Abstract
On a Shimura curve, the reduction modulo a prime $p$ of the Hecke correspondence $T(p)$ yields the congruence relation $\Pi\cup\Pi'$ with $\Pi$ being the graph of the Frobenius mapping from the Shimura curve modulo $p$ to itself, and $\Pi'$ its transpose. Starting with a curve $C$ of genus $g \geq 2$ over $\mathbb{F}_p$ together with a subset $\mathfrak{S}\subset C(\mathbb{F}_{p^2})$, Ihara studied the liftability to characteristic $0$ of $\Pi\cup\Pi'$ so that $\Pi$ and $\Pi'$ are separated outside $\mathfrak{S}$ in the lifting. In some case, Ihara obtained the uniqueness of the liftability to characteristic $0$ and gave some necessary and sufficient condition, described by some differential form on $C$, for $(C,\mathfrak{S})$ to be liftable to modulo $p^2$. In this paper, in case when $C$ is defined over $\mathbb{F}_{p^2}$, we compute complete tables of such $(C,{\mathfrak S})$ liftable to modulo $p^2$ for $g=2$ and $3\leq p \leq 13$ using computer, and as an application of this uniqueness, we identify some particular Shimura curve by its equation.
Citation
Akira KURIHARA. "On Congruence Relations and Equations of Shimura Curves." Tokyo J. Math. 42 (2) 525 - 550, December 2019. https://doi.org/10.3836/tjm/1502179308