Written in Q-form, one has
This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane. However, the ratio of two solutions of the hypergeometric equation is also known as a Schwarz triangle map.
At least four methods to find the j-function inverse can be given.
Dedekind defines the j-function by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the fundamental domain:
where (Sƒ)(x) is the Schwarzian derivative of ƒ with respect to x.
- J. Harnad and J. McKay, Modular solutions to equations of generalized Halphen type, Proc. R. Soc. London A 456 (2000), 261–294,
- (Provides a readable introduction, some history, references, and various interesting identities and relations between solutions)
- J. Harnad, Picard–Fuchs Equations, Hauptmoduls and Integrable Systems, Chapter 8 (Pgs. 137–152) of Integrability: The Seiberg–Witten and Witham Equation (Eds. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000)).