On Modular Forms, Hecke Operators, Replication and Sporadic Groups

In the first part of this thesis we find all congruence subgroups of PSL2(R) and respective weights for which the corresponding space of cusp forms is one-dimensional. We compute generators for those spaces. In the second part we establish a connection between the Hecke Algebra of Γ0(2)+ and the group 2 · B, the double cover of the Baby Monster group. Namely, we find a new form of replication, 2A-replication, that is reflected in the power map structure of 2 · B. This is very similar to the fact that usual replication reflects the power map structure in the Monster group. We use a vertex operator algebra and a Lie algebra that were constructed by H¨ohn and see that the McKay-Thompson series for 2 ·B satisfy 2A-replication identities. This also simplifies the computations made by H¨ohn to identify every McKay-Thompson series as a Hauptmodul by using generalized Mahler recurrence relations. This strategy follows in spirit Borcherd’s proof of the original Moonshine Conjectures. We also extend these ideas to Γ0(3)+ and 3·F3+. However, even though the generalization is straightforward there are McKay-Thompson series that have irrational coefficients for which our replication formulas don’t work