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Discrete groups, Mumford curves and theta functions. (English) Zbl 0789.14020
The author studies discrete subgroups of \(\text{PGL} (2,K)\) where \(K\) is a complete non-archimedean valued field. With such a subgroup \(\Gamma\) one can associate a rigid analytic space \(\Omega \subset \mathbb{P}^ 1_ K\) on which \(\Gamma\) acts.
One of the main results of the paper is a classification of the quotient spaces \(X=\Omega/ \Gamma\) corresponding to the possible cases for \(\Gamma\). – Other results about automorphic functions on \(\Omega\) (with respect to \(\Gamma)\) are generalizations of earlier results by the same author.
A number of examples is calculated. In these examples the author gives a connection between discrete subgroups of quaternions and certain Shimura curves.

MSC:
14G20 Local ground fields in algebraic geometry
14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
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