Green function, Painlevé VI equation, and Eisenstein series of weight one

Abstract

We study the problem: How many singular points of a solution λ(t) to the Painleve VI equation with parameter ( 8 , −1 8 , 1 8 , 3 8 ) might have in C \ {0, 1}? Here t0 ∈ C \ {0, 1} is called a singular point of λ(t) if λ(t0) ∈ {0, 1, t0, ∞}. 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 6= 6, we calculate the number of the real j-values of zeros of the Eisenstein series EN 1 (τ; k1, k2) of weight one, where (k1, k2) runs over [0, N − 1]2 with gcd(k1, k2, N) = 1. The geometry of the critical points of the Green function on a flat torus Eτ , as τ varies in the moduliM1, plays a fundamental role in our analysis of the Painleve IV equation. In particular, the conjectures raised in [22] on the shape of the domain Ω5 ⊂ M1, which consists of tori whose Green function has extra pair of critical points, are completely solved here.