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).
We go forwards: we compute instead the mod p reduction of the GL_2(Q_p)-Banach spaces attached to such a representation. This Banach space consists of certain functions on Q_p modulo certain p-adic logarithmic functions. We shall explain some properties of these functions from a high-school perspective (Taylor expansions, differentiability etc).
We then go backwards: we use a newly stated Iwahori mod p LLC to identify the mod p reductions of these Banach spaces in terms of explicit mod p representations of the Galois group of Q_p.
If time permits, we shall explain how these reduction computations in the semi-stable world allow one to prove an outstanding zig-zag conjecture on the reductions of crystalline representations of slopes up to (p-1)/2.

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.