Differential equations satisfied by modular forms

Abstract: A classical result known since the nineteenth century asserts that if $F(z)$ is a modular form of weight $k$ and $t(z)$ is a nonconstant modular function on a Fuchsian subgroup of $SL(2,\mathbb{R})$ of the first kind, then $F(z), zF(z),... z^kF(z)$, as (multi-valued) functions of $t$, are solutions of a $k+1$-st order linear ordinary differential equations with algebraic functions of t as coefficients. This result constitutes one of the main sources of applications of modular forms to other branches of mathematics. In this talk, we will give a quick overview of this classical result and explain some of its applications in number theory.

number theoryrepresentation theory

Audience: researchers in the topic

SJTU number theory seminar