Cycle Integrals of the Parson Poincar\'e Series and Intersection Angles of Geodesics on Modular Curves

TL;DR

This paper establishes a geometric formula connecting cycle integrals of Parson's modular integrals to intersection angles of geodesics on modular curves, extending classical and recent formulas in the field.

Contribution

It provides a new geometric formula for cycle integrals of Parson's weight 2k modular integrals, linking them to geodesic intersection angles on modular curves.

Findings

Derived a formula relating cycle integrals to intersection angles

Extended classical hyperbolic Poincaré series results

Built upon recent geometric formulas for modular integrals

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.