Uniformization of algebraic curves by discrete arithmetic subgroups of $PGL_2(k_w)$ with compact quotients

We consider curves having either a uniformization by the upper half-plane or a Mumford uniformization by discrete arithmetic subgroups of $PGL_2(k_w)$ corresponding to quaternion algebras with center $k$, with $k$ a global field of (possibly nonzero) characteristic $p$, $k$ being totally real if $p = 0$; $k_w$ is the completion of $k$ with respect to a valuation $w$ which is real or non-Archimedean. The principal result is a theorem that in characteristic $p = 0$ the curves corresponding to algebras related in a certain sense coincide.
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