On Ramanujan's Modular Equations and Hecke Groups

TL;DR

This paper explicitly determines the degrees of polynomial modular equations related to Ramanujan's identities and explores their connection with Hecke groups in Ramanujan's theories of signatures 2, 3, and 4.

Contribution

It provides an explicit method to find the degrees of polynomial modular equations and links these equations to Hecke groups within Ramanujan's framework.

Findings

Degrees of polynomial modular equations are explicitly determined.

Established a relation between Hecke groups and Ramanujan's modular equations.

Enhanced understanding of the structure of modular equations in Ramanujan's theories.

Abstract

Inspired by the work of S. Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is no developed theory about how to find the degrees of these polynomial modular equations explicitly. In this paper, we determine the degrees of the polynomial modular equations explicitly and study the relation between Hecke groups and modular equations in Ramanujan's theories of signatures 2, 3, and 4.