Title: The Emerton-Gee stack.


Abstract: In the last decades it has become evident that the study of algebraic families of Galois representation is very useful to various problems in number theory.

The goal of this course is to describe the Emerton-Gee stack, which is a moduli space of $p$-adic representations of a local Galois group. We shall aim to describe some of the classical material before going into the stack, its structure and properties, as well as the motivation for its construction. We shall explain some applications, like the problem of finding crystalline lifts of a Galois representation. 

If time permits, we shall describe the conjectural picture regarding the stack in the setting of the Langlands program for GL_2(Q_p), and the relation between the analytic and Banach theory in this setting.

References:

M. Emerton, T. Gee: Moduli stacks of e'tale (\Phi, \Gamma)-modules and the existence of crystalline lifts, Princeton University Press (2022).

M. Emerton, T. Gee, E. Hellman: An introduction to the categorical $p$-adic Langlands program, IHES notes.

M. Emerton, T. Gee: Moduli stacks of e'tale (\Phi, \Gamma)-modules, a survey (lectured delivered in Bonn, to be found on Emerton's home page)

Tentative plan: 

1) We plan to give a lengthy introduction (at least two meetings, maybe 3) formulating the main results, talking about motivation, and working out rank 1 and 2 examples. 

2) Once we start sketching the constructions, we shall give background on 

(a) (phi-gamma) modules and Galois representations 
(b) stacks in general. 

On (a) we shall give all the definitions, examples and main theorems, but no proofs. One of us gave a Kazhdan seminar just on (a) with full details 7 years ago. Of course, we do not expect people to know it, so we shall review the topic. How much to devote to generalities on (b) (stacks) depends on the demand from the audience. Right now we plan one lecture as a crash course on the topic.