Vector Valued Modular Forms

Modular forms and their generalizations are one of the most central
concepts in number theory. It took almost 300 years to cultivate the
mathematics lying behind the classical (i.e. scalar) modular forms. All of
the famous modular forms (e.g. Dedekind eta function) involve a multiplier,
this multiplier is a 1-dimensional representation of the underlying group.
This suggests that a natural generalization will be matrix valued
multipliers, and their corresponding modular forms are called vector valued
modular forms. These are much richer mathematically and more general than
the (scalar) modular forms.

In my talk, I will define and classify vector valued modular forms
associated to any arbitrary multiplier.  The connection between
vector-valued modular forms and Fuchsian differential equations, and the
consequences of this connection will be explained.