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Recently jointly with C.L. Chai, Z.J. Chen, T.J. Kuo and C.L. Wang, we have developed a theory to connect different subjects such as (multiple) Green functions, mean field equations, Lamé equation, hyperelliptic curves, modular forms and Painlevé VI equation. Among others in those joint works, the premodular forms Z (n) r,s (τ) occupies the central role. We will discuss its role from the aspects of differential geometry (Green function), conformal geometry (mean field equations) and the classical integral Lamé equation. The key issue is to study the function properties of Z (n) r,s (τ) such as the simple zero properties and the asymptotics as τ → +∞. Unexpectedly, Painlevé VI is the essential tool for the proof of these properties. The Okamoto transformation for Painlevé VI is used to set up the induction step. This is the key for many difficult computation. We also show how to apply the Painlevé properties to study the degenerate curves for branch points of the hyperelliptic curve Y (τ), n = 2.

Green function, mean field equation and Painlevé VI equation