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

Abstract

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion 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 PGL 2 (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 in [2]. This extends results of Alsina-Bayer [1] to the p-adic context.