Abstract:
In this thesis, we investigate the arithmetic properties of regularized Petersson inner
products and Fourier coefficients of harmonic Maass forms. We study traces of cycle
integrals of modular objects over infinite geodesics, their interactions, and interplay with
Fourier coefficients of harmonic Maass forms and L-functions. Moreover, we examine the
regularized Petersson inner products of weakly holomorphic and meromorphic modular
forms, linking them to invariants of both real and imaginary quadratic fields, their
arithmetic and algebraic characteristics, their generating series, and their associations
with the divisors of modular forms.
