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On Modular Forms, Hecke Operators, Replication and Sporadic Groups


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.

Text (application/pdf)
Matias_PhD_F2014.pdf - Accepted Version


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.
Date:28 August 2014
Thesis Supervisor(s):Cummins, Chris
ID Code:978919
Deposited On:26 Nov 2014 14:26
Last Modified:19 Nov 2018 14:47
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