On level 2 Modular differential equations

TL;DR

This paper investigates a specific class of modular differential equations related to level 2 subgroups, deriving explicit solutions and conditions for modularity using Schwarzian equations and classical modular functions.

Contribution

It introduces a novel approach to solving level 2 modular differential equations via Schwarzian equations and provides explicit solutions in terms of classical modular functions.

Findings

Derived conditions for solutions to be modular functions

Explicit expressions for solutions in terms of classical modular functions

Used equivariant functions and representation theory of level 2 subgroups

Abstract

In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $\Gamma_0(2)$. Our approach centers on solving the associated Schwarzian equation $\displaystyle \{h, z\} = 2F(z)$, where $\{h, z\}$ represents the Schwarzian derivative of a meromorphic function $h$ on $\mathbb{H}$. We derive conditions under which the solutions to this equation are modular functions for subgroups of the modular group and provide explicit expressions for these solutions in terms of classical modular functions. Key tools in our analysis include the theory of equivariant functions on the upper half-plane and the representation theory of level 2 subgroups of the modular group.