Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal O}a^{-1}$ with $a \in {\mathrm GL}_2({\Bbb R})$. Then it is seen that both ${\cal O}[\tau,1]^t$ and ${\cal O}'[\tau,1]^t$ are lattices in ${\Bbb C}^2$ of ${\Bbb Z}$-rank $4$.

## Question: How can one make a natural isomorphism ${\Bbb C}^2/{\cal O}[\tau,1]^t \cong {\Bbb C}^2/{\cal O}'[\tau,1]^t$ ?

This is concerned on the moduli theoretic viewpoint of Shimura curve $C$ associated quaternion algerba $D$ and its maximal order ${\cal O}$. The above question arose to make up the ambiguity of ${\cal O}$ up to inner automorphisms od $D$.