Arithmetic of regularized inner products and Fourier coefficients of harmonic Maass forms

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.