In the summer semester of 2019 (SS 2019), Prof. Dr. Jan Kohlhaase and I organised a learning seminar on p-adic geometry and rigid cohoomology at Universität Duisburg-Essen. The aim of this seminar was to learn the finite-dimensionality of rigid cohomology in [Ked06]. We first developed the theory of rigid analytic spaces, first established by Tate. In the third talk, we learnt (vaguely) some variations of p-adic geometry. Afterwards, we introduced the concept of rigid cohomology and proved the finite-dimensionality of rigid cohomology (for smooth case).

Time: Thursday 10:15 - 11:45 Room: WSC-N-U-3.05

(There are some possibilities of changing time and room. News will be announced and renewed below.)

• Talk 1. (02. May.) Rigid analytic space I. Speaker: Luca Mastella.

  • Contents: Tate algebras, affinoid algebras, affinoid subdomains. Reference: [BGR] and [Tia16]

• Talk 2. (09. May.) Rigid analytic space II. Speaker: Luca Mastella. (different time: 16:15-17:45)

  • Contents: G-topology, rigid analytic spaces and Tate's acyclic theorem. Reference: [Tia16]

• Talk 3. (16. May.) Variations of p-adic geometry. Speaker: Jan Kohlhaase. (different room: WSC-S-3.14)

  • Contents: Rigid analytification, Berkovich spaces, adic spaces. Reference: for example [Con07] and [Wed12]

Reading material before Talk 4.: Spectral sequence, hypercohomology, de Rham cohomology. Reference: [Ked07]

• Talk 4. (05. Jun.) Rigid Cohomology I. Speaker: Ju-Feng Wu. (different time: Wednesday at 2:00/ different room: WSC-S-3.14)

  • Contents: Dagger algebras, (\sigma, \nabla)-modules, overconvergent F-isocrystals, definition of rigid cohomology. Reference: [Ked06]

• Talk 5. (27. Jun.) Rigid Cohomology II. Speaker: Ju-Feng Wu.

  • Contents: The proof of the finite-dimensionality of rigid cohomology for smooth cases. Reference: [Ked06]

  • Here is a note for these two talks, including a little summary of the first two talks. Use with your own risk.

• Talk 6. (03. Jul.) Application. Speaker: Luca Dall’Ava. (different time: Wednesday at 2:00/ different room: WSC-S-3.14)

  • Contents: Counting points on curves over finite field via rigid cohomology. Reference: [Edi03]

References

[BGR] Siegfried Bosch, Ulrich Güntzer, and Reinhold Remmert. Non-Archimedean Analysis. Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg, 1984.

[Con07] Brian Conrad. Several approaches to non-archimedean geometry. Note for the Arizona Winter School 2007. Available at: http://swc.math.arizona.edu/aws/2007/ConradNotes11Mar.pdf.

[Edi02] Bas Edixhoven. Point counting after Kedlaya. EIDMA-Stieltjes Graduate course, Leiden. 2003. Available at: https://www.math.leidenuniv.nl/~edix/oww/mathofcrypt/carls_edixhoven/kedlaya.pdf

[Ked06] Kiran S. Kedlaya. “Finiteness of rigid cohomology with coefficients”. In: Duke Math. J. 134.1 (July 2006), pp. 15–97. doi: 10.1215/S0012-7094-06-13412-9.

[Ked07] Kiran Kedlaya. p-adic cohomology: from theory to practice. Note for the Arizona Winter School 2007. Available at: http://swc.math.arizona.edu/aws/2007/KedlayaNotes11Mar.pdf. 2007.

[Tia16] Yichao Tian. Introduction to Rigid Geometry. Lecture note for the course Advanced Topics in Algebraic Geometry - Introduction to Rigid Geometry, at Universität Bonn. 2016.

[Wed12] Torsten Wedhorn. Adic Spaces. Available at: http://math.stanford.edu/~conrad/Perfseminar/refs/wedhornadic.pdf