Equations of hyperelliptic Shimura curves
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We describe a method for computing equations of hyperelliptic Shimura curves attached to indefinite quaternion algebras over Q and Atkin–Lehner quotients of them. It exploits Cerednik–Drinfeld 's non-archimedean uniformization of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points, as introduced in Molina [‘Ribet bimodules and specialization of Heegner points’, Israel Journal of Mathematics]. We provide a list of equations of Shimura curves and quotients of them obtained by our method that had been conjectured by Kurihara
CitationMolina, S. Equations of hyperelliptic Shimura curves. "Proceedings of the London Mathematical Society", 2012, vol. 3, núm. 105, p. 891-920.