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Abstract:
In 1939 Rademacher derived a conditionally convergent series
expression for the modular j-invariant, and used this expression—the
first Rademacher sum—to verify its modular invariance. We will
explain how to attach Rademacher sums to an arbitrary group
commensurable with the modular group, and we will demonstrate how the
automorphy of the resulting functions reflects the geometry of the
group in question.
In the case of a group of genus zero the relationship is particularly
striking. On the other hand, of the properties of the groups of
isometries of the hyperbolic plane that arise in moonshine, the genus
zero property is perhaps the most elusive. We will illustrate how
Rademacher sums shed light on this property by using them to formulate
a characterization of the discrete groups of monstrous moonshine.
A physical interpretation of the Rademacher sums comes into view when
we consider black holes in the context of three dimensional quantum
gravity. This observation, together with the application of Rademacher
sums to moonshine, amounts to a new connection between moonshine,
number theory and physics, and promises applications in all three
fields.
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