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 Scholzeā€™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.

Class info

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

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.