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Uniformizing functions for certain Shimura curves, in the case \(D=6\). (English) Zbl 1158.11031
Summary: According to Shimura, the curve \(X_6\) associated to the rational quaternion algebra of discriminant \(D = 6\) has a canonical model defined over \({\mathbb Q}\). In the present article, we determine series expansions for the components of a uniformizing function \(j_6\) of \(X_6\). The function \(j_6\) is an analog of the elliptic modular function \(j\), which corresponds to the split quaternion algebra \(\mathbf {M}(2,\mathbb Q)\), of discriminant \(D=1\). Nevertheless, the functions \(j\) and \(j_6\) present notable differences. The function \(j\) is automorphic under the modular group \(\text{PSL}(2, \mathbb Z)\), which is a triangle Fuchsian group endowed with parabolic transformations. A fundamental domain arises from a symmetric region obtained by reflecting the triangle with vertices \(i, \exp{(2\pi i)/3}, \infty\) in the imaginary axis. The function \(j_6\) is automorphic for a quadrilateral Fuchsian group without parabolic transformations, \(\overline{\varGamma}_6 \subseteq \text{PSL}(2, \mathbb R)\). A fundamental domain arises from a symmetric region obtained by reflecting a hyperbolic quadrilateral under a hyperbolic line. The lack of cusps in this case prevents the use of Fourier series expansions.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
33E30 Other functions coming from differential, difference and integral equations
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