Families of Elliptic Functions, Realizing Coverings of the Sphere, with Branch-Points and Poles of Arbitrary Multiplicities

Abstract

We investigate smooth one-parameter families of complex tori over the Riemann sphere. The main problem is to describe such families in terms of projections of their branch-points. Earlier we investigated the problem for the case where, for every torus of the family, there is only one point lying over infinity. Here we consider the general case. We show that the uniformizing functions satisfy a partial differential equation and derive a system of differential equations for their critical points, poles, and moduli of tori. Based on the system we suggest an approximate method allowing to find an elliptic function uniformizing a given genus one ramified covering of the Riemann sphere.