NCMW - Introduction to the p-adic local Langlands program
Speakers and Syllabus
Syllabus to be covered in terms of modules of 6 lectures each:
Name of the Speaker with their Affiliation |
No. of Lectures |
Detailed Syllabus |
Prof. Shalini Bhattacharya Indian Institute of Science Education and Research, Tirupati |
3 |
Mod p representations of GL_2 over a local field In this course, I will summarize some basic results on mod p representations of GL_2(F) over an algebraic closure of F_p, where F is a finite extension of Q_p. By the work of Barthel-Livne’ one divides the irreducible representations of GL_2(F) into one-dimensional characters, principal series, special series and supersingular modules. Later on, Breuil described the supersingular modules explicitly for GL_2(Q_p), completing the classification of irreducible representations mod p of GL_2(Q_p) with a central character. This classification can be considered a necessary step in the construction of the mod p Local Langlands correspondence for GL_2(Q_p). |
Prof. Andrea Dotto Univerity of Cambridge |
6 |
Localization of smooth mod p representations of GL_2(Q_p) A central result in the p-adic Langlands correspondence for GL_2(Q_p) is Paskunas's classification of the blocks of the category A of smooth representations of this group in characteristic p. In recent joint work with Emerton and Gee we introduced a scheme X that parametrizes these blocks, and we extended A to a stack of abelian categories on the Zariski site of X, satisfying many additional properties. This course will explain the background and content of these results, and possibly discuss some applications towards the categorical form of the p-adic Langlands correspondence. |
Prof. Eknath Ghate Tata Institute of Fundamental Research, Mumbai |
6 |
Reductions of semi-stable representations using the Iwahori mod p LLC In this course, we shall explain how to compute the reduction of a two-dimensional semi-stable representations of the Galois group of Q_p of weight at most p+1. Such representations occur in the theory of modular forms at primes p exactly dividing the level of the form (if the character of the form at p is trivial). |
Prof. Karol Koziol, Baruch College, City University of New York |
6 |
Iwahori-Hecke algebras The goal of this course will be to discuss the role of the pro-p-Iwahori-Hecke algebra in the Langlands program, with particular focus on the case of mod p coefficients. I will start by explicitly defining the algebra, and giving the presentation in terms of generators and relations. I will then talk about how it is related to mod p representations of p-adic reductive groups, and discuss when this relation breaks. If time permits, I will talk about derived aspects of the mod p Langlands program, and how we can enhance the pro-p-Iwahori-Hecke algebra into an Ext-algebra. |
Dr. Mihir Sheth IISc Bengaluru |
3 |
Diagrams and the mod p local Langlands correspondence Diagrams provide a handy tool to construct mod p representations of GL_2 over p-adic fields with control on their K-socles. This has several implications for the mod p local Langlands correspondence for GL_2. In this course, I will recall the theory of diagrams due to Paskunas and Breuil-Paskunas, and then talk about its application to the construction of supersingular representations. If time permits, I will briefly explain the work of Dotto-Le which canonically relates local Galois representations to certain diagrams using the mod p local-global compatibility. |
References:
- L. Barthel, R. Livne, Irreducible modular representations of GL2 of a local field, J. Number Theory (1995).
- C. Breuil, Representations and Galois and GL_2 in characteristic p, Columbia University lecture notes (2007).
- C. Breuil, Sur quelques représentations modulaires et p-adiques de GL2(Q_p), Compos. Math. (2003).
- C. Breuil, V. Paskunas, Towards a modulo p Langlands correspondence for GL_2, Mem. Amer. Math. Soc. (2012).
- A. Chitrao, An Iwahori theoretic mod p Local Langlands Correspondence (https://arxiv.org/pdf/2311.02919.pdf)
- A. Chitrao, E. Ghate, Reductions of semi-stable representations using the Iwahori mod p Local Langlands Correspondence. (https://arxiv.org/pdf/2311.03740.pdf)
- A. Dotto, T. Gee, M. Emerton, Localization of smooth p-power torsion representations of GL_2(Q_p) (https://https://arxiv.org/pdf/2207.04671.pdf)
- A. Dotto and D. Le, Diagrams in the mod p cohomology of Shimura curves, Compos. Math. (2021).
- E. Ghate, Zig-zag for Galois representations (https://arxiv.org/pdf/2211.12114.pdf)
- E. Ghate, M. Sheth, Diagrams and mod p representations of p-adic groups, expository article in 'Perfectoid spaces', Springer Nature (2022).
- F. Herzig, p-modular representations of p-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. (2015).
- R. Ollivier, P. Schneider, The modular pro-p Iwahori-Hecke Ext-algebra, Representations of reductive groups (2019).
- V. Paskunas, Coefficient systems and supersingular representations of GL_2(F), Mem. Soc. Math. Fr. (2004).
- M. Sheth, On irreducible supersingular representations of GL_2(F), Pacific J. Math. (2022).
- M-F. Vigneras, Representations modulo p of the p-adic group GL(2,F), Compos. Math. (2004).
- M-F. Vigneras, The pro-p-Iwahori Hecke algebra of a reductive p-adic group I, Compos. Math. (2016).
Names of the tutors with their affiliations:
- Dr.Anand Chitrao (HRI, Prayagraj)
- Dr.Ravitheja Vangala (IISc Bengaluru)
- Dr.Arindam Jana (IISER Berhampur)
- Dr.Rishabh Agnihotri (IISc Bengaluru)
Time Table
Time Table:
Day | Date | Lecture 1 (9.30–11.00) |
Tea (11.05-11.25) |
Lecture 2 (11.30–1.00) |
Lunch (1.05–2.25) |
Lecture 3 (2.30–3.30) |
Snacks (3.35-3:55) |
Discussion (4.00-5.00) |
(name of the speaker) | (name of the speaker) | (name of the speaker) | (Name of the tutor) | |||||
Tues | 17/09/24 | SB | EG | UKA | SB: RA, RV | |||
Wed | 18/09/24 | KK | MS | SD | KK: AC, AJ | |||
Thu | 19/09/24 | EG | SB | ARK | EG: AJ, AC | |||
Fri | 20/09/24 | KK | MS | SV | MS: RA, RV | |||
Sat | 21/09/24 | EG | AD | AG | EG: AJ, AC | |||
Sun | 22/09/24 | rest | ||||||
Mon | 23/09/24 | KK | AD | AK | KK: AC, AJ | |||
Tue | 24/09/24 | KK | AD | SOD | AD: RV, RA | |||
Wed | 25/09/24 | EG | MS | MS: RV, RA (Discussion) |
Full forms for the abbreviations of speakers and tutors:
KK : Prof. Karol Koziol
MS: Dr. Mihir Seth
AD: Prof. Andrea Dotto
EG: Prof. Eknath Ghate
SB: Prof. Shalini Bhattacharya
AC: Dr. Anand Chitrao
RV: Dr. Ravitheja Vangala
AJ: Dr. Arindam Jana
RA: Dr. Rishabh Agnihotri
UKA: Prof. U. K. Anandavardhanan
SD: Prof. Shaunak Deo
SV: Prof. Sandeep Varma
ARK: Prof. Arvind Kumar
SOD: Prof. Soumyadip Das
AG: Prof. Abhik Ganguli
AK: Prof. Aditya Karnataki