5.33 (1976)

(Y. Ihara). Consider the quaternion algebra $Q$ with norm $f = x^2 - \tau y^2 - \rho z^2 + \rho \tau u^2$, $\rho, \tau \in \mathbb{Z}$. Assume that $f$ is indefinite and of $\mathbb{Q}$-rank 0, i. e. $f = 0$ for $x, y, z, u \in \mathbb{Q}$ implies $x = y = z = u = 0$. Consider $Q$ as the algebra of the matrices
$$X = \begin{pmatrix} x + \sqrt{\tau}y & \rho(z + \sqrt{\tau}u) \\ z - \sqrt{\tau}u & x - \sqrt{\tau}y \end{pmatrix}$$ with $x, y, z, u \in \mathbb{Q}$. Let $p$ be a prime, $p \nmid \rho \tau$. Consider the group $G$ of all $X$ with $x, y, z, u \in \mathbb{Z}^{(p)}$, $\operatorname{det} X = 1$, where $\mathbb{Z}^{(p)} = \{m/p^t \mid m, t \in \mathbb{Z}\}$.

Conjecture: $G$ has the congruence subgroup property, i. e. every non-central normal subgroup $N$ of $G$ contains a full congruence subgroup $N(\mathfrak{a}) = \{X \in G \mid X \equiv E \pmod{\mathfrak{a}}\}$ for some $\mathfrak{a}$. Notice that the congruence subgroup property is independent of the matrix representation of $Q$.