Computing CM Points on Shimura Curves Arising from Cocompact Arithmetic Triangle Groups

  • John Voight
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

Abstract

Let \(\Gamma \subset PSL_2({\mathbb R})\) be a cocompact arithmetic triangle group, i.e. a Fuchsian triangle group that arises from the unit group of a quaternion algebra over a totally real number field. The group Γ acts on the upper half-plane \({\mathfrak{H}}\); the quotient \(X_{\mathbb C}=\Gamma \backslash {\mathfrak{H}}\) is a Shimura curve, and there is a map \(j:X_{\mathbb C} \to {\mathbb P}^1_{\mathbb C}\). We algorithmically apply the Shimura reciprocity law to compute CM points \(j(z_D) \in {\mathbb P}^1_{\mathbb C}\) and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how this method works in practice.

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • John Voight
    • 1
  1. 1.Magma Group, School of Mathematics and StatisticsUniversity of SydneyAustralia

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