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

Abstract

Recently, Hong, Mertens, Ono and Zhang proved a conjecture of Căldăraru, 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.

9 pages

References in corpus (1)

• Proof of the elliptic expansion Moonshine Conjecture of Căldăraru, He, and Huang