This website requires JavaScript.

# CM points on Shimura curves via QM-equivariant isogeny volcanoes

Dec 2022

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 quaternionicmultiplication and extra level structure. In particular, we show that anisogeny-volcano approach to CM points on these curves, in analogy to recentwork 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 byconsidering CM components of QM-equivariant isogeny graphs over$\overline{\mathbb{Q}}$. This approach provides an algorithmic description ofthe CM locus on $X_0^D(N)_{/\mathbb{Q}}$ for $D$ a rational quaterniondiscriminant and $\text{gcd}(D,N) = 1$, providing for any imaginary quadraticorder $\mathfrak{o}$ a count of all $\mathfrak{o}$-CM points on$X_0^D(N)_{/\mathbb{Q}}$ with each possible residue field. This allows for adetermination of all primitive residue fields and primitive degrees of$\mathfrak{o}$-CM points on $X_0^D(N)_{/\mathbb{Q}}$, and in particular allowsfor 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, weleverage computations of these least degrees towards determining the existenceof sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$.

Q1论文试图解决什么问题？
Q2这是否是一个新的问题？
Q3这篇文章要验证一个什么科学假设？
0