On Ramanujan’s inversion formulas

Abstract: In [Cooper, PLMS 2009], using various identities of modular forms, Cooper re-derives Ramanujan’s inversion formulas for elliptic integrals of signatures~\(2,3,4\) and \(6\), and using these together with arithmetic-geometric means, he also re-establishes certain classical transformation formulas for hypergeometric functions as byproducts. In this note, we reinterpret these results from the perspective of the orbifold uniformization of modular curves and as an application, give a new proof to those transformation formulas, as well as derive a novel analogous transformation formula.