The group $SL_2(\mathbb{Z})$ act on the projective spaces $P(\mathbb{Z}_p)$ and the upper half of the complex plane $\mathbb{H}$ by linear fractional transformations. I am wondering whether there is a direct connection between these actions.
For example: Is there a natural projection $\mathbb{H}\rightarrow P(\mathbb{Z}_p)$ or an embedding $P(\mathbb{Z}_p)\rightarrow\mathbb{H}$ which agrees with the action? Also, the points of $P(\mathbb{Z}_p)$ could be placed as the vertices of a $p$-gon plus a point at infinity; is there a connection with the disk model of $\mathbb{H}$ so that the natural action on the disk induces the action on $P(\mathbb{Z}_p)$?
The intermediate is the projective line over $\mathbb{Q}$. There is a natural projection from the projective line over $\mathbb{Q}$ onto $P(\mathbb{Z}_p)$, and also the projective line over $\mathbb{Q}$ is naturally embedded into the boundary of $\mathbb{H}$.