Suppose that we have a quaternion algebra $D$ over a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We assume that $D$ splits everywhere at finite places of $K$ and at one infinite place of $K$. (Hence, $D$ is ramified at $[K \colon {\Bbb Q}] -1$ infinite places.) Suppose we have an Eichler ordre ${\cal O}_N$ of level $N$ in $D$. ${\cal O}_N$ is defined as the intersection of two maximal ordres of $D$ and level $N$ is defined as the index of ${\cal O}_N$ in any maximal ordre which contains ${\cal O}_N$. We shall fix one isomorphism $\phi \colon D \otimes_{\Bbb Q} {\Bbb R} \cong M_2({\Bbb R})$. Take ${\cal O}_N^{(1)} \subset {\cal O}_N$ which consists of norm $1$ elements.
Q. Is Shimura curve $C \colon = {\Bbb H}/{\cal O}_N^{(1)*}$ well-defined? That is, ${\cal O}_N$ is defined uniquely up to conjugate in $D$? I.e., type number of ${\cal O}_N$ is one.
By the way, I have learned that $C$ is actually well-defined as follows: Shimura curve over ${\Bbb C}$ is just a quotient space $C \colon= D^* \backslash D_A^*/\widehat{{\cal O}_N}^*F_{\infty}^*C$ for a maximal compact subgroup $C$. So it is a union of $a \widehat{{\cal O}_N} a^{-1} \cap D^* \backslash {\Bbb H}$ for a running over a complete representative set of $D^* \backslash D_A^*/{\widehat{R}}^*D_{\infty}^*$. That is, all adelic conjugacy classes of ${\cal O}_N$ appears once.
I cannot very well understand this explanation, although I know it is correct. My biggest question is that when one speaks of uniformisation, one must once and for all choose arithmetic Fuchsian group up to conjugacy. Hence the uniquness of Eichler ordre ${\cal O}_N$ in $D$ must be ensured somehow in ordre to give the precise definition of Shimura curve for our quaternion algebra.