Shimura curve (awaiting review)

Let $(O,\mu)$ be a polarized quaternion order in an indefinite division quaternion algebra over $\Q$ of discriminant $D$. Let $\mathcal{H}^\pm \colonequals \C \smallsetminus \R$ be the union of the complex upper and lower half planes. Since $B$ is indefinite, we may fix an embedding $B^\times \hookrightarrow \GL_2(\R)$, and so $B^\times$ acts on $\mathcal{H}^\pm$ by Möbius transformations.

We define \[ \widehat{B} \colonequals B\otimes_{\Z} \widehat{\Z}, \] \[ \widehat{O} \colonequals O\otimes_{\Z} \widehat{\Z}. \] Then $\widehat{O}^\times$ is an open compact subgroup of $\widehat{B}^\times$. Let $N$ be a positive integer. We define \[ \Gamma(\widehat{O},N) \colonequals \ker(\widehat{O}^\times \to \modstar{O}{NO}). \] We form the double coset space \[ X(O;N)^{\text{big}}(\C) \colonequals B^\times\backslash (\mathcal{H}^\pm \times \widehat{B}^\times)/\Gamma(\widehat{O},N). \] Here, the group actions are as follows: for $\beta \in B^\times$, $\tau \in \mathcal{H}^\pm$, $\widehat{b} \in \widehat{B}^\times$, and $\widehat{\gamma} \in \Gamma(\widehat{O},N)$, we define \[ \beta(\tau,\widehat{b})\widehat{\gamma} = (\beta\cdot \tau,\beta\widehat{b}\widehat{\gamma}). \]

By work of Shimura and Deligne, $X(O;N)^{\text{big}}(\C)$ is canonically the complex points of a (possibly geometrically disconnected) algebraic curve $X(O;N)^{\text{big}}$ over $\Q$, which we call the full level $N$ Shimura curve associated to $O$. For example, when $N = 1$, the level $1$ Shimura curve associated to $O$ can be identified with the quotient \[ X(O;1)^{\text{big}}(\C) = O^\times\backslash \mathcal{H}^\pm. \]

Let $H \le \Aut_{\pm\mu}(O)\ltimes \widehat{O}^\times$ be an open compact subgroup of the enhanced semidirect product of level $N$. Then $H$ acts on $X(O;N)^{\text{big}}$ by a finite group of automorphisms over $\Q$. We let \[ \mathfrak{X}_H \colonequals [H\backslash X(O;N)^{\text{big}}] \] be the corresponding stack quotient. The (quaternionic) Shimura curve of level $H$ is the associated coarse space $X_H$.

When $O$ is maximal, we write $X(D;1) \colonequals X(O;1)^{\text{big}}$. When $O$ is an Eichler order of level $N$, we write $X_0(D;N) \colonequals X(O;1)^{\text{big}}$.