Green function, Painlevé VI equation, and Eisenstein series of weight one
type: Item
instance of: scholarly article
title: Green function, Painlevé VI equation, and Eisenstein series of weight one
zbMATH Open document ID: 1390.34242
author: Zhijie Chen
author: Ting-Jung Kuo
author: Chang-Shou Lin
author: Chin-Lung Wang
published in: Journal of Differential Geometry
publication date: 2018-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1518490817
review text: 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.
reviewed by: Tsvetana Stoyanova
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 6846977
zbMATH Keywords: Painlevé \(VI\) equation
zbMATH Keywords: Green function
zbMATH Keywords: Eisenstein series of weight one
zbMATH Keywords: singular points
zbMATH Keywords: critical points
MaRDI profile type: Publication
OpenAlex ID: W2787063500
Wikidata QID: Q115165839
DOI: 10.4310/JDG/1518490817
description: scientific article; zbMATH DE number 6846977