Rigid Analysis

Abstract

Rigid Analysis is the p-adic analogue of the classical complex geometry. After Hensel discovered the p-adic numbers in 1893, attempts were made to formulate a theory of analytic functions over Q_p . Initially, the question of interest had been to find out if there existed an analog of the theory of classical functions over the field of complex numbers. But then as Algebraic Geometry developed and was applied to number theory, there was a need for a good theory of analytic functions. Modern non-Archimedean geometry was born in 1961 when J. Tate, motivated by the question of characterising elliptic curves with bad reduction, gave a seminar at Harvard with the title "Rigid Analytic Spaces". The theory was subsequently further developed by Kiehl, Remmert, Grauert, Gerritzen, among others. It was apparent from the beginning that rigid geometry was much closer to algebraic geometry than to complex analysis. This algebro-geometric view was worked upon by Raynaud. In this thesis, we give an exposition to Rigid Geometry (in the first five chapters), and then introduce the theory of Formal Geometry.

In the last chapter, we introduce the Ramification Theory of Local Fields. In particular, we introduce the so-called APF extensions and give a characterization of the strictly APF extensions.