Chai, Ching-Li; Lin, Chang-Shou; Wang, Chin-Lung Mean field equations, hyperelliptic curves and modular forms. I. (English) Zbl 1327.35116 Camb. J. Math. 3, No. 1-2, 127-274 (2015). In this article, the authors consider the following singular Liouville equation with parameter \(\rho \in \mathbb R_{>0}\) \[ -\Delta u+e^u=\rho \delta_0\quad\text{on } E, \tag{1} \] where \(E\) is a flat torus, \(\delta_0\) denotes the Dirac delta measure at the zero point and the parameter \(\rho\) is an integer multiple of \(4\pi\). The authors study the cases where \(\rho = 4\pi(2n+1)\) and \(\rho = 8\pi n\) for a non-negative integer \(n\) and explore the connection between the solvability of the Liouville equation (1), the classical Lamé equation and modular forms. Reviewer: Said El Manouni (Berlin) Cited in 4 ReviewsCited in 44 Documents MSC: 35J75 Singular elliptic equations 35J08 Green’s functions for elliptic equations 14H70 Relationships between algebraic curves and integrable systems 33E05 Elliptic functions and integrals 35R06 PDEs with measure Keywords:singular Liouville equation; Dirac delta measure PDFBibTeX XMLCite \textit{C.-L. Chai} et al., Camb. J. Math. 3, No. 1--2, 127--274 (2015; Zbl 1327.35116) Full Text: DOI arXiv