Patching and Galois Theory

Summary

Galois theory over ℂ (x) is well-understood as a consequence of Riemann's Existence Theorem, which classifies the algebraic branched covers of the complex projective line. The proof of that theorem uses analytic and topological methods, including the ability to construct covers locally and to patch them together on the overlaps. To study the Galois extensions of k(x) for other fields k, one would like to have an analog of Riemann's Existence Theorem for curves over k. Such a result remains out of reach, but partial results in this direction can be proved using patching methods that are analogous to complex patching, and which apply in more general contexts. One such method is formal patching, in which formal completions of schemes play the role of small open sets. Another such method is rigid patching, in which non-archimedean discs are used. Both methods yield the realization of arbitrary finite groups as Galois groups over k(x) for various classes of fields k, as well as more precise results concerning embedding problems and fundamental groups. This manuscript describes such patching methods and their relationships to the classical approach over C, and shows how these methods provide results about Galois groups and fundamental groups.

1. Introduction

This article discusses patching methods and their use in the study of Galois groups and fundamental groups. There is a particular focus on Riemann's Existence Theorem and the inverse Galois problem, and their generalizations (both known and conjectured). This first section provides an introduction, beginning with a brief overview of the topic in Section 1.1. More background about Galois groups and fundamental groups is provided in Section 1.2. Section 1.3 then discusses the overall structure of the paper, briefly indicating the content of each later section.