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Shimura curve computations. (English) Zbl 1010.11030

Buhler, J. P. (ed.), Algorithmic number theory. 3rd international symposium, ANTS-III, Portland, OR, USA, June 21-25, 1998. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1423, 1-47 (1998).
Summary: The author gives some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. He illustrate these methods by working out several examples in varying degrees of detail. For instance, he computes coordinates for all the rational CM points on the curves \(\chi^*(1)\) associated with the quaternion algebras over \(\mathbb{Q}\) ramified at \(\{2,3\}\), \(\{2,5\}\), \(\{2,7\}\) and \(\{3,5\}\). He concludes with a list of open questions that may point the way to further computational investigation of these curves.
For the entire collection see [Zbl 0891.00022].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11Y35 Analytic computations
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