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

## TL;DR

This paper establishes a bijection between CM points on certain Shimura curves and classes of $p$-adic binary quadratic forms, extending classical results to the $p$-adic setting and providing explicit computations for families of such forms.

## Contribution

It introduces a novel correspondence between CM points on Shimura curves and classes of $p$-adic binary quadratic forms, extending prior results to the $p$-adic context.

## Key findings

- Bijection between CM points and $p$-adic quadratic form classes
- Explicit computation of $p$-adic quadratic forms for infinite families of Shimura curves
- Extension of classical results to the $p$-adic setting

## 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 obtain 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 in a previous paper of Amor\'os-Milione. This extends results of Alsina-Bayer to the $p$-adic context.