CM points on Shimura curves via QM-equivariant isogeny volcanoes

TL;DR

This paper investigates CM points on Shimura curves with quaternionic multiplication, using isogeny-volcano techniques to classify and count such points, and explores the existence of sporadic CM points.

Contribution

It introduces an isogeny-volcano approach for describing CM points on Shimura curves, extending previous methods from modular curves to more general cases.

Findings

Counted all CM points with a specified order on these curves.

Determined primitive residue fields and degrees of CM points.

Provided evidence for sporadic CM points on Shimura curves.

Abstract

We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the $D=1$ case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on $X_0^D(N)_{/\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$.