Shimura曲线上的超特殊点
Superspecial Points on Shimura Curves
AI总结：
本文研究Shimura曲线模素数的约化，给出其超特殊点存在性判据，计算两类超特殊点数量，推广相关经典公式。
AI中文摘要：
设$X$是附属于不定四元数$\boldsymbol{Q}$-代数$B$且带有极大序$O_B$的Shimura曲线，本文研究$X$模任意素数$p$的约化$X\bigotimes \boldsymbol{F}_p$，尤其关注其超特殊轨迹。我们给出$X$上$\boldsymbol{F}_q$-有理超特殊点存在性的显式判据；此外，通过Eichler类数公式和Selberg迹公式，分别计算几何超特殊点的数量与$\boldsymbol{F}_p$-有理超特殊点的数量。作为关键要素，我们对附属于超特殊$O_B$-阿贝尔曲面的Dieudonné模进行分类，该分类推广了Ribet关于秩2可允许四元数双模的分类，具体做法是去掉可允许假设。这些结果推广了Deuring关于$\boldsymbol{F}_p$上 supersingular椭圆曲线的显式公式，并给出$\boldsymbol{F}_p$上主极化超特殊阿贝尔曲面的Ibukiyama-Katsura公式的Shimura曲线类似物。
英文摘要：
Let $X$ be the Shimura curve attached to an indefinite quaternion $\mathbb{Q}$-algebra $B$ with a maximal order $O_B$. This paper investigates the reduction $X\otimes \mathbb{F}_p$ of $X$ modulo an arbitrary prime $p$, focusing particularly on its superspecial locus. We give an explicit criterion for the existence of superspecial $\mathbb{F}_q$-rational points on $X$. Furthermore, we compute both the number of geometric superspecial points and the number of $\mathbb{F}_p$-rational superspecial points, through the Eichler class number formula and the Selberg trace formula. As a key ingredient, we classify the Dieudonné modules attached to superspecial $O_B$-abelian surfaces, which generalizes Ribet's classification of admissible quaternion bimodules of rank $2$ by dropping the admissible hypothesis. These results generalize Deuring's explicit formula for supersingular elliptic curves over $\mathbb{F}_p$ and give the Shimura-curve analogue of the Ibukiyama-Katsura formulas for principally polarized superspecial abelian surfaces over $\mathbb{F}_p$.
