Topics in Number Theory, Fall 2016

This will be a course in p-adic Hodge theory, with an emphasis on new geometric methods and relative p-adic Hodge theory. The course will be divided into two roughly equal parts. In the first part, we'll prove the de Rham comparison theorem for the cohomology of proper smooth rigid analytic spaces; along the way, we'll develop the theory of rigid analytic geometry from the point of view of adic spaces. In the second part, we'll introduce period domains and moduli spaces of mixed characteristic local shtukas for GL_{n}, following Scholze's Berkeley course. The goal here will be to discuss ~~ongoing~~ work of the lecturer proving a conjecture of Harris on the cohomology of these spaces in certain cases. Throughout the course, perfectoid spaces will recur as a tool, and we'll develop this theory as needed. There will be regular homework assignments: out of necessity, a good deal of important material will be presented in the form of extended homework problems. (Graduate students, please note that your homeworks won't be collected or graded in any way, and that "grades" for the course will be calculated by the usual recipe for advanced classes like this.)

Meeting time: MW 2:40-3:55 in Math 307

**Homework 1 (Huber rings, adic spaces, and rigid geometry)**

(please let me know if you see any strange typos or mistakes in these problems!)

References

Conrad, Several approaches to non-archimedean geometry

Huber, A generalization of formal schemes and rigid analytic spaces

Huber, Etale cohomology of rigid analytic varieties and adic spaces

Faltings, *p*-adic Hodge theory

Scholze, *p*-adic Hodge theory for rigid analytic varieties

Scholze, *p*-adic geometry (Berkeley notes)

Kedlaya-Liu, Relative *p*-adic Hodge theory, I & II

Fargues-Fontaine, Courbes et fibres vectoriels en theorie de Hodge p-adique

DH, Notes on diamonds

DH, Moduli of local shtukas and Harris's conjecture, I

Tentative Schedule

**Part I**

Lecture 1 (Sept. 7): Huber rings and adic spaces

Lecture 2 (Sept. 12): Adic spaces and rigid geometry

Lecture 3 (Sept. 14): Adic spaces and rigid geometry, cont'd

Lecture 4 (Sept. 19): Perfectoid spaces: a user's guide

Lecture 5 (Sept. 21): The pro-etale site; tricks of the trade

Lecture 6 (Sept. 26): Reminder on Fontaine rings; statement of de Rham comparison & Hodge-Tate decomposition

Lecture 7 (Sept. 28): Finiteness d'àpres Kiehl et Scholze; primitive comparison

Lecture 8 (Oct. 3): Begin proof of de Rham comparison: period sheaves and their properties; semi-primitive comparison

Lecture 9 (Oct. 5): Structural de Rham sheaves; the Poincare lemma

Lecture 10 (Oct. 10): Geometric acyclicity of OB_{dR}; finish proof of de Rham comparison

Lecture 11 (Oct. 12): Odds and ends

**Part II**

...