I'd like to show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is a variety. Is it even true that $\mathbb{H} = \{ x + iy : y > 0 \}$ is an algebraic variety? There's no metric, so it's just the half of the affine plane $\mathbb{A}^2$.
Do we have that $\{ q \in \mathbb{Q} : q > 0\}$ is a variety? We have that $\{ xy = 1 \} \subset \mathbb{A}^2$ is a variety, and there's a group action such as $(x,y) \mapsto (-x,-y)$ preserving that curve.
Then we have the group action of $\text{SL}_2(\mathbb{Z})$ (which are integer-valued matrices) - is the quotient space of an algebraic variety another one?
I'd like to be able to say that $\big[\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}\big](\mathbb{Q})$ is a variety over $K = \mathbb{Q}$ and discuss the rational points.
Depending on which theorems we admit the problem is straightfoward. It's well-known that a modular curve is an algebraic curve.
We know this is an instance of a Shimura Variety. This is like assuming the result we want to prove. See also: Hilbert modular surfaces,