论文标题
CM通过QM-量管式的降低火山上的Shimura曲线点
CM points on Shimura curves via QM-equivariant isogeny volcanoes
论文作者
论文摘要
我们在shimura曲线上研究CM点$ x_0^d（n）_ {/\ Mathbb {q}} $和$ x_1^d（n）_ {/\ Mathbb {q}} $，参数化具有quaternionic乘法和额外级别结构的abelian abelian表面。通过类似于Clark和Clark的工作，在$ d = 1 $的模块化曲线情况下，通过类似于Clark和Clark的作品来获得一般级别的CM点的描述。这允许在此曲线上按指定顺序进行所有点的计数，并确定所有原始残基字段和此类点的原始程度在$ x_0^d（n）_ {/\ mathbb {q}} $上。我们利用$ x_0^d（n）_ {/\ mathbb {q}} $上的零星CM点的计算来实现零星CM点的存在。
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 $X_0^D(N)_{/\mathbb{Q}}$.
