On Ramanujan's Modular Equations and Hecke Groups

Md. Shafiul Alam

Published 2021-05-27Version 1

We study the relation between Hecke groups and the modular equations in Ramanujan's theories of signature 2, 3 and 4. The solution $(\alpha,\beta)$ to the generalized modular equation satisfies a polynomial equation $P(\alpha,\beta)=0$ and we determine the degree in each of $\alpha$ and $\beta$ of the polynomial $P(\alpha,\beta)$ in terms of the indices of Hecke subgroups. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(\alpha,\beta)$ and $(1-\beta,1-\alpha)$ satisfy the same polynomial equation.