Superspecial Points on Shimura Curves

Abstract

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 $\mathbb{F}_p$.