Hypergeometric solutions to Schwarzian equations
Modular forms have a great application area in various parts of mathematics. So there is no surprise that there exist some problems dealing with differential equations. On the other hand, it is well known that Eisenstein series are ``the most natural'' examples of modular forms. When we combine these two, we can consider the following modular differential equation: \N\[\Ny'' +s E_4 y=0 ,\N\]\Nwhere \(E_4\) is weight \(4\) Eisenstein series and \(s=\pi^2 r^2\) with \(r=n/m\) being a rational number in reduced form such that \(m \geq 7\). Furthermore, we can associate this differential equation with a Schwarzian equation as \N\[\N\{ h, \tau \}=2s E_4.\N\]\NThen in the paper under review, the authors solve this Schwarzian equation in terms of the Gauss hypergeometric series by using the theory of equivariant functions on the upper half-plane and the \(2\)-dimensional vector-valued modular forms. This work can be considered as a completion work of a previous one by one of the authors.
scientific article; zbMATH DE number 7937717
scientific article