Universität zu Köln

Löbrich, Steffen (2018). On Divisors, Congruences, and Symmetric Powers of Modular Forms. PhD thesis, Universität zu Köln.

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Abstract

This thesis contains research articles on various topics in the theory of modular forms. First we investigate divisors of meromorphic modular forms of higher level using polar harmonic Maass forms of weight 2. We continue by relating these Maass forms to modular forms associated to imaginary quadratic fields. We show that the Fourier coefficients of these functions are given by traces of singular moduli and compute their regularized inner products. After that we investigate p-adic properties of the Fourier coefficients of generalized eta-quotients. In particular, we show that these coefficients cannot satisfy any linear congruences whose residues do not fulfill certain quadratic equations. Finally we construct polynomials for motivic L-functions that generalize the well-known period polynomials for Hecke eigenforms. We show that these polynomials have almost all of their zeros on the complex unit circle and that the zeros tend to be equidistributed as the level or the weight of the motive are sufficiently large.