BIMSA-YMSC p-ADIC WORKSHOP THE EMERTON–GEE STACK & BEYOND

Organizers:Yong Suk Moon, Koji Shimizu

Time:March 10th – 12th (Friday – Sunday)

Venue:BIMSA, Room 1110

 

Purpose & Goal:

The theory of (φ, Γ)-modules provides powerful techniques for studying p-adic Galois representations of a p-adic field. Recently, Emerton and Gee [EG23] constructed the moduli stack of (φ, Γ)-modules in the Banach case. The moduli stack is now often referred to as the Emerton–Gee stack and has brought new perspectives to the Galois deformation theory. This year, Emerton, Gee, and Hellmann circulated a survey paper [EGH22] on their forthcoming work on the moduli stack of (φ, Γ)-modules in the analytic case. The latter moduli is an Artin stack in rigid analytic geometry and will play a crucial role in the categorification of the p-adic Langlands program.
The workshop aims to study these moduli stacks of (φ, Γ)-modules by following [EG23, EGH22]. We plan to include discussions on formal algebraic stacks and rigid analytic Artin stacks, which are expected to become a standard tool in p-adic geometry in a few years.

 

Schedule:

 

Friday, March 10

10:30 - 12:00

Koji Shimizu (YMSC)- Introduction

12:00 - 13:00

Lunch

13:00 - 14:30

Shizhang Li (MCM)- Formal algebraic stacks I

14:50 - 16:20

Shizhang Li (MCM)- Formal algebraic stacks II

16:40 - 18:10

Yupeng Wang (MCM)- Moduli stacks of φ-modules

 

Saturday, March 11

09:00 - 09:30

Welcome remarks & group photo

09:30 - 11:00

Zhiyou Wu (Peking University)- Moduli stacks of (φ, Γ)-modules
11:15 - 12:30 Yong Suk Moon (BIMSA)- Crystalline and semistable moduli stacks I
12:30 - 13:30 Lunch
13:30 - 15:15 Heng Du (YMSC)- Crystalline and semistable moduli stacks II
15:45 - 17:30 Yong Suk Moon (BIMSA)- Families of extensions
17:30 - 18:30 Dinner
18:30 - 20:30 Heng Du (YMSC)- Crystalline lifts and the finer structure of (Xd)red

 

Sunday, March 12

09:00 - 10:30 Zicheng Qian (MCM)- Geometric Breuil–M´ezard conjecture
10:45 - 12:00 Lei Fu (YMSC)- Rigid analytic stacks
12:00 - 13:00 Lunch
13:00 - 14:15 Shanxiao Huang (Peking University)- Theory of (φ, Γ)- modules over the Robba ring
14:30 - 16:00 Jiahao Niu (CAS)- Rigid analytic stacks of (φ, Γ)-modules I
16:15-18:00 Koji Shimizu (YMSC)- Rigid analytic stacks of (φ, Γ)-modules II

 

Topics: Below is subject to change.

(1) Extended introduction
     Give an extended introduction, which also covers [EG23, §2, §7].
(2) Formal algebraic stacks I
     Review algebraic spaces and algebraic stacks in the scheme case.
(3) Formal algebraic stacks II
     Discuss [Eme] and [EG23, Appendix A].
(4) Moduli stacks of φ-modules
     [EG23, §3.1]. Explain the arguments in [PR09, EG21].
(5) Moduli stacks of (φ, Γ)-modules
     [EG23, §3.2-3.6]. Introduce the moduli stack Xd of (φ, Γ)-modules, and show that it is an Ind-algebraic stack.
(6) Crystalline and semistable moduli stacks I
     [EG23, §4, Appecdix F]. Explain the Breuil–Kisin–Fargues GK-modules.
(7) Crystalline and semistable moduli stacks II
     [EG23, §4, Appecdix F]. Discuss the stacks of crystalline and semistable Breuil–Kisin–Fargues modules.
(8) Families of extensions
     [EG23, §5] (see also [EG]). Use the Herr complex and relevant topics as tools to give a proof of Theorem 5.5.12  (the representability of the moduli stack Xd).

(9) Crystalline lifts & the finer structure of (Xd)red
      [EG23, §6]. Discuss some more properties of Xd and (Xd)red, and prove the existence of crystalline lifts.
(10) Geometric Breuil–M´ezard conjecture
      [EG23, §8]. Give a general introduction to the original Breuil–M´ezard conjecture. Then explain the formulation in terms of semistable and crystalline moduli stacks, and discuss relevant results.
(11) Rigid analytic stacks
       [EGH22, 5.1.7-5.1.15]. Discuss rigid analytic Artin stacks. The goal is to give an idea behind [EGH22, Prop. 5.1.15]. If possible, explain other approaches in the cited literature.

(12) Theory of (φ, Γ)-modules over the Robba ring
     [EGH22, §5]. Discuss Explain fundamental results necessary for later talks.
(13) Rigid analytic stacks of (φ, Γ)-modules I
       [EGH22, §5.1, 5.2]. Explain important properties and discussions about Xd.
(14) Rigid analytic stacks of (φ, Γ)-modules II
       [EGH22, §5]. Discuss the compactification, (potential) relation to the trianguline variety, and local model. Also cover some of the interesting parts in [EGH22, §6-9].

 

References

[EG] Matthew Emerton and Toby Gee, Moduli stacks of ´etale (φ, Γ)-modules: errata, available at https://www.ma.imperial.ac.uk/~tsg/Index_files/moduli-errata.pdf
[EG21] ‘Scheme-theoretic images’ of morphisms of stacks, Algebr. Geom. 8 (2021), no. 1, 1–132.
[EG23]  Moduli stacks of ´ Etale (ϕ, Γ)-modules and the existence of crystalline lifts, Annals of Mathematics Studies, vol. 215, Princeton University Press, Princeton, NJ, 2023.
[EGH22] Matthew Emerton, Toby Gee, and Eugen Hellmann, An introduction to the categorical p-adic Langlands program, arXiv e-prints, arXiv:2210.01404v2.
[Eme] Matthew Emerton, Formal algebraic stacks, available at https://math.uchicago.edu/~emerton/pdffiles/formal-stacks.pdf
[PR09] G. Pappas and M. Rapoport, Φ-modules and coefficient spaces, Mosc. Math. J. 9 (2009), no. 3, 625–663, back matter.