P-adic variation in arithmetic geometry: a survey

Abstract

The main goal of this survey is to provide a general overview of the theme of p-adic variation, both from a historical and technical view point. We start off with Kummer’s work and Iwasawa’s treatment of cyclotomic fields, which eventually paved the way to the modern p-adic variational techniques. These methods have proved extremely powerful and enabled us to gain access to some of the most important problems in mathematics, such as the Bloch-Kato conjectures and Langlands’ Programme. We will point at a variety of concrete applications in this vein.