Shimura curve computations
Noam D. Elkies
Abstract
We give some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them. We then illustrate these methods by working out several examples in varying degrees of detail. For instance, we compute coordinates for all the rational CM points on the curves "X(1)" associated with the quaternion algebras over Q ramified at 2,3, 2,5, 2,7, and 3,5. We conclude with a list of open questions that may point the way to further computational investigation of these curves.
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