## V5A4 Selected topics in algebraic geometry: The de Rham comparison for rigid-analytic varieties - summer term 2023

Contact:

Dr. Johannes Anschütz

Endenicher Allee 60 ยท Zimmer 4.027

**Tel.:** 0228-73-62216

** E-mail:** ja(add @math.uni-bonn.de)

## Time and place

Thursday, 10-12h, Zeichensaal

First lecture: Thursday, 6.4.2023

End of lecture period: 14.7.2023

No lecture on: 18.5., 1.6., 8.6., 13.7.

Discussion session: ** Monday, 14-16h, N0.008 (starting on 12.6.). For the exercise session on 5.6. no room is available. Hence, the exercise group meets in front of the Nebengebäude and improvises.**

## Content

The de Rham comparison theorem for proper, smooth schemes over p-adic fields yields an isomorphism of two different cohomological invariants, namely p-adic étale cohomology and de Rham cohomology. Thus, it implies a non-trivial condition on the Galois representations appearing in cohomology (they are "deRham"), which is crucial for the Fontaine-Mazur conjecture. A major difficulty in establishing the de Rham comparison is the absence of a (naive) version of the Poincaré lemma for rigid-analytic spaces. Following Scholze's work on p-adic Hodge theory, a major goal of the course will be to establish a non-trivial modification of the Poincaré lemma via perfectoid spaces, and thus also to provide an introduction to the latter.
Prerequisites are some understanding of étale cohomology of schemes (sites etc.) and topological algebra (completions of rings, valuations etc.). Knowledge of rigid-analytic varieties or adic spaces is helpful, but not strictly necessary.

My handwritten lecture notes as well as Thomas Manopulo's live texed notes can be found here.

## Literature

- Scholze: p-adic Hodge theory for rigid-analytic varieties.
- Scholze: Perfectoid spaces.
- Scholze, Weinstein: Berkeley lectures on p-adic geometry.

## Exam

To attend the oral examinations you have to register via Basis, and then arrange an appointment with me in Juli. The oral exams will take place on 26.7., or 28.9.-29.9..

*Last modified: April 2023, Johannes Anschütz*

## News

EMS Prize 2024 for Jessica Fintzen

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Jessica Fintzen wins Cole Prize

Regula Krapf receives university teaching award

Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts

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Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society

Peter Scholze elected as Foreign Member of the Royal Society