CM points on Shimura curves via QM-equivariant isogeny volcanoes
We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parameterizing abelian surfaces with quaternionic multiplication and extra level structure. In particular, we show that an isogeny-volcano approach to CM points on these curves, in analogy to recent work of Clark and Clark--Saia in the $D=1$ case of elliptic modular curves $X_0(M,N)_{/\mathbb{Q}}$ and $X_1(M,N)_{/\mathbb{Q}}$, is possible by considering CM components of QM-equivariant isogeny graphs over $\overline{\mathbb{Q}}$. This approach provides an algorithmic description of the CM locus on $X_0^D(N)_{/\mathbb{Q}}$ for $D$ a rational quaternion discriminant and $\text{gcd}(D,N) = 1$, providing for any imaginary quadratic order $\mathfrak{o}$ a count of all $\mathfrak{o}$-CM points on $X_0^D(N)_{/\mathbb{Q}}$ with each possible residue field. This allows for a determination of all primitive residue fields and primitive degrees of $\mathfrak{o}$-CM points on $X_0^D(N)_{/\mathbb{Q}}$, and in particular allows for a computation of the least degree of a CM point on $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, ranging over all orders. As an application, we leverage computations of these least degrees towards determining the existence of sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$.