Math 6370, Fall 2020: An elementary but modern introduction to p-adic Hodge theory
Classical Hodge theory studies the relation between singular and de Rham cohomology for complex algebraic varieties. By analogy, p-adic Hodge theory studies the relation between different cohomology theories for p-adic varieties (etale, de Rham, crystalline, and now prismatic!). Introduced by Tate in the 60s then developed by Fontaine, Faltings, and others in the 70s and 80s, p-adic Hodge theory became a pillar of modern number theory in the 90s through its central role in the modularity theory of Galois representation (Wiles, Fontaine-Mazur, etc.).
Over the past decade, the language and scope of p-adic Hodge theory has undergone a fundamental transformation through the introduction of the Fargues-Fontaine curve and Scholze’s theory of perfectoid spaces. This course will present an elementary introduction to p-adic Hodge theory that takes these advances into account. For the most part, we will only assume knowledge from graduate algebra (at the level of 6310-6320).
In particular, we will not assume prior knowledge of algebraic number theory, etale cohomology, or the p-adic numbers. Instead, basic concepts in these areas will be developed or blackboxed as needed, and the emphasis will be placed on illustrations through concrete examples derived from elliptic curves. Some background in the theory of algebraic curves and/or Riemann surfaces will be helpful, but not strictly necessary, for the course.
(Edit -- the prismatic learning seminar will take place in Spring) This course will be complemented by a more advanced learning seminar on prismatic cohomology, which will begin a month or so after the semester begins. Attendance of the learning seminar is not required for students in the course.
Fall 2020, Tu/Th 12:25pm-1:45pm (mountain)
Office hours Th 4:00pm-5:00pm (mountain)
Class and office hours are in the same Zoom; if you would like the zoom details please write me at: seanDOThoweATutahDOTedu
References and other resources
- Serre - Local Fields / Corps Locaux.
First few chapters hit basic material on discrete valuation rings and ramification theory, plus some complements on Dedekind domains that we won't cover at all
- Tate - p-divisible groups
- Fontaine - Le corps des periodes p-adiques
- Fontaine - Sur certains types de representations p-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti-Tate
- Brinon and Conrad's notes
No longer a "modern" source but good examples, some exercises, and works out computations/proofs.
- Serin Hong's notes
I have not looked closely at these but I have heard they are nice, and certainly they are much closer to the perspective of this course than Brinon-Conrad's notes.
- 2015 Fargues notes
Gives some more explanation of construction of Fargues-Fontaine curve
- Kedlaya's 2017 AWS course notes
More detailed introductory source on adic spaces and the Fargues-Fontaine curve
Lecture recordings and notes
TeXed notes temporarily (...hopefully...) removed because these did not keep up with the class, but I am hoping to compile at least the exercises and definitions from the lectures at some point to repost here (after the course has finished). Handwritten notes and videos are available below for most lectures.
- 2020-12-03, Lecture 28. Notes. Video (youtube).
- 2020-12-01, Lecture 27. Notes. Video (youtube).
- 2020-11-24, Lecture 26. Notes. Video (youtube).
- 2020-11-19, Lecture 25. Notes. Video (youtube).
- 2020-11-17, Lecture 24. Notes. Video (youtube).
- 2020-11-12, Lecture 23. Notes. No video - I made too many mistakes and am embarassed to post it! You're better off with the notes.
- 2020-11-10, Lecture 22. Notes. Video (youtube).
- 2020-11-05, Lecture 21. Notes. Video (youtube).
- 2020-11-03, Lecture 20. Notes. Video (youtube).
- 2020-10-29, Lecture 19. Notes. Forgot to press record!
- 2020-10-27, Lecture 18. Notes. Video (youtube).
- 2020-10-22, Lecture 17. Notes. Video (youtube).
- 2020-10-20, Lecture 16. Notes. Video (youtube).
- 2020-10-15, Lecture 15. Notes. Video (youtube).
- 2020-10-13, Lecture 14. Notes. Video (youtube).
- 2020-10-08, Lecture 13. Notes. Video (youtube).
- 2020-10-06, Lecture 12. Notes. Video (youtube).
- 2020-10-01, Lecture 11. Notes. Video (youtube).
- 2020-09-29, Lecture 10. Notes. Video (youtube).
- 2020-09-24, Lecture 9. Notes. Video (youtube).
- 2020-09-22, Lecture 8. Notes. Video (youtube).
- 2020-09-17, Lecture 7. Notes. Video (youtube).
- 2020-09-15, Lecture 6. Notes. Incorrectly configured the recording again, and even biffed the audio too this time, sorry!
- 2020-09-10, Lecture 5. Notes. Video (youtube).
- 2020-09-08, No lecture. Treepocalypse, no class.
- 2020-09-03, Lecture 4. Notes. Video (youtube).
- 2020-09-01, Lecture 3. Video (youtube). Unfortunately I seem to have somehow irrecoverably deleted the notes...
- 2020-08-27, Lecture 2. Notes. Video (youtube).
- 2020-08-25, Lecture 1. Notes. Audio. Unfortunately I configured the recording incorrectly, so I cannot distribute the video, but you can simulate the experience by listening to the audio while scrolling through the notes!