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Uniformization of certain Shimura curves. (English) Zbl 1036.11026

Crespo, Teresa (ed.) et al., Differential Galois theory. Proceedings of the workshop, Bȩdlewo, Poland, May 28–June 1, 2001. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 58, 13-26 (2002).
Certain Shimura curves occur as moduli spaces of principally polarized abelian surfaces endowed with quaternionic multiplication. In this paper the author presents an approach to the uniformization of such Shimura curves by using automorphic forms obtained by integration of nonlinear differential equations. This method involves a differential construction of the modular \(j\)-function as well as a differential operator of order three.
For the entire collection see [Zbl 1007.00030].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
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