I began reading about modular forms and I had a question. So, I know that $SL(2,\mathbb{Z})$ is mapped to the upper half complex plane using a function. The way I see it, the reason we map it to the upper plane and not the lower plane is because of the following. Assume we have a matrix $g=\begin{pmatrix} a & b \\ c & d\\ \end{pmatrix}\in SL(2,R)$ and a point $z\in \mathbb{C}$ then
$$\text{Im } gz=\text{Im }\frac{az+b}{cz+d}=\frac{ad-bc(\text{Im }z)}{|cz+d|^2}$$ Since the determinant of $g$ is 1 and the denominator is positive, then for a positive $\text{Im }z$ it must be the case that the points are translated to the upper plane which is positive. What happens if we had $g\in GL(2,R)$? when the determinant is -1, does it get translated to the lower half-plane?
FYI. I'm an undergrad so I'll probably have a hard time understanding a grad-level explanation.