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

Piermarco Milione

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arXiv · 1703.01133

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

Piermarco Milione

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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 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.

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Piermarco Milione. 2017-03-03. CM points on Shimura curves and $p$-adic binary quadratic forms. https://arxiv.org/abs/1703.01133

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