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arXiv · 1707.03644

Mumford curves and Mumford groups in positive characteristic

Harm H. Voskuil·Marius van der Put
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Abstract

A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.

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Harm H. Voskuil, Marius van der Put. 2017-07-12. Mumford curves and Mumford groups in positive characteristic. https://arxiv.org/abs/1707.03644

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