Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$? I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular functions, and modular forms resp. I have also seen the importance of $D:=\{z\in \mathbb{H} : -\frac 12 \leq \Re(z) \leq \frac 12 , |z|\geq 1 \}$, where $\mathbb{H}$ is the upper half plane, and the proof that $\gamma \in\Gamma$ sends $z\in\mathbb{H}$ to $\gamma z \in D$, and also that the points on the boundary of $D$ are the unique points where $\gamma z=z$, so that there are three points at which the stablilizer subgroup of $SL_2(\mathbb{Z})$ is not the identity matrix.
My question is regarding the theorem that seems to follow on from these observations.
We are given a function, $f$, that is meromorphic at $p$, and we define $v_p(f)$ to be  $k$, where $f(z)=(z-p)^k+\sum_{n=k+1}^\infty a_n(z-p)^n$. We also define $e_p$ to be $\frac{\#Stab(p)\subset SL_2(\mathbb{Z})}{2}$, which can take values of $1,2$ or $3$.
Finally, if $f$ is weakly modular of weight $k$, define $v_{\infty}(f):=k$ s.t. $f=a_kq^k+\sum_{n=k+1}^\infty a_nq^n, a_k\neq 0$
The theorem states:
$$v_{\infty}(f)+\sum_{p\:\in \:\mathbb{H}\:/\:SL_2(\mathbb{Z})}  \frac{v_p(f)}{e_p}=\frac{k}{12}$$
My main problem is that I don't understand the definition of $\mathbb{H}/SL_2(\mathbb{Z})$. Is it the set given by the action of $SL_2(\mathbb{Z})$ on $\mathbb{H}$? Is it merely all the points at which $f$ is meromorphic? 
I ask because it is used repeatedly later on and although I have a sense of what it is I don't quite get it.
 A: Typically, it's actually written $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ since we usually write the group action on the left. I will do so here.
Then $\text{SL}_2(\mathbb Z) \backslash \mathbb{H}$ is the set of elements in $\mathbb{H}$ quotiented out be the relation that $z \sim \gamma z$ for all $\gamma \in \text{SL}_2(\mathbb Z)$. Some more naturally identify this with the "cannonical" fundamental domain
$$ \mathcal{D} = \left\{ z | -\frac{1}{2} \leq \text{Re} z \leq \frac{1}{2}, \lvert z \rvert \geq 1 \right\}. $$
There is a more sophisticated point of view that makes this a bit better, too. We can identify $\mathbb{H}$ with the matrix $\begin{pmatrix} y&x \\ 0&1 \end{pmatrix}$ appearing in the Iwasawa decomposition of matrices in $\text{GL}_2(\mathbb{R})$. That is, every matrix $g \in \text{GL}_2(\mathbb{R})$ can be written uniquely as
$$
g = \begin{pmatrix} y&x \\ 0&1 \end{pmatrix} \Theta R
$$
where $\Theta \in \text{O}_2(\mathbb{R})$ is a (uniquely determined) orthogonal matrix and $R = \begin{pmatrix} r&0 \\ 0&r \end{pmatrix}$ is a diagonal matrix with $r \neq 0$.
In this view,
$$ \mathbb{H} \simeq \text{GL}_2(\mathbb{R}) /( \text{O}_2(\mathbb{R}) \cdot\mathbb{R}^\times). $$
We may now quotient on the left by $\text{SL}_2(\mathbb{Z})$ as an actual group quotient. It is remarkable (and not immediately obvious) that this quotient is the exact same as identification under the standard action, but it is.
