Upper half-plane

Remarks

The action is well defined because $$\operatorname{Im}\left(\frac{a\tau + b}{c\tau + d}\right) = \frac{(ad - bc) \operatorname{Im} \tau}{\abs{c\tau + d}^2},$$ which is positive when $ad - bc = 1$ and $\operatorname{Im} \tau > 0$. The holomorphic bijections of $\mathbb{H}$ onto itself are exactly these Möbius transformations, and they form the group $\mathrm{PSL}_2(\R)$. The map $\tau \mapsto (\tau - i)/(\tau + i)$ is a biholomorphic map of $\mathbb{H}$ onto the open unit disc. With the metric $ds^2 = (dx^2 + dy^2)/y^2$, where $\tau = x + iy$, the upper half-plane is a model of the hyperbolic plane, and $\mathrm{PSL}_2(\R)$ acts by isometries. Modular forms are functions on $\mathbb{H}$.