Speakers:
Naoki Imai (University of Tokyo)
Yoichi Mieda (University of Tokyo)
Course Assistants:
Koji Shimizu (UC Berkeley)
Alex Youcis (UC Berkeley)
Morning Lecture Series:
Abstract:
The local Shimura variety attached to a p-adic reductive group G (with some additional data) is a geometric object whose etale cohomology conjecturally realizes the local Langlands correspondence for G. The Lubin-Tate space, the universal deformation space of a 1-dimensional formal group of height n, is the first example of the local Shimura variety, in which case G=GL(n). Actually, the etale cohomology of the Lubin-Tate space played an essential role in the proof of the local Langlands correspondence for GL(n) by Harris-Taylor. For general G, the construction of the local Shimura variety is more involving; it requires the theory of perfectoid and the Fargues-Fontaine curve. The goal of this lecture series is to give a definition of the local Shimura variety for a general group and formulate a conjecture on relation between its etale cohomology and the local Langlands correspondence. First we will begin with the Lubin-Tate space and its relation to the local Langlands correspondence. We also give a description of the Lubin-Tate space as a perfectoid space. After that we briefly sketch the theory of the Fontaine-Fargues curve, and construct local Shimura varieties as a moduli space of modifications of vector bundles over the curve.
Contents:
Afternoon Working Group:
Aim:
The afternoon sessions will provide participants with an opportunity to work on exercises, read materials, and get more details or clarifications from the speakers. Active participation by all participants is strongly encouraged.
Short Lectures and Presentations:
We plan to spare some time for short lectures and presentations by the course assistants and participants. For example, there will be an informal introduction to rigid geometry, adic spaces and perfected spaces on Monday aimed at younger students and non-experts. The schedule will be decided in the course of the workshop.
Problem Sets and Notes:
Problem Set by Mieda (updated on Tuesday)
Notes on adic spaces by Youcis
References:
Caraiani and Scholze, On the generic part of the cohomology of compact unitary Shimura varieties (available here)
Carayol, Nonabelian Lubin-Tate theory (available here)
Colmez, Espaces de Banach de dimension finie (available here)
Fargues, G-torseurs en théorie de Hodge p-adique (preprint available here)
Fargues, Geometrization of the local Langlands correspondence: an overview (preprint available here)
Fargues and Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-adique
Harris and Taylor, The geometry and cohomology of some simple Shimura varieties (available here)
Kedlaya and Liu, Relative p-adic Hodge theory: Foundations
Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes (available here)
Rapoport, Non-Archimedean period domains (available here)
Rapoport and Viehmann, Towards a theory of local Shimura varieties (available here)
Rapoport and Zink, Period spaces for p-divisible groups (available here)
Scholze, Etale cohomology of diamonds (available here)
Scholze, Perfectoid spaces (available here)
Scholze, On the p-adic cohomology of the Lubin-Tate tower, with an appendix of Michael Rapoport (available here)
Scholze and Weinstein, Moduli of p-divisible groups (available here)
Scholze and Weinstein, Berkeley lectures on p-adic geometry (available here)