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

Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin, Chin-Lung Wang
2018 Journal of differential geometry
We study the problem: How many singular points of a solution λ(t) to the Painlevé VI equation with parameter ( 1 8 , −1 8 , 1 8 , 3 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
more » ... 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, we calculate the number of the real j-values of zeros of the Eisenstein series E N 1 (τ; 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 τ , as τ varies in the moduli M 1 , plays a fundamental role in our analysis of the Painlevé IV equation. In particular, the conjectures raised in [22] on the shape of the domain Ω 5 ⊂ M 1 , which consists of tori whose Green function has extra pair of critical points, are completely solved here. 8 , −1 8 , 1 8 , 3 8 ) . We say that two solutions λ r,s (t) and λ r ,s (t) give (or belong to) the same solution if λ r ,s (t) is the analytic continuation of λ r,s (t) along some closed loop in C \ {0, 1}. In §4, we will prove that λ r,s and λ r ,s give the same solution to PVI ( 1 8 , −1 8 , 1 8 , 3 8 ) if and only if (s , r ) ≡ (s, r) · γ mod Z 2 for some matrix γ ∈ Γ(2) = {γ ∈ SL(2, Z) | γ ≡ I 2 mod 2}. In this paper, we are mainly concerned with the question of smoothness of solutions to PVI ( 1 8 , −1 8 , 1 8 , 3 8 ) , and the location of its singular points. Notice