Rigid Analytic Methods

Abstract

The solution of the inverse problem of Galois theory over the field ℂ(t) was achieved by a blend of topological and analytical methods. A similar approach is possible for any ground field complete with respect to a non-archimedean valuation. The suitable analytic structures are provided by the so-called rigid analytic spaces. They satisfy a GAGA-principle, which makes it possible to recover algebraic structures from analytic constructions. This replaces the Riemann Existence Theorem in the complex case. In the first paragraph we collect some definitions and results on rigid analytic geometry and sketch a proof of the GAGA-principle for covers of the rigid analytic projective line.