A reciprocity law for prime geodesics

Abstract

We continue the study begun by Sarnak of closed geodesics on Γ⧹ H , Γ the modular group. A closed geodesic p corresponds to a hyperbolic conjugacy class {γ} in Γ, and an eigenvalue ε of an element of this class determines a real quadratic field Q (ε). On the other hand, a prime integer q determines a covering surface Γ 0( q )⧹ H of Γ⧹ H . The reciprocity law relates the behavior of the geodesic p when lifted to Γ 0( q )⧹ H to the splitting type of q in Q (ε).