The Fargues-Fontaine Curve and p-Adic Hodge Theory

This survey paper is based, in part, on the author’s lectures at the summer school that was held at Tata Institute’s ICTS in Bangalore in August 2019, and in part on a web-seminar that was held at the Hebrew University in the Spring Semester of 2020. I would like to thank the organizers of the summer school and the participants of both events for their contribution. Special thanks go to David Kazhdan for leading the seminar at the Hebrew University. The goal of this survey is to explain the main results of [5]. To be able to do so in a reasonable amount of space we have sacrificed generality and omitted many interesting topics, but we did choose to give some background, whenever we felt it was necessary. The reader should always refer to the book by Fargues and Fontaine for details, missing explanations, and other developments. We have also benefitted from the excellent Bourbaki seminar by Morrow [16], which we recommend as a starting point for anybody encountering the topic for the first time. The informal style of the lectures, especially in their later sections, where they became increasingly sketchy, was also kept in the printed version. Needless to say, none of the results surveyed here are due to the author, but errors, as much as they have escaped my attention, are all original errors.