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

2017/03/03 by Milione, Piermarco
#11E08 #11G15 #11G18 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1703.01133

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 obtain by the action of a discrete and cocompact subgroup of PGL₂(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 a previous paper of Amorós-Milione. This extends results of Alsina-Bayer to the p-adic context.

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