Spectral theory and the Eigenvariety machine

TL;DR

This paper provides an accessible introduction to the eigenvariety machine using adic spaces, combining non-archimedean functional analysis with modern geometric techniques for studying eigenvarieties.

Contribution

It offers a modern reinterpretation of the eigenvariety machine within the framework of adic spaces, expanding the theoretical toolkit for non-archimedean geometry.

Findings

Clarifies the construction of eigenvarieties using adic spaces

Integrates non-archimedean functional analysis into eigenvariety theory

Provides foundational insights for further research in p-adic geometry

Abstract

These are extended lecture notes for a mini course at the Spring School on Non-Archimedean Geometry and Eigenvarieties held at Heidelberg University in March 2023. The goal of the course is to explain a modern take on the eigenvariety machine in the language of adic spaces. For this we build on the theory developed in the first week of the school. We also explain some basic selective non-archimedean functional analysis.