On Congruence Relations and Equations of Shimura Curves

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.