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

Abstract

The behavior and the location of singular points of a solution to Painlevé 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 Painlevé VI equation with parameter $(\frac{1}{8}, \frac{-1}{8}, \frac{1}{8}, \frac{3}{8})$ might have in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$? Here $t_0 \in \mathbb{C} \setminus \lbrace 0, 1\rbrace$ is called a singular point of $\lambda (t)$ if $\lambda (t_0) \in \lbrace 0, 1, t_0, \infty \rbrace$. 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 $\mathbb{C} \setminus \lbrace 0, 1\rbrace$. (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 $\mathbb{C} \setminus \lbrace 0, 1\rbrace$}. (iii) Any Riccati solution has singular points in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$. (iv) For each $N \geq 5$ and $N \neq 6$, we calculate the number of the real $j$-values of zeros of the Eisenstein series $\mathfrak{E}^N_1 (\tau ; k_1, k_2)$ of weight one, where $(k_1, k_2)$ runs over ${[0, N-1]}^2$ with $\mathrm{gcd}(k_1, k_2, N) = 1$.

The geometry of the critical points of the Green function on a flat torus $E_{\tau}$, as $\tau$ varies in the moduli $\mathcal{M}_1$, plays a fundamental role in our analysis of the Painlevé VI equation. In particular, the conjectures raised in “Elliptic functions, Green functions and the mean field equations on tori” [C.-S. Lin and C.-L. Wang, Annals of Math. 172 (2010), no. 2, 911–954] on the shape of the domain $\Omega_5 \subset \mathcal{M}_1$, which consists of tori whose Green function has extra pair of critical points, are completely solved here.