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CM points on Shimura curves and $p$-adic binary quadratic forms

Volume 183 / 2018

Piermarco Milione Acta Arithmetica 183 (2018), 237-256 MSC: Primary 11G18; Secondary 11E08, 11G15. DOI: 10.4064/aa170221-25-11 Published online: 21 May 2018


We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtained by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite family of Shimura curves studied by Amorós and Milione (2018). This extends some results of Alsina–Bayer (2004) to the $p$-adic context.


  • Piermarco MilioneDepartment of Mathematics and Systems Analysis
    School of Science
    Aalto University
    Espoo, Finland

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