CM points on Shimura curves via QM-equivariant isogeny volcanoes

We study CM points on the Shimura curves $X_0^D(N{)}_{\mathrm{/}\mathbb{Q}}$ and $X_1^D(N{)}_{\mathrm{/}\mathbb{Q}}$, 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 $X_0^D(N{)}_{\mathrm{/}\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D(N{)}_{\mathrm{/}\mathbb{Q}}$.