FAMILIES OF ELLIPTIC FUNCTIONS AND UNIFORMIZATION OF COMPLEX TORI WITH A UNIQUE POINT OVER INFINITY

Abstract

We investigate the problem of describing a one-parametric family of elliptic functions which uniformizes a given family of ramified coverings of the Riemann sphere with maximal possible ramification over infinity. We find a PDE for the family of functions and use it to deduce a system of ODEs for their critical points.

Keywords

• Riemann surface
• complex torus
• elliptic function
• uniformization