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Uniformization of triangle modular curves. (English) Zbl 1185.11028

A Fuchsian group \(\Gamma\) of signature \((0; e_1,e_2,e_3),\) where \(e_1,e_2,e_3\) are positive integers or infinity, satisfying the inequalities \(e_1 \geq e_2 \geq e_3\) and \[ \frac{1}{e_1}+\frac{1}{e_2}+\frac{1}{e_3}<1, \] is called a triangle Fuchsian group of type \((e_1,e_2,e_3)\).
The authors restrict to the case of modular triangle groups and they present a new and unified computation of the uniformizing functions for any modular triangle group not using the classical representations via Dedekind’s eta function, but following some other ideas of R. Dedekind [J. Reine Angew. Math. 83, 265–292 (1877; JFM 09.0353.03)]. In order to achieve their purpose they need to find fundamental domains and presentations of the modular triangle groups, differential equations satisfied by the uniformizing functions and local uniformizing parameters.
The presentation has the advantage that the uniformizing functions are obtained without any previous knowledge of other special functions. The method has also been applied, by the scientific group of Bayer, to Fermat and Shimura curves.

MSC:

11F03 Modular and automorphic functions
11F06 Structure of modular groups and generalizations; arithmetic groups
11F30 Fourier coefficients of automorphic forms
11G18 Arithmetic aspects of modular and Shimura varieties
11G35 Varieties over global fields

Citations:

JFM 09.0353.03
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