GREEN FUNCTION, PAINLEVE VI EQUATION, AND EISENSTEIN SERIES OF WEIGHT ONE

The behavior and the location of singular points of a solution to Painleve VI equation could encode important geometric properties. For example, Hitchin's formula indicates that singular points of algebraic solutions are exactly the zeros of Eisenstein series of weight one. In this paper, we study the problem: How many singular points of a solution lambda(t) to the Painleve VI equation with parameter (1/8, -1/8, 1/8, 3/8) might have in C \ {0, 1}? Here t(0) is an element of C \ {0, 1} is called a singular point of lambda(t) if lambda(t(0)) is an element of {0, 1, t(0), infinity}. Based on Hitchin's formula, we explore the connection of this problem with Green function and the Eisenstein series of weight one. Among other things, we prove: (i) There are only three solutions which have no singular points in C \ {0, 1}. (ii) For a special type of solutions (called real solutions here), any branch of a solution has at most two singular points (in particular, at most one pole) in C \ {0, 1}. (iii) Any Riccati solution has singular points in C \ {0, 1}. (iv) For each N >= 5 and N not equal 6, we calculate the number of the real j-values of zeros of the Eisenstein series E-1(N) (tau; k(1), k(2)) of weight one, where (k(1), k(2)) runs over [0, N - 1](2) with gcd(k(1), k(2), N) = 1. The geometry of the critical points of the Green function on a flat torus E-tau, as tau varies in the moduli M-1, plays a fundamental role in our analysis of the Painleve VI equation. In particular, the conjectures raised in [23] on the shape of the domain Omega(5) subset of M-1, which consists of tori whose Green function has extra pair of critical points, are completely solved here.