Theory of Vector-Valued Modular Forms

Modular forms and their generalizations are one of the most central concepts in number theory. It took almost 200 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 this talk, a story of vector valued modular forms for any genus zero Fuchsian group of the first kind will be told. The connection between vector-valued modular forms and Fuchsian differential equations will be explained.