An Algorithm for Numerically Inverting the Modular $j$-function

TL;DR

This paper presents a novel method for numerically inverting the modular j-function using polar harmonic Maass forms, providing a new approach to an old inverse problem in complex analysis.

Contribution

It introduces a direct approach to invert the modular j-function by analyzing the Fourier expansion of a specialized polar harmonic Maass form.

Findings

Successfully inverts the modular j-function numerically.

Connects the inverse problem to the theory of Maass forms.

Provides a new perspective on classical modular function inversion.

Abstract

The modular $j$-function is a bijective map from $X_0(1) \setminus \{\infty\}$ to $\mathbb{C}$. A natural question is to describe the inverse map. Gauss offered a solution to the inverse problem in terms of the arithmetic-geometric mean. This method relies on an elliptic curve model and the Gaussian hypergeometric series. Here we use the theory of polar harmonic Maass forms to solve the inverse problem by directly examining the Fourier expansion of the weight $2$ polar harmonic Maass form obtained by specializing the logarithmic derivative of the denominator formula for the Monster Lie algebra.