Thanks a lot coudy for the nice answer and Beardon's reference.

Thanks @YCor for the clarification. So what I thought was indeed wrong.

Hi @Anton, if your triangle group has three finite angles, then it seems to me that it will be co-compact. Therefore, this means that you need to choose one angle which has to be infinite. But I thought that all such triangle groups would be commensurable to a conjugate of $PSL_2(Z)$, so may be I was wrong.

Hi @YCor. I see, so you are appealing here to some kind of uniformization result for non-compact Riemann surfaces. In practice is it possible to give an explicit example of such a group $\Gamma$, by explicit I mean: explicit generators where the entries of each matrix can be viewed as the zeros of some "reasonable function" (e.g. hyper-geometric functions, or some function which satisfies a simple differential equation) ?

Hi @Andreas, sure, I mean commensurable. I'll reedit it.

There should still exist some kind of uniform proof involving the theory of formal groups...

In order to treat the case where $p O_K=\mathfrak{p}\bar{\mathfrak{p}}$ one may use the following ad-hoc observation: The map $E----> E/E[\mathfrak{p}] \pmod{\wp}$ is the Frobenius post composed with an isomorphism with kernel $E[\mathfrak{p}]$. S2 follows from that.