flip side, there are natural lifts $r:G\to M_{\mathbb{A}}$ and $r_P:P_{\mathbb{A}}\to M_{\mathbb{A}}$ through which we view $G$ and $P_{\mathbb{A}}$ as subgroups of $M_{\mathbb{A}}$.

For any two fractional ideals ${\mathfrak}{x}, {\mathfrak}{y}$ of $\mathbb{Q}$ such that ${\mathfrak}{xy}\subseteq\mathbb{Z}$, congruence subgroups are defined by the following respective subgroups of $G_p, G_{\mathbb{A}}$, and $G$: $$\begin{aligned}
    D_p[{\mathfrak}{x}, {\mathfrak}{y}] :&= \{x\in G_p\mid a_x, d_x\in M_n(\mathbb{Z}_p), b_x\in M_n({\mathfrak}{x}_p), c_x\in M_n({\mathfrak}{y}_p)\}, \\
    D[{\mathfrak}{x}, {\mathfrak}{y}] :&= Sp_n(\mathbb{R})\prod_p D_p[{\mathfrak}{x}, {\mathfrak}{y}], \\
    \Gamma[{\mathfrak}{x}, {\mathfrak}{y}] :&= G\cap D[{\mathfrak}{x}, {\mathfrak}{y}].\end{aligned}$$ Typically these will take the form $\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ for certain fractional ideals ${\mathfrak}{b}$ and integral ideals ${\mathfrak}{c}$.

One of the key differences in the theory of half-integral weight modular forms is in the congruence subgroups one considers. The factor of automorphy involved can only be defined for a certain subgroup ${\mathfrak}{M}\leq M_{\mathbb{A}}$, and any congruence subgroups $\Gamma$ must therefore be contained in ${\mathfrak}{M}$. This subgroup, ${\mathfrak}{M}$, is defined via the theta series and is given by $$\begin{aligned}
    C_p^{\theta} :&= \{\xi\in D_p[1, 1]\mid (a_{\xi}b_{\xi}^T)_{ii}\in 2\mathbb{Z}_p, (c_{\xi}d_{\xi}^T)_{ii}\in 2\mathbb{Z}_p, 1\leq i\leq n\}, \\
    C^{\theta} :&= Sp_n(\mathbb{R})\prod_p C_p^{\theta}, \\
    {\mathfrak}{M} :&= \{\sigma\in M_{\mathbb{A}}\mid {\mathop{\rm pr}\nolimits}(\sigma)\in P_{\mathbb{A}}C^{\theta}\}.\end{aligned}$$ Typically we shall take ${\mathfrak}{b}$ and ${\mathfrak}{c}$ such that $\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]\leq \Gamma[2, 2]\leq {\mathfrak}{M}$.

The spaces defined supra act on each other as 