Non-Archimedean Geometry and Eigenvarieties

This book offers an introduction to the theory of adic spaces, with applications to the geometry of automorphic forms. It is comprised of expanded lecture notes for six mini-courses delivered by the contributing authors at the Spring School on “Non-Archimedean Geometry and Eigenvarieties”, held in March 2023 at Heidelberg University.

The book is divided into two parts. The first part provides a focused, comprehensive and mostly self-contained introduction to the theory of adic spaces, with chapters contributed by John Bergdall, Katharina Hübner, and Christian Johansson. The second part demonstrates the theory through two key applications; perfectoid spaces, explained in a chapter by Ben Heuer, and eigenvarieties, covered in chapters by Judith Ludwig and James Newton.

Designed for researchers and students with some background in algebraic geometry, this book serves as an accessible entry point into the theory. Numerous examples, illustrations, and carefully designed exercises are included throughout to help readers understand the concepts and build intuition before moving on to more advanced literature.