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Abstract

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.