This is a series seminar organized by MCM on p-adic Langlands Program at MCM110 on each Friday. If you have any questions, please feel free to contact us (mcmoffice@math.ac.cn).
Yiwen Ding
Yongquan Hu
Liang Xiao
Each Friday, MCM110
Upcoming talks:
Date: December 27, 2024
Time: 10:00-12:00
Speaker: Prof. Liang Xiao (BICMR)
Title: TBA
Abstract: TBA
Date: January 3, 2025
Time: 10:00-12:00
Speaker: Prof. Liang Xiao (BICMR)
Title: TBA
Abstract: TBA
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Date: September 6, 2024
Time: 10:00-12:00
Speaker: Prof. Yiwen Ding (BICMR)
Title: Extra locally algebraic constituents and locally analytic p-adic Langlands program (1)
Abstract: We associate a locally analytic GLn(Qp)-representation pi_1(rho) to an n-dimensional generic non-critical crystalline representation rho of the absolute Galois group of Qp, which reciprocally determines rho. A key feature of pi_1(rho) is that it contains unexpected extra locally algebraic constituents with a large multiplicity. When rho is associated to p-adic automorphic representations, we show under mild hypotheses that pi_1(rho) is a subrepresentation of the GLn(Qp)-representation globally associated to rho.
The lectures are planed as follows. In the first talk, we recall the background of the locally analytic p-adic Langlands program, and state the main results.
Date: September 13, 2024
Time: 10:00-12:00
Speaker: Prof. Yiwen Ding (BICMR)
Title: Extra locally algebraic constituents and locally analytic p-adic Langlands program (2)
Abstract: The second talk will be purely on Galois side. We discuss a higher intertwining phenomenon, which roughly speaking shows that the crystalline rho is determined by certain paraboline deformations of rho.
Date: September 20, 2024
Time: 10:00-12:00
Speaker: Prof. Yiwen Ding (BICMR)
Title: Extra locally algebraic constituents and locally analytic p-adic Langlands program (3)
Abstract: In the third talk, we will construct pi_1(rho), and show that the correspondence between rho and pi_1(rho) is a one-to-one correspondence.
Date: September 27, 2024
Time: 10:00-12:00
Speaker: Prof. Yiwen Ding (BICMR)
Title: Extra locally algebraic constituents and locally analytic p-adic Langlands program (4)
Abstract: In the last talk, we will prove the local-global compatibility, showing that pi_1(rho) is a subrepresentation of the global Gl_n(Qp)-representation. If time permits, we will discuss the phenomenon of extra locally algebraic constituents in some other cases.
Date: October 11, 2024
Time: 10:00-12:00
Speaker: Prof. Haoran Wang (Capital Normal University)
Title: Factorization of the p-adic etale cohomology of the Drinfeld tower, after Colmez--Dospinescu--Niziol
Abstract: We summarize the main results of the magnificent paper of Colmez--Dospinescu--Niziol (Forum Math., Pi 11 (2023), e16 1–62.), and possibly sketch the proofs.
Date: October 18, 2024
Time: 10:00-12:00
Speaker: Dr. Arnaud Vanhaecke (MCM, CAS)
Title: Cohomology of p-adic étale local systems on the coverings of Drinfeld's half plane
Abstract: We explain how to extend the results of Colmez Dospinescu and Niziol to arbitrary Hodge-Tate weights. To achieve this, one needs to consider p-adic étale cohomology of the universal local system and its symmetric powers on Drinfeld's tower. The main observation is that these local systems are « isotrivial opers » on a curve, which allows for defining and computing their proétale cohomology. A striking difference with the trivial coefficient case is the appearance of potentially semi-stable 2-dimensional non-cristabelian Galois representations in the cohomology. After explaining how syntomic cohomology of isotrivial opers on a curve works, I will apply it to Drinfeld's universal local system and show how it relates to its étale cohomology before computing the multiplicities of Galois representations in it. Finally, I will sketch how to adapt the factorisation result of Colmez Dospinescu and Niziol to this case.
Date: November 1, 2024
Time: 9:30-10:30
Speaker: Prof. Benjamin Schraen (Université Claude Bernard Lyon 1)
Title: Locally analytic p-adic representations, coherent sheaves and patching I
Abstract: One of the goals of the p-adic Langlands Program is to study the p-adic representations of p-adic Lie groups in the completed cohomology of Shimura varieties. The patching techniques of Taylor-Wiles-Kisin are a powerful method to study these representations. For instance they can be used to prove "multiplicity one" theorems for p-adic automorphic eigenforms. The key step is often to prove that some module is finite free. The purpose of these two talks is to investigate situations where we don't have multiplicity one, in the context of locally analytic representations. In this case the modules coming from patching methods are not free. It is then interesting to determine their isomorphism class. This can be done for the group U(3) using some functor constructed by Roman Bezrukavnikov. In the first talk, I will recall the context and the results and in the second talk I will focus on proofs in the case of the group U(3). This is a joint work with Eugen Hellmann and Valentin Hernandez.
Date: November 1, 2024
Time: 11:00-12:00
Speaker: Prof. Pierre Colmez (Institut de Mathématiques de Jussieu, CNRS)
Title: On the pro-étale cohomology of p-adic analytic I
Abstract: We will give a series of lectures on our work on the pro-étale cohomology of p-adic analytic spaces: comparison theorems, geometrization, Poincaré duality.
Date: November 8, 2024
Time: 9:30-10:30
Speaker: Prof. Benjamin Schraen (Université Claude Bernard Lyon 1)
Title: Locally analytic p-adic representations, coherent sheaves and patching II
Abstract: One of the goals of the p-adic Langlands Program is to study the p-adic representations of p-adic Lie groups in the completed cohomology of Shimura varieties. The patching techniques of Taylor-Wiles-Kisin are a powerful method to study these representations. For instance they can be used to prove "multiplicity one" theorems for p-adic automorphic eigenforms. The key step is often to prove that some module is finite free. The purpose of these two talks is to investigate situations where we don't have multiplicity one, in the context of locally analytic representations. In this case the modules coming from patching methods are not free. It is then interesting to determine their isomorphism class. This can be done for the group U(3) using some functor constructed by Roman Bezrukavnikov. In the first talk, I will recall the context and the results and in the second talk I will focus on proofs in the case of the group U(3). This is a joint work with Eugen Hellmann and Valentin Hernandez.
Date: November 8, 2024
Time: 11:00-12:00
Speaker: Prof. Pierre Colmez (Institut de Mathématiques de Jussieu, CNRS)
Title: On the pro-étale cohomology of p-adic analytic II
Abstract: We will give a series of lectures on our work on the pro-étale cohomology of p-adic analytic spaces: comparison theorems, geometrization, Poincaré duality.
Date: November 22, 2024
Time: 10:00-12:00
Speaker: Prof. Liang Xiao (BICMR)
Title: Selected topics on cohomology of (phi, Gamma)-modules
Abstract: In this introductory lecture, I will first briefly recall various versions of (phi, Gamma)-modules and the Herr complex that computes the cohomology of (phi, Gamma)-modules, which in turn can be indentified with the Galois cohomology. Then, I will explain a proof of the finiteness results of (phi, Gamma)-cohomology over rigid analytic space, due to Liu-Kedlaya, which greatly simplifies the argument in Kedlaya-Pottharst-Xiao. Towards the later part of the talk, I will switch to discuss the more mysterious psi operator, and explain how one might use it to construct explicitly the cocycles representing a cohomology class.
Date: November 29, 2024
Time: 10:00-11:00
Venue: Online (Zoom ID: 3329836068 Password: mcm1234)
Speaker: Dr. Zhongyipan Lin (Northwestern University)
Title: Generic Breuil-Mézard cycles
Abstract: Breuil-Mézard cycles are closely related to the congruence of modular forms and modularity lifting and have thus been extensively studied in the past few decades. In this talk, I'll interpret (many cases of) the Breuil-Mézard conjecture as the Kazhdan-Lusztig inversion formula through the lens of geometric representation theory. In particular, we can compute the Breuil-Mézard cycles algorithmically. I'll present computational results about Breuil-Mézard cycles for the small rank groups and share interesting observations and speculations. This joint work with Bao Le Hung.
Date: December 6, 2024
Time: 10:00-12:00
Speaker: Prof. Liang Xiao (BICMR)
Title: Reduction mod p of triangulline (phi, Gamma)-modules over halo region, I: introduction
Abstract: This is the first in a series of talks, in which Yong-Suk Moon and I will explain a joint work in progress with John Bergdall and Brandon Levin. We try to understand the reduction "modulo $p$" of an etale triangulline (\phi, \Gamma)-modules of rank two whose parameters are in the halo region, and relate this to the Emerton--Gee stack for (\phi, \Gamma)-modules in characteristic $p$. In this first talk, I will explain the overall structure of the question at hand and compute the universal extension for the triangulline (phi, \Gamma)-modules.
Date: December 13, 2024
Time: 10:00-12:00
Speaker: Prof. Yong-Suk Moon (BIMSA)
Title: Reduction mod p of triangulline (phi, Gamma)-modules over halo region II
Abstract: This is the 2nd in a series of talks on a joint work in progress with John Bergdall, Brandon Levin and Liang Xiao. After discussing a motivation towards studying the reduction of rank-2 etale triangulline $(\phi, \Gamma)$-modules over the halo region, we will talk about relevant local charts for Emerton-Gee stack.
Date: December 20, 2024
Time: 10:00-12:00
Speaker: Prof. Yiwen Ding (BICMR)
Title: Extra locally algebraic constituents and locally analytic p-adic Langlands program (5)
Abstract: This is subsequent to our previous talks, where we associate a locally analytic GLn(Qp)-representation pi_1(rho) to an n-dimensional generic non-critical crystalline representation rho of the absolute Galois group of Qp. We study the internal structure of pi_1(rho). And we discuss how it may be related with the filtered phi-module Dcris(rho) in some functorial ways. This is a joint work with Christophe Breuil in progress.
