Given an imaginary quadratic extension $K$ of $\mathbb Q$, we classify the maximal nonelementary subgroups of the Picard modular group $\operatorname{ PU}(1,2;\mathcal O_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. We prove that these maximal $\mathbb R$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $\Delta$ of the corresponding $\mathbb R$-circle lies in $\mathbb N-\{ 0\}$, then the stabilizer arises from the quaternion algebra $\Big(\!\begin{ array}{ c} \Delta\,,\, |D_K|\\\hline\mathbb Q\end{ array} \!\Big)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb C)$.
Categories: math.NT math.CV