Inversion formulas for the j-function around elliptic points

Abstract

Recently, Hong, Mertens, Ono, and Zhang [8] proved a conjecture of Căldăraru, He, and Huang [4] 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.