Cycle Integrals of the Parson Poincaré Series and Intersection Angles of Geodesics on Modular Curves

Abstract: We prove a geometric formula for the cycle integrals of Parson's weight 2k modular integrals in terms of the intersection angles of geodesics on modular curves. Our result is an analog for modular integrals of a classical formula for the cycle integrals of certain hyperbolic Poincar\'e series, due to Katok. On the other hand, it extends a recent geometric formula of Matsusaka and Duke, Imamo={g}lu, and T\'oth for the cycle integrals of weight 2 modular integrals.