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Etale coverings of a Mumford curve

Marius Van Der Put — 1983

Annales de l'institut Fourier

Let the field K be complete w.r.t. a non-archimedean valuation. Let X / K be a Mumford curve, i.e. the irreducible components of the stable reduction of X have genus 0. The abelian etale coverings of X are constructed using the analytic uniformization Ω X and the theta-functions on X . For a local field K one rediscovers G . Frey’s description of the maximal abelian unramified extension of the field of rational functions of X .

The class group of a one-dimensional affinoid space

Marius Van Der Put — 1980

Annales de l'institut Fourier

A curve X over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space Y is trivial if and only if Y is a subspace of P 1 . As a consequence, X has locally a trivial class group if and only if the stable reduction of X has only rational components.

Families of linear differential equations related to the second Painlevé equation

Marius van der Put — 2011

Banach Center Publications

This paper is a sequel to [vdP-Sa] and [vdP]. The two classes of differential modules (0,-,3/2) and (-,-,3), related to PII, are interpreted as fine moduli spaces. It is shown that these moduli spaces coincide with the Okamoto-Painlevé spaces for the given parameters. The geometry of the moduli spaces leads to a proof of the Painlevé property for PII in standard form and in the Flaschka-Newell form. The Bäcklund transformations, the rational solutions and the Riccati solutions for PII are derived...

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