Hypergeometric series, modular linear differential equations, and vector-valued modular forms

TL;DR

This paper surveys the theory of vector-valued modular forms, their relation to modular differential equations, and explores connections with hypergeometric series through numerical examples.

Contribution

It provides a comprehensive overview of vector-valued modular forms and demonstrates their links to hypergeometric series via explicit examples.

Findings

Connections between modular forms and hypergeometric series in dimensions 2 and 3

Numerical examples illustrating the theory

Insights into modular differential equations over the three-punctured sphere

Abstract

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.