Green function, Painlevé VI equation, and Eisenstein series of weight one
In this paper, the authors study the connection between the Green function and the Eisenstein series of weight one from one hand and the number of the singular points on \(\mathbb{C}\,\backslash\,\{0, 1\}\) of a solution \(\lambda(t)\) of the sixth Painlevé equation with parameters \(\left(\frac{1}{8}, - \frac{1}{8}, \frac{1}{8}, \frac{3}{8}\right)\) from other hand. During this paper the point \(t_0\in\mathbb{C}\,\backslash\,\{0, 1\}\) is a singular point of the solution \(\lambda(t)\) if \(\lambda(t_0)\in\{0, 1, \infty, t_0\}\). Exploiting the pointed connection, they obtain a series of results. In particular using the Hitchin solutions and the additional missing Riccati solutions of the elliptic form of the pointed Painlevé equation, the authors prove that there are only three smooth, for all \(t\in\mathbb{C}\,\backslash\,\{0, 1\}\), solutions of this equation. The authors also prove that any Riccati solution has singularities on \(\mathbb{C}\,\backslash\,\{0, 1\}\) and that any branch of a real solution \(\lambda(t)\) has at most two singular points (at most one pole) over \(\mathbb{C}\,\backslash\,\{0, 1\}\). In addition, when \(\lambda(t)\) is an algebraic solution, they calculate the number of the real \(j\)-values of zeros of the associate Eisenstein series of weight one. In order to obtain the above results, the authors develop the conceptions of the last two authors [\textit{C.-S. Lin} and \textit{C.-L. Wang}, Ann. Math. (2) 172, No. 2, 911--954 (2010; Zbl 1207.35011)] of the geometry of the critical points of the Green function on a flan torus \(E_{\tau}\) with \(\tau\) in the moduli \(\mathcal{M}_1=\mathbb{H}/SL(2, \mathbb{Z})\), \(\mathbb{H}\) is the upper half plane.
scientific article; zbMATH DE number 6846977