Categorical Langlands Learning Seminar

Goals: Continuing from winter, the plan for this quarter will be focused on discussing the Banach case of the conjectural categorical correspondence as in Section 6 of the IHES notes, specifically we will focus on understanding the categorical conjecture via looking at GL_1 and GL_2(Qp) case of the conjecture.

Set-up: There will be weekly talks Friday 14:00 - 15:30 Location TBA. There will also be an option to attend via Zoom -- (please contact Samanda for the Zoom link). With the speakers' persmission, recordings of the talk will be uploaded to this page. Please let one of the organisers know if you want to be added on the mailing list.

Organisers: Matthew Emerton (emerton (at) math (dot) uchicago (dot) edu), Samanda Zhang (samanda (at) uchicago (dot) edu)

Key Reference:
An Introduction to Categorical p-adic Langlands program by Emerton, Gee and Hellmann (EGH)

General (Tentative) Outline for this quarter: All the cases of the conjectural categorical correspondence below are considered solely in the Banach case.
1. Samanda Zhang -- GL_1 Q_p)
2. Wei Yao -- GL_1(F) for F a finite extension over Q_p
3. Chengyang Bao and Ray Li -- GL_2 (Q_p)
4. TBA -- GL_2(Q_{p^f})

Mar 22: Ray Li -- Example of various open and closed substacks of the EG stack and crystalline stack

Recording
References:
1. "Moduli Stacks of (phi, Gamma)-modules: A survey" by Emerton and Gee
2. "Moduli Stacks of Etale (phi, Gamma)-modules and the Existence of Crystalline Lifts" by Emerton and Gee

Mar 29: Ray Li -- Example of various open and closed substacks of the EG stack and crystalline stack Contd

Apr 5: Matthew Emerton -- Overview

Recording

Apr 12: Samanda Zhang -- GL_1(Q_p) case

Recording
References:
1. An Introduction to Categorical p-adic Langlands program by Emerton, Gee and Hellmann (EGH), Section 6 and 7
2. "Moduli Stacks of Etale (phi, Gamma)-modules and the Existence of Crystalline Lifts" by Emerton and Gee, Chapter 7

Apr 19: Wei Yao -- GL_1(F) case (F finite extension over Q_p)

Apr 26: Chengyang Bao -- Classical p-adic local Langlands correspondence for GL_2(Q_p)

Abstract: In the proof of the categorical Langlands correspondence for GL_2(Q_p), Dotto--Emerton--Gee construct the sheaf L_oo, which is a generalization of Colmez's D^{\natural} \boxtimes P^1. In this talk, I will review the classical p-adic local Langlands correspondence and describe the functor attaching to a (phi,gamma)-module a GL2(Qp) unitary admissible Banach representation and vice versa. For most of the talk, very little prelim is assumed other than knowing the definition of etale phi-gamma modules.
References:
1. Breuil's ICM Talk The emerging p-adic Langlands programme
2. An Introduction to Categorical p-adic Langlands program by Emerton, Gee and Hellmann (EGH)
3. Representations de GL_2(Q_p) et (phi, Gamma)-modules by Colmez

May 3: Ray Li -- Representations of GL_2(Q_p) over F_p

Abstract: I will discuss the representation theory of GL2(Qp) over Fp. I will try to give a feeling for what the irreducible representations look like, and state the main theorems that play a role in categorical Langlands for GL2(Qp).
References:
1. Irreducible module representations of GL_2 of a local field by L. Barthel, R. Livné
2. Localization of smooth p-power torsion representations of GL_2(Q_p) by A. Dotto, M. Emerton, T. Gee

May 10: Matthew Emerton -- Categorical p-adic Langlands for GL_2(Q_p)

Abstract: I will sketch the construction of the categorical p-adic Langlands correspondence for GL_2(Q_p), following ongoing (hopefully nearly finished!) work of Andrea Dotto, Toby Gee, and myself.
Recording

May 17: Matthew Emerton -- A DGA approach to categorical p-adic LL for GL_2(Q_p)

Abstract: I will outline a different approach to proving the theorem I explained last week (or at least, the generic part of it). It applies arguments with derived Hecke algebras and DGA methods (and patching!), rather than Colmez's functor and the results of Paskunas and Johansson--Newton--Wang-Erickson.

Winter 2024 Plan

General Outline for this quarter: Below outline the order of the topics will be discussed in this quarter (and possibly next quarter), and the corresponding speaker,
1. Michael Barz -- Formal Scheme and Algebraic Stack
2. Wei Yao -- (phi, Gamma)-modules over finite Zp-algebra and its relation to Galois representations
3. Abhijit Mudigonda and Samanda Zhang -- (phi, Gamma)-modules over general ring and definition of the Emerton-Gee stack (EG stack)
4. Chengyang Bao -- Picture of \mathcal{X}_red for GL_2(Qp) and structure of components \mathcal{X}(sigma) for generic sigma in the dimension 2 case
5. Ray Li -- Example of various open and closed substacks of \mathcal{X} and crystalline stack

Jan 8: Matthew Emerton -- Overview

Recording

Jan 15: MLK Holiday

Jan 22: Michael Barz -- Formal Schemes and Algebraic Stacks

Abstract: The Emerton-Gee stack is, as the name implies, a formal algebraic stack, and not a scheme. There are therefore two independent sets of phenomena that one must get used to in order to use the EG stack: formal scheme phenomena, and stack-y phenomena. We will try to start by giving a quick tour of the (quite formidable) formalism, and then focus on giving some concrete examples. Most of the talk will be focused on algebraic stacks.
Recording (Due to technical problems, the first few minutes of the talk wasn't recorded)
References:
1. Artin's "The Implicit Function Theorem in Algebraic Geometry" PDF page 22 - 45
2. Emerton-Gee "'Scheme-theoretic images' of morphisms of stacks"
3. For examples of stacks, https://math.columbia.edu/~dejong/seminar/examples-stacks.pdf
4. The Stacks Project , especially Chp 65, 94, 98, and 87.

Jan 29: Wei Yao -- (phi, Gamma)-modules over finite Zp-algebra and its relation to Galois representations

Recording
References:
1. "Moduli Stacks of (phi, Gamma)-modules: A survey" by Emerton and Gee, Lecture 1, 3-5
2. "Galois Representations and (phi, Gamma)-Modules" by Schneider

Feb 5: Abhijit Mudigonda-- (phi, Gamma)-modules over general ring and definition of the Emerton-Gee stack I

Recording
References:
1. "An introduction to the theory of p-adic representations" by Berger
2. "Galois Representations and (phi, Gamma)-modules -- Course given at IHP in 2010" by Berger
3. "Moduli Stacks of (phi, Gamma)-modules: A survey" by Emerton and Gee

Feb 12: Samanda Zhang -- (phi, Gamma)-modules over general ring and definition of the Emerton-Gee stack II

Recording
References:
1. "An Introduction to the theory of p-adic representations" by Berger
2. "Moduli Stacks of (phi, Gamma)-modules: A survey" by Emerton and Gee , Lecture 1, 3-5
3. "Moduli Stacks of Etale (phi, Gamma)-modules and the Existence of Crystalline Lifts" by Emerton and Gee , Chapter 2 and 3.1, 3.2
4. Ray Li's Topic Proposal
5. IHES 2022 "An Introduction to the p-adic Langlands program" Lecture 2 (by Gee) and 3 (by Emerton)

Feb 19: Chengyang Bao -- The Irreducible Components of the reduced EG stack for GL2(Qp)

Feb 26: Chengyang Bao -- The Irreducible Components of the reduced EG stack for GL2(Qp) Contd.

Fall 2023 Plan

Goals: The plan for this quarter is having a list of overview talks on backgrounds relevant to learning p-adic Langlands. In particular, the goal is to reach some understanding of the six-authors' paper Patching and the p-adic Local Langlands Correspondence .

Sep 28: Matthew Emerton -- Overview

Oct 2: Chengyang Bao -- Classical Local Langlands for GL_2 and GL_n; Aaron Slipper -- Bernstein Center

(Hand-written) Notes by Aaron on Bernstein Center.
References (Note that LL stands for Local Langlands and BC stands for Bernstein Center):
1. (LL) "Some introductory notes on the Local Langlands Correspondence" by Buzzard
2. (BC) "Ottawa Lectures on Representations of Reductive p-adic Groups" edited by Cunningham and Nevins . Rouche's article (first chapter) gives a complete description of the Bernstein's theorem and the Bernstein Center.
3. (BC) MathOverflow post on Bernstein Center of a p-adic reductive group
4. (BC) "Some algebras of essentially compact distributions of a reductive p-adic group" by Moy and Tadić
5. (BC) "The Bernstein Center in terms of invariant locally integrable functions" by Moy and Tadić
6. (BC) "Le "centre" de Bernstein" by Bernstein and Deligne

Oct 9: Aaron Slipper -- Bernstein Center contd.; Casimir Kothari -- Modular Curves and Shimura Varieties

Notes by Casimir on Shimura Varieties
References (Note that M stands for Modular Curves, Modular Forms and Hecke Operators and S stands for Shimura Varieties):
1. (M) "A First Course in Modular Forms" by Diamond and Shurman
2. (M) "Modular forms and modular curves" by Diamond and Im
3. (M) Online Course on Mazur's Theorem by Snowden
4. (M) "Attaching Galois representations to modular forms" by Porat
5. (S) "An example-based introduction to Shimura varieties" by Lan
6. (S) "Introduction to Shimura Varieties" by Milne

Oct 16: Ray Li -- Completed Cohomology and its relation to p-adic Langlands

Notes and Recording

Oct 23: Abhijit Mudigonda -- Jacquet-Langlands

Notes and Recording

Oct 30: Abhijit Mudigonda -- Jacquet-Langlands contd; Kun Liu and Pawel Poczobut -- p-adic Hodge Theory

Notes and Recording of Abhijit's Talk

Nov 6: Kun Liu and Pawel Poczobut -- p-adic Hodge Theory

Notes
References

Nov 13: Aisosa Efemwonkieke -- Galois Deformation

References
1. Mazur, B. (1989). Deforming Galois Representations . In: Ihara, Y., Ribet, K., Serre, JP. (eds) Galois Groups over Q.
2. Modular Forms and Fermat’s Last Theorem , Chapter 8 An introduction to the Deformation Theory of Galois Representations.
3. Schlessinger, Michael. “Functors of Artin Rings.” Transactions of the American Mathematical Society 130, no. 2 (1968): 208–22.

Nov 20: Thanksgiving break

Nov 27: Samanda Zhang -- Taylor-Wiles-Kisin Patching I

Recording
References
1. "Modularity Lifting Theorems and the case of GL1" by Gal Porat (Majority of the talk materials are from this.)
2. Toby Gee's Video 4-Lectures Series on Modularity Lifting Theorems from AWS 2013 (for overview)
3. The lecture notes corresponding to Toby Gee's AWS course
4. Arbeitsgemeinschaft Report: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups In particular, David Savitt's talk notes is the GL1 case with l_0 =0 and Preston Wake's notes is for the more general case.

Dec 4: Jinyue Luo -- Taylor-Wiles-Kisin Patching II

Recording
References
1. Modular Forms and Fermat's Last Theorem edited by Cornell, Silverman and Stevens
2. Fermat's Last Theorem by Darmon, Diamond and Taylor

End of Fall Quarter