Geodesic
Uniformization of algebraic curves by discrete arithmetic subgroups of $PGL_2(k_w)$ with compact quotients
Sbornik. Mathematics, Tome 29 (1976) no. 1, pp. 55-78

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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. Bibliography: 10 titles. 
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     author = {I. V. Cherednik},
     title = {Uniformization of algebraic curves by discrete arithmetic subgroups of $PGL_2(k_w)$ with compact quotients},
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     volume = {29},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_29_1_a4/}
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I. V. Cherednik. Uniformization of algebraic curves by discrete arithmetic subgroups of $PGL_2(k_w)$ with compact quotients. Sbornik. Mathematics, Tome 29 (1976) no. 1, pp. 55-78. http://geodesic.mathdoc.fr/item/SM_1976_29_1_a4/