# Cycle integrals of meromorphic modular forms and Siegel theta functions

## TL;DR

This paper explores mathematical properties of modular forms and their connections to theta functions in number theory.

## Contribution

The paper introduces new results on rational cycle integrals of meromorphic modular forms and their relation to Siegel theta functions.

## Key findings

- Certain linear combinations of meromorphic modular forms yield rational cycle integrals.
- Cycle integrals of Siegel theta functions are expressed using Hecke’s indefinite theta functions.

## Abstract

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.