Berkovich spaces

Meets: W 13.15-15.00 in von Neumann 1.023.

Starts: 20.04.2016.


(pdf version)

The main goal of the semester is to introduce ourselves to Berkovich spaces and some of their arithmetic applications, including Mordellic bounds on curves. We'll ease ourselves into nonarchimedean analysis by first overviewing Tate's theory of rigid analytic spaces, and then move on to a detailed study of Berkovich spaces. Throughout we'll be looking to curves over valued fields and their analytifications as our primary source of examples, though we will discuss some higher dimensional geometry as the need arises.

I have as yet been able to find a good single source. There are some definitive but sometimes overwhelming references for rigid geometry, and usually the only available reference for Berkovich spaces are his papers. Bosch's book is probably the best introduction/reference for the general theory of rigid geometry and its connection to formal geometry, while Fresnel--van der Put is good for some in-depth geometric examples.

Here are some notes available online that may provide some more useful descriptions:

Lecture topics and notes

20.04.2016: Overview (Ben).
27.04.2016: Tate algebra and affinoids (Ugur).
04.05.2016: Rigid spaces and analytification (Gregor). Notes.
11.05.2016: Uniformization of elliptic curves and the Tate curve (Irene).
18.05.2016: Formal models and Raynaud's theorem (Ben).
25.05.2016: Berkovich spaces I (Eva).
01.06.2016: Berkovich spaces II (Emre).
08.06.2016: ---Kramer birthday conference---
15.06.2016: Skeleta of Berkovich curves (Tanya).
22.06.2016: ---Poster Session---.
06.07.2016: p-adic integration (Antereep).
13.07.2016: Chabauty's method (Wouter).