BIMSA-YMSC p-adic Workshop: the Emerton–Gee stack and beyond
The theory of (φ, Γ)-modules provides powerful techniques for studying padic Galois representations of a p-adic field. Recently, Emerton and Gee [EG19] 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 [EGH] 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 spaces of (φ, Γ)-modules in the Banach case and analytic case in [EG19, EGH]. 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.
Remarks:
(1) We comply with the pandemic prevention instructions of the local government, BIMSA, and Tsinghua University. In particular, the following conditions apply for participation:
(a) Participants cannot have lived, stayed in or traveled to medium and high-risk areas and streets in the past month.
(b) Participants need to provide the “Green Code” (a negative nucleic acid test result within 48 hours) via the Health Kit (Jian Kang Bao) and the “Green Code” of the communication big data travel card when entering the conference site.
(c) Participants have not left Beijing for the last 14 days (before check-in).
(2) The number of participants is limited due to the venue.
The workshop aims to study these moduli spaces of (φ, Γ)-modules in the Banach case and analytic case in [EG19, EGH]. 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.
Remarks:
(1) We comply with the pandemic prevention instructions of the local government, BIMSA, and Tsinghua University. In particular, the following conditions apply for participation:
(a) Participants cannot have lived, stayed in or traveled to medium and high-risk areas and streets in the past month.
(b) Participants need to provide the “Green Code” (a negative nucleic acid test result within 48 hours) via the Health Kit (Jian Kang Bao) and the “Green Code” of the communication big data travel card when entering the conference site.
(c) Participants have not left Beijing for the last 14 days (before check-in).
(2) The number of participants is limited due to the venue.
Organizer
Date
10th ~ 12th March, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday,Saturday,Sunday | 09:00 - 20:30 | 1110 | - | - | - |
Syllabus
TBA. Note that this is an intensive workshop. We plan to spend at least 6 hours on lectures and discussions every day. A tentative outline of the talks is given below:
(1) Extended introduction
Give an extended introduction, which also covers [EG19, §2, §7].
(2) Formal algebraic stacks
Longer talk (possibly 2 talks). Review algebraic spaces and algebraic stacks in the scheme case. Then discuss [Eme] and [EG19, Appendix A].
(3) Moduli stack of φ-modules
[EG19, §3.1]. Explain the arguments in [PR09, EG21].
(4) Moduli stack of (φ, Γ)-modules
[EG19, §3.2-3.6]. Introduce the moduli stack Xd of (φ, Γ)-modules, and show that it is an Ind-algebraic stack.
(5) Crystalline and semistable moduli stacks
[EG19, §4, Appecdix F]. Longer talk (possibly 2 talks). The first half addresses the Breuil–Kisin–Fargues GK-modules, and the second covers the corresponding stacks.
(6) Families of extensions
[EG19, §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).
(7) Crystalline lifts & the finer structure of (Xd)red
[EG19, §6]. Discuss some more properties of Xd and (Xd)red, and prove the existence of crystalline lifts.
(8) Geometric Breuil–M´ezard conjecture
[EG19, §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.
(9) Analytic moduli stacks of (φ, Γ)-modules I
[EGH, §5]. Discuss (φ, Γ)-modules over the Robba ring. Explain fundamental results necessary for later talks.
(10) Analytic moduli stacks of (φ, Γ)-modules II
[EGH, §5]. Discuss rigid analytic Artin stacks. If possible, explain other approaches in the literature. Then explain the rigid analytic stack of (φ, Γ)-modules and its representability.
(11) Analytic moduli stacks of (φ, Γ)-modules III
[EGH, §5]. Discuss the compactification, (potential) relation to the trianguline variety, and local model.
(12) Expectations, conjectures, examples, and future directions
[EGH, §6-9]. Lead discussions on future directions.
(1) Extended introduction
Give an extended introduction, which also covers [EG19, §2, §7].
(2) Formal algebraic stacks
Longer talk (possibly 2 talks). Review algebraic spaces and algebraic stacks in the scheme case. Then discuss [Eme] and [EG19, Appendix A].
(3) Moduli stack of φ-modules
[EG19, §3.1]. Explain the arguments in [PR09, EG21].
(4) Moduli stack of (φ, Γ)-modules
[EG19, §3.2-3.6]. Introduce the moduli stack Xd of (φ, Γ)-modules, and show that it is an Ind-algebraic stack.
(5) Crystalline and semistable moduli stacks
[EG19, §4, Appecdix F]. Longer talk (possibly 2 talks). The first half addresses the Breuil–Kisin–Fargues GK-modules, and the second covers the corresponding stacks.
(6) Families of extensions
[EG19, §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).
(7) Crystalline lifts & the finer structure of (Xd)red
[EG19, §6]. Discuss some more properties of Xd and (Xd)red, and prove the existence of crystalline lifts.
(8) Geometric Breuil–M´ezard conjecture
[EG19, §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.
(9) Analytic moduli stacks of (φ, Γ)-modules I
[EGH, §5]. Discuss (φ, Γ)-modules over the Robba ring. Explain fundamental results necessary for later talks.
(10) Analytic moduli stacks of (φ, Γ)-modules II
[EGH, §5]. Discuss rigid analytic Artin stacks. If possible, explain other approaches in the literature. Then explain the rigid analytic stack of (φ, Γ)-modules and its representability.
(11) Analytic moduli stacks of (φ, Γ)-modules III
[EGH, §5]. Discuss the compactification, (potential) relation to the trianguline variety, and local model.
(12) Expectations, conjectures, examples, and future directions
[EGH, §6-9]. Lead discussions on future directions.
Reference
[EG] Matthew Emerton and Toby Gee, Moduli stacks of ´etale (φ, Γ)-modules: errata.
[EG19] ____, Moduli stacks of ´etale (φ, Γ)-modules and the existence of crystalline lifts, 2019.
[EG21] ____, ‘Scheme-theoretic images’ of morphisms of stacks, Algebr. Geom. 8 (2021), no. 1, 1–132. MR 4174286
[EGH] Matthew Emerton, Toby Gee, and Eugen Hellmann, An introduction to the categorical p-adic langlands program.
[Eme] Matthew Emerton, Formal algebraic stacks.
[PR09] G. Pappas and M. Rapoport, Φ-modules and coefficient spaces, Mosc. Math. J. 9 (2009), no. 3, 625–663, back matter. MR 2562795
[EG19] ____, Moduli stacks of ´etale (φ, Γ)-modules and the existence of crystalline lifts, 2019.
[EG21] ____, ‘Scheme-theoretic images’ of morphisms of stacks, Algebr. Geom. 8 (2021), no. 1, 1–132. MR 4174286
[EGH] Matthew Emerton, Toby Gee, and Eugen Hellmann, An introduction to the categorical p-adic langlands program.
[Eme] Matthew Emerton, Formal algebraic stacks.
[PR09] G. Pappas and M. Rapoport, Φ-modules and coefficient spaces, Mosc. Math. J. 9 (2009), no. 3, 625–663, back matter. MR 2562795