Myrberg's numerical uniformization of hyperelliptic curves

Mika Seppälä

Abstract

The problem of numerical uniformization of an algebraic plane curve C defined by an affine polynomial equation P(x,y)=0 consists of:

Finding a domain G in the Riemann sphere and a discontinuous group G of Möbius transformations acting in G such that G/G = C
Finding an explicit form of the uniformizing mapping p : G-> C.

By the Uniformization Theorem, we know that such a presentation is possible. Explicit constructions for uniformization have been difficult to find in spite of huge efforts to solve this problem already for more than hundred years ago.

In this paper we construct an algorithmic way to pass from the polynomial P defining a hyperelliptic curve C to a uniformization of C. We do this in three steps:

We construct first a Schottky uniformization for hyperelliptic real algebraic M-curves C, i.e., hyperelliptic curves with real branch points.
Using quasiconformal mappings we next construct a deformation of the above Schottky uniformization. This deformation yields a Fuchsian uniformization.
To get rid of the assumption that all the branch points are real, we finally apply another quasiconformal deformations.

The first step is due to P. J. Myrberg (1920). The modifications of steps 2 and 3 allow us to get better results for a wider class of curves.

Content

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Introduction

Preliminaries

Iteration of the opening of the slots

The opening mapping

Opening the slots

Schottky Uniformization

Myrberg's algorithm

Fuchsian uniformization of hyperelliptic M-curves

Uniformization of general hyperelliptic curves

Bibliography

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