Inversion Formulas for the $j$-function Around Elliptic Points

TL;DR

This paper extends inversion formulas for the modular j-function around elliptic points i and ρ, connecting them with hypergeometric functions and Ramanujan's theories in signatures 2, 3, 4, and 6.

Contribution

It generalizes previous results by providing new inversion formulas for the j-function at elliptic points using hypergeometric functions and Ramanujan's theories in multiple signatures.

Findings

Derived inversion formulas for j-function around i and ρ

Connected j-function expansions with hypergeometric functions

Extended Ramanujan's theories to signatures 4 and 6

Abstract

Recently, Hong, Mertens, Ono and Zhang proved a conjecture of C\u{a}ld\u{a}raru, He, and Huang that expresses the Taylor series of the modular $j$-function around the elliptic points $i$ and $\rho=e^{\pi i/3}$ as rational functions arising from the signature 2 and 3 cases of Ramanujan's theory of elliptic functions to alternative bases. We extend these results and give inversion formulas for the $j$-function around $i$ and $\rho$ arising from Gauss' hypergeometric functions and Ramanujan's theory in signatures 4 and 6.