# Spring School on: Non-archimedean Geometry and eigenvarieties

## March 6, 2023 – March 17, 2023

Families of *p*-adic automorphic forms are well studied objects of arithmetic geometry since the pioneering work of Hida and Coleman. Their study resulted in the definition of geometric objects, called eigenvarieties, that parametrize systems of Hecke eigenvalues of *p*-adic automorphic forms. Conversely, the rich geometry of these varieties gives insights about *p*-adic (and thereby also about classical) automorphic forms. Recent techniques from perfectoid geometry, locally analytic representation theory and the point of view of the *p*-adic Langlands program give new insights and impulses.

The spring school will give an introduction to both *p*-adic automorphic forms and eigenvarieties as well as the necessary background in *p*-adic analytic geometry. The courses will be complemented by research talks that will focus on recent developments in the area.

The first week of the spring school will focus on *p*-adic analytic geometry, the analogue of complex analytic geometry over *p*-adic base fields. We will study classical rigid analytic spaces from the point of view of adic spaces and introduce perfectoid spaces. The second week will focus on *p*-adic automorphic forms and eigenvarieties. We will introduce and compare several approaches to *p*-adic automorphic forms.