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Families of Elliptic Functions, Realizing Coverings of the Sphere, with Branch-Points and Poles of Arbitrary Multiplicities
Lobachevskii Journal of Mathematics Pub Date : 2020-12-30 , DOI: 10.1134/s1995080220110153
S. Nasyrov

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.



中文翻译:

椭圆函数族,实现球面的覆盖,具有任意多重性的分支点和极点

摘要

我们研究了在黎曼球面上的光滑的一参量复杂的托里族。主要问题是用分支点的投影来描述此类族。较早前,我们针对以下问题进行了研究:对于家庭的每个圆环,只有一个点位于无限远处。这里我们考虑一般情况。我们表明,均匀化函数满足偏微分方程,并为其临界点,极点和托里模量导出了一个微分方程系统。基于该系统,我们建议一种近似方法,该方法允许找到使给定属均匀化黎曼球面的一个椭圆覆盖的椭圆函数。

更新日期:2020-12-30
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