Lectures on moduli of p-adic Galois representations and local models

Lecture Notes

I plan to type (probably rough) lecture notes. It will appear here later.

Schedule & Location

10 am - 12 pm, July 22-26 (with break 10:50 - 11:10 am)
Building 108, Room 320 (Reading Room), UNIST 

Overview

The main subjects of this course are the moduli stack of p-adic Galois representations (so-called the Emerton-Gee stack) and the theory of local models (developed by Le-Le Hung-Levin-Morra). The Emerton-Gee stack has its significance in the Langlands program. First, its local geometry is equivalent to the geometry of p-adic Galois deformation rings. Second, it provides a framework for the categorical p-adic local Langlands program recently proposed by Emerton-Gee-Hellmann. Therefore, it is desirable to understand the geometry of the Emerton-Gee stack. Although it is a highly complicated geometric object, the theory of local models provides concrete projective algebraic varieties whose singularities model those of the Emerton-Gee stacks. We will introduce these two subjects as well as some related topics. We will focus on conceptual understanding with a brief discussion of technical details.

I would like to thank Prof. Chol Park and Ulsan National Institute of Science and Technology for the opportunity to give this lecture series.

Tentative plan

Lecture 1
The Emerton-Gee stack is a moduli stack of p-adic Galois representations. We will discuss p-adic Galois representations and some p-adic Hodge theory. We also discuss the notions of moduli and deformation. After that, we will define the Emerton-Gee stack and study its basic properties.

Lecture 2
Serre weight conjectures and Breuil-Mézard conjectures are two motivations behind the construction of the Emerton-Gee stack. We will discuss these conjectures and their relationship to the geometry of the Emerton-Gee stack. 

Lecture 3
We introduce the theory of local models.  We first discuss Breuil-Kisin modules and their connection to Galois representations. This will lead us to objects in the geometric representation theory (certain affine Grassmannians and affine Schubert varieties), from which we build local models. We will finish the lecture by discussing singularities of local models. 

Lecture 4
We study the special fiber of local models. In particular, we classify its irreducible components and discuss how to study its geometry.  

Lecture 5
We will discuss some applications of local models to Serre weight conjectures.

References

Main references are the book of Emerton-Gee and the paper by Le-Le Hung-Levin-Morra. There are survey papers for both articles: [EG], [LLH].