an explicit formula for C0 [11], [16]. A related problem is to count closed geodesics subject to a constraint on the periods ∫ γ ω, where ω 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 → C of weight 2m with respect to Γ. In [22], Shimura introduced a period of a weight 2m automorphic form f over a closed geodesic γ, which we shall denote by rm(f, γ). Zelditch showed that f has an “asymptotic period” (f) over a fixed homology class α. More precisely, writing C(T, α) = {γ : l(γ) ≤ T, [γ] = α},