How can we identify the modular curve $X_1(N)$ with $\textrm{GL}_2(\mathbb{Q}) \setminus \textrm{GL}_2(\mathbb{A})/K_{\mathbb{R}}K_f$?

Question

Let $N$ be an integer, and let $\Gamma$ be the principal congruence subgroup of level $N$:

$$\Gamma = \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \textrm{SL}_2(\mathbb{Z}) : a \equiv d \equiv 1 \pmod N; b \equiv c \equiv 0 \pmod{N}\}$$

Then $\Gamma$ acts on the upper half plane $\mathcal H = \{ z \in \mathbb{C} : \textrm{Re}(z) > 0 \}$. The modular curve $X_1(N)$ is the space of orbits $\Gamma \setminus \mathcal H$.

The notes I'm reading (https://www.math.ubc.ca/~cass/research/pdf/miyake.pdf) talk about a "natural embedding"

$X_1(N) \rightarrow \textrm{GL}_2(\mathbb{Q}) \setminus \textrm{GL}_2(\mathbb{A})/K_{\mathbb{R}}K_f$

What is this natural embedding? By strong approximation, it is supposed to be a bijection. What else is it besides a bijection? A homeomorphism? An isomorphism of $\Gamma$-sets?

Answer 1

Start with a modular form $f \in M_k(\Gamma)$, for simplicity $ \Gamma = SL_2(\mathbb{Z})$ and set $$F(\gamma) = |\det(\gamma)|^{k/2} (ci+d)^{-k} f(\gamma \cdot i)$$ where $\gamma = {\scriptstyle\begin{pmatrix} a & b \\ c & d \end{pmatrix}}) \in GL_2(\mathbb{Q}), \gamma\cdot i =\frac{ai+b}{ci+d}$.

Thus $F$ is a function $GL_2(\mathbb{Q}) \to \mathbb{C}$ which is : left $\Gamma$ invariant and left-right $GL_1(\mathbb{Q})$ invariant.

Then show that $\lim_{n \to \infty} F(\gamma_n)$ is well-defined when $\gamma_n$ converges in $GL_2(\mathbb{Q}_v)$. For $v= \infty$ it is obvious. Also for $g ={\scriptstyle\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}}\in SO_2(\mathbb{R})$ then $g \cdot i = i$ thus $F(\gamma g) = e^{-i k \theta}F(\gamma)$. For $v = p$ use that $F$ is left $SL_2(\mathbb{Z}_p)$ invariant.

Thus $F$ is now a function on each $GL_2(\mathbb{Q}_v)$ which is : left $SL_2(\mathcal{O}_v)$ invariant (for $v = p, \mathcal{O}_v = \mathbb{Z}_p$, for $v = \infty, \mathcal{O}_v = \mathbb{Z}$) and left-right $GL_1(\mathbb{Q}_v)$ invariant and transforms as $F(\gamma g) = e^{-i k \theta}F(\gamma)$ for $g \in SO_2$.

With the correct topology on $GL_2(\mathbb{A}_\mathbb{Q})$, $F$ is now a function on it.

Say a points of $X_1(N)_\mathbb{Q}$ are of the form $\Gamma_1(N) \gamma SO_2(\mathbb{Q})\cdot i$ for some $ \gamma \in GL_2(\mathbb{Q})$, it maps to $A_\mathbb{Q}\Gamma_1(N) \gamma SO_2(\mathbb{A}_\mathbb{Q})\cdot i$ thus to $A_\mathbb{Q}\Gamma_1(N) \gamma SO_2(\mathbb{A}_\mathbb{Q})$.

For $\Gamma= \Gamma_1(N)$ a congruence subgroup, it works the same way except for the few $p| N$.

Or something like that.