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. Let H6 be the rational quaternion algebra of discriminant D = 6, which we consider embedded into M(2, R). Let Γ6 ⊆ SL(2, R) be the group of units of norm 1 in a maximal order O6 ⊆ H6 of discriminant D = 6. The quotient group Γ 6 = Γ6/{±1} ⊆ PSL(2, R) is a co-compact Fuchsian group whose action on the upper half-plane H gives rise to a nonsingular complete curve X6 of genus zero. According to Shimura, the curve X6 has a canonical model defined over Q. Our purpose is to compute a complex uniformizing function t6 of X6 relying on the fundamental domain for Γ 6 obtained in [1], and to derive from it a canonical model, j6. The function j6 is an analog of the elliptic modular function j, which is obtained from the split quaternion algebra M(2, Q), of discriminant D = 1. However, the functions j and j6 present notable differences. The function j is automorphic under the modular group PSL(2, Z), which is a triangle Fuchsian group endowed with parabolic transformations. The function j6 is automorphic for a quadrilateral Fuchsian group without parabolic transformations, Γ 6 ⊆ PSL(2, R). The lack of cusps in this case prevents the use of Fourier series expansions.

Uniformizing functions for certain Shimura curves, in the case D=6