On Larcher subgroups and Fourier Coefficients of modular forms.

The Theory of Moonshine was initiated more than three decades ago to explore the interaction between the sporadic simple groups and the Fourier coefficients of modular functions on Moonshine-type groups. Even though this theory now involves a larger variety of diverse concepts and structures, the interplay between these two subjects remains the central theme of this theory. The purpose of this thesis was motivated by Moonshine, but it is within the second domain. This work is two-fold, covering the structure of a class of congruence subgroups and the computation of the Fourier coefficients of modular forms defined on genus-zero moonshine-type subgroups. The purpose of the first part of this thesis is to compute some invariants of a family of congruence subgroups containing Larcher subgroups. These subgroups initially were defined by Larcher to prove his result on the cusp widths of congruence subgroups. However it later turned out that these subgroups have an interesting role in the classification of genus-zero and genus-one torsion-free congruence subgroups. The second part of this thesis is a generalization of the recurrence formulae which were first established by Bruinier, Kohnen and Ono for the Fourier coefficients of the modular forms of the full modular group. Shortly after, similar recurrences were found by some other authors for some genus-zero congruence subgroups of the full modular group. Using a different technique, this work finds the universal recursive formulae satisfied by the Fourier coefficients of any meromorphic modular form on any genus-zero subgroup of SL(2, [Special characters omitted.] ) commensurable with SL(2, [Special characters omitted.]