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Uniformizing real hyperelliptic \(M\)-curves using the Schottky-Klein prime function. (English) Zbl 1214.30033

Bobenko, Alexander I. (ed.) et al., Computational approach to Riemann surfaces. Berlin: Springer (ISBN 978-3-642-17412-4/pbk; 978-3-642-17413-1/ebook). Lecture Notes in Mathematics 2013, 183-193 (2011).
Using the simple conformal map \[ z(\zeta)=\frac{\omega^2(\zeta,-1)+\omega^2(\zeta,1)}{\omega^2(\zeta,-1)-\omega^2(\zeta,1)}, \] where \(\omega(\zeta,\cdot)\) is the Schottky-Klein prime function associated with the relevant uniformizing groups, the authors present a numerical Schottky uniformization (of Fuchsian type) of a real hyperelliptic \(M\)-curve \[ y^2 =\prod_{j=1}^{2g+2} (x - e_j), \tag{1} \] where the constants \(e_j\), \(j=1,\dots,2g+2\), are distinct real numbers.
The Schottky-Klein prime function is used to construct an explicit conformal slit mapping from a fundamental region associated with a classical Schottky group to the region exterior to the slits (or branch cuts) of the curve (1). Illustrative examples for \(g=3\) and \(g=4\) are given.
The consideration of the Schottky-Klein prime function is a novel ingredient in the present method. It is an important mathematical object associated with any compact Riemann surface, and it can facilitate the solution of other problems on conformal mappings and uniformization. For example, in terms of it, it is possible to derive a rather classical Schwarz-Cristoffel formula to give conformal mappings from multiply connected circular regions to arbitrary multiply connected polygonal regions.
For the entire collection see [Zbl 1207.14002].

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
14P15 Real-analytic and semi-analytic sets
30F10 Compact Riemann surfaces and uniformization

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