Galois Representations and (Phi, Gamma)-Modules

A detailed and self-contained introduction to a key part of local number theory, ideal for graduate students and researchers.

Peter Schneider (Author)

9781107188587, Cambridge University Press

Hardback, published 20 April 2017

156 pages, 20 exercises
23.5 x 15.8 x 1.4 cm, 0.36 kg

'Much of this material is available in the literature, but it has never been presented so cleanly and concisely in a single place before … In this book, Schneider has done a remarkable job of displaying the beauty and power of perfectoid theoretic techniques. His text is sure to occupy and satisfy the attention of students and researchers working on Galois representations, or those who suspect that perfectoid-style techniques might be relevant for their work. We recommend Schneider's text to anyone with even a passing interest in the perfectoid revolution initiated by Scholze.' Cameron Franc, Mathematical Reviews

Understanding Galois representations is one of the central goals of number theory. Around 1990, Fontaine devised a strategy to compare such p-adic Galois representations to seemingly much simpler objects of (semi)linear algebra, the so-called etale (phi, gamma)-modules. This book is the first to provide a detailed and self-contained introduction to this theory. The close connection between the absolute Galois groups of local number fields and local function fields in positive characteristic is established using the recent theory of perfectoid fields and the tilting correspondence. The author works in the general framework of Lubin–Tate extensions of local number fields, and provides an introduction to Lubin–Tate formal groups and to the formalism of ramified Witt vectors. This book will allow graduate students to acquire the necessary basis for solving a research problem in this area, while also offering researchers many of the basic results in one convenient location.

Preface
Overview
1. Relevant constructions
2. (?L, ?L-modules)
3. An equivalence of categories
4. Further topics
References
Notation
Subject index.

Subject Areas: Number theory [PBH], Algebra [PBF]