Abstract

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.