Geometric deduction of the solutions to modular equations

TL;DR

This paper introduces a geometric method to prove solutions to generalized modular equations originally presented by Ramanujan, avoiding complex identities for theta series and hypergeometric functions.

Contribution

It provides a novel geometric proof technique for Ramanujan's modular equations, independent of traditional hypergeometric and theta series identities.

Findings

Geometric proofs of Ramanujan's modular equations are feasible.

The approach simplifies understanding of solutions to modular equations.

The method offers an alternative to complex analytical identities.

Abstract

In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series and hypergeometric functions. We offer a geometric approach to the proof of those formulae. We emphasize that our proofs are geometric and independent of such identities.