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On Modular Forms, Hecke Operators, Replication and Sporadic Groups

Title:

On Modular Forms, Hecke Operators, Replication and Sporadic Groups

Farinha Matias, Rodrigo (2014) On Modular Forms, Hecke Operators, Replication and Sporadic Groups. PhD thesis, Concordia University.

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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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Farinha Matias, Rodrigo
Institution:Concordia University
Degree Name:Ph. D.
Program:Mathematics
Date:28 August 2014
Thesis Supervisor(s):Cummins, Chris
ID Code:978919
Deposited By: RODRIGO FARINHA MATIAS
Deposited On:26 Nov 2014 14:26
Last Modified:19 Nov 2018 14:47
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