As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)$ (for example $a=b=\sqrt{2},F=\mathbb{Q}[\sqrt{2}]$), while in many papers, compact arithmetic surfaces are restricted to the case $a,b\in\mathbb{Z},F=\mathbb{Q}$.
Hecke operators always play important roles: for $m\in\mathbb{N}^+$, put $R(m)=\{\alpha\in \mathbb{H}^{a,b}_\mathbb{Z}\mid \alpha\overline{\alpha}=m\}$, then $R(1)\backslash R(m)$ is finite. Since $R(m)$ acts naturally on the upper half plane $\mathbb{H}$, we have the Hecke operator $T_m\colon L^2(R(1)\backslash\mathbb{H})\to L^2(R(1)\backslash\mathbb{H})$ given by \begin{equation} T_mf(z)=\sum_{\alpha\in R(1)\backslash R(m)}f(\alpha z). \end{equation} Could we define Hecke operators analogously by replacing $\mathbb{H}^{a,b}_\mathbb{Z},m\in\mathbb{N}^+,R(m)$ with $\mathbb{H}^{\sqrt{2},\sqrt{2}}_{\mathbb{Z}[\sqrt{2}]},m\in\mathbb{Z}[\sqrt{2}]^+,R(m)=\{\alpha\in \mathbb{H}^{\sqrt{2},\sqrt{2}}_{\mathbb{Z}[\sqrt{2}]}\mid \alpha\overline{\alpha}=m\}$ respectively?
