20-11-2021

2020-2021学年春季学期讨论班时间安排在每周四 19：30-20：30，以线上形式进行，腾讯会议会议 ID固定：671 806 023

Talk 11, Dec 16 (周四) 2021 10：00-11：00 腾讯会议： 850-171-298

Title: Potential Automorphy for $GL_n$

Speaker：Lie Qian （Stanford university）

Abstract: We prove that under mild condition for the residual representation, any ordinary $l$-adic representation of the absolute Galois group $G_F$ of a CM number field $F$ can be made automorphic when restricted to some subgroup $G_{F'}$. The result gives a much larger class of potential automorphic Galois representation than previously known in the sense that most previous results works with groups like $GSp_n$, or a compatible family of Galois representation. The family of Dwork motives is the main object we study. Along the way, we also prove an interesting ordinarity result concerning the cohomologies (viewed as local $p$-adic Galois representation) associated to certain fibres of that family.

Talk 10, Nov 25 (周四) 2021 19：00-20：00 腾讯会议：123 660 251

Title: Top Fourier coefficients of certain automorphic representations of GL(n)

Speaker：许宾（四川大学）

Abstract: Fourier coefficients of automorphic forms play an important role in the study of automorphic representations. In this talk, we will recall some basics on Fourier coefficients attached to nilpotent orbits, and then introduce some results on the top Fourier coefficients of automorphic representations of GL(n).

Talk 9, May 21 （周五）2021，10点-11点

Title:酉群的Fourier-Jacobi周期积分

Speaker: Hang XUE (University of Arizona)

Abstract：我们解释酉群商Fourier-Jacobi型周期积分的定义，关于它的Gan-Gross-Prasad猜想以及它背后的表示论意义. 我们将重点阐明它与Bessel型周期积分的关系.

Talk 8，May 13（周四）2021

Title: Rational structures for continuous cohomologies of general linear groups

Speaker: Dongwen Liu (浙江大学)

Abstract: We propose a definition of rational structures for the continuous cohomologies of Archimedean general linear groups. It is defined in terms of parabolic induction, based on Delorme’s method, and has potential application towards period relations of Rankin-Selberg L-functions. Joint work with Prof. Binyong Sun.

Talk 7， May 7 （周五）, 2021 公共教学二楼 2105

（注意：该报告为线下报告，时间为周五14：00-15：00）

Title: Parabolic ordinary families of p-adic automorphic representations

Speaker: Yiwen Ding (北京国际数学中心)

Abstract: We use Emerton’s parabolic ordinary part functor to build certain families of p-adic automorphic representations. We establish some basic properties of these families, including the density of classical points and an estimate of the Krull dimension.

Talk 6， April 22, 2021 18:15-19:15, 19:30-20:30

Title: On the Mod p Cohomology for GL_2

Speaker: Haoran WANG (Tsinghua University)

Abstract: Let F be a totally real field unramified at all places above p and D be a quaternion algebra over F which splits at exactly one infinite place. The mod p cohomology of the Shimura curve ssociated to D has a great interest from the point of view of the mod p Langlands program. Let v be a fixed place of F over p. I will report on our results on the smooth admissible representations of GL_2(F_v) occurring in the corresponding Hecke eigenspaces of the mod p cohomology, when the local Galois representation at v is non-semisimple and sufficiently generic. This is a joint work with Yongquan Hu.

Title: On a generalization of Colmez’s functor

Speaker: Yongquan HU(MCM)

Abstract: In 2005, Colmez defined an exact functor from the category of finite length admissible smooth representation of GL_2(Q_p) over a field of characteristic p to the category of finite length continuous representations of the absolute Galois group of Q_p. This functor has played a crucial role in the p-adic Langlands program for GL_2(Q_p). In this talk, I wil review the construction of Colmez’s functor, and discuss a generalization due to Breuil. If time permits, I will explain the proof of the exactness of this generalized functor. This is a joint work with Breuil, Herzig, Morra and Schraen.

Talk 5 April 15, 2021

Title: Integral p-adic Hodge theory of formal schemes in low ramification

Speaker: Yu MIN (晨兴数学中心)

Abstract: In this talk, I will briefly review the theory of prismatic cohomology and talk about some results about the module structure of prismatic cohomology groups. Then I will discuss their applications to the study of integral comparison theorem and the degeneration of the (integral) Hodge-to- de Rham spectral sequence.

Talk 4 April 8,2021

Title: Slopes of modular forms and geometry of eigencurves

Speaker: Bin Zhao (晨兴数学中心)

Abstract: In this talk, I will first explain the motivation to study the slopes of modular forms, particularly how it is related with the study of Galois representations. Then I will explain a conjecture raised by Bergdall and Pollack which gives an effective algorithm to compute the slopes of modular forms. In a joint work in progress with Ruochuan Liu, Nha Truong and Liang Xiao, we prove this conjecture and give some important consequences of this conjecture. If time allows, I will also explain how our work on the slopes of modular forms can help to study the geometry of eigencurves.

Talk 3 March 25,2021

Title: On characteristics elements of Selmer groups in noncommutative Iwasawa theory

Speaker: Mengfai LIM (华中师范大学)

Abstract: In this talk, we begin by reviewing the formulation of main conjecture in noncommutative Iwasawa theory for an elliptic curve and, in particular, the characteristic elements in the noncommutative settings. We then discuss some properties of these elements.

Talk2 ：

March 18, 2021

Title: Representation of small quantum groups and affine Springer fibers

Speaker: Peng SHAN (清华大学)

Abstract：I will explain a work in progress in which we study the link between the center of small quantum groups and cohomology of affine Springer fibers, Joint with R. Bezrukavnikov, P. Boixeda-Alvarez and Eric Vasserot.

Talk 1：

March 11, 2021

Title: Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Speaker: Miaofen CHEN (华东师范大学)

Abstract: Let K be a p-adic field. Let \rho be a n-dimensional continuous absolutely irreducible mod p representation of the absolute Galois group of K. The Kisin variety is a projective scheme which parametrizes the finite flat group schemes over the ring of integers of K with generic fiber \rho satisfying some determinant condition. The connected components of the Kisin variety is in bijection with the connected components of the generic fiber of the flat deformation ring of \rho with given Hodge-Tate weights. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K is totally ramfied with n=3 or the determinant condition is of a very particular form. We also give counterexamples to show Kisin's conjecture does not hold in general. This is a joint work with Sian Nie.

Talk 1:

Title: Homological conjectures related to Hochschild cohomology

Speaker: Guodong ZHOU ( 华东师范大学)

Time： 3rd Septembre 10:00-11:00

Tencent Metting ID: 391 659 331

Abstract: We will present the state of art of several homological conjectures related to Hochschild cohomology.

Talk2：

Title： A survey of shafarevich conjecture for hyperkahler varieties.

Speaker:李志远（上海数学中心）

Time：2020年9月10日

Tencent Meeting ID: 482 979 007

Abstract: The classical Shafarevich conjecture is about thefiniteness of isomorphism classes of varieties with good reduction over numberfields. In higher dimension case, Andre had verified the conjecture for verypolarized hyperkahler varieties. In this talk, I will survey the recentprogress for unpolarized hyper-Kahler varieties of arbitrary dimension.

Talk3：

Title： Basepoint-freeness of primitivepolarizations on Abelian varieties

Speaker:江智（上海数学中心）

Time：2020年9月17日

Tencent Meeting ID:106 321 103

Abstract:Fujita’s basepoint-freeness conjecture remains open in dimension >5. We will discuss some refined version of Fujita type equation on abelian varieties.

Talk4：

Title：GL_2(Q_p)-ordinary families and automorphy lifting

Speaker:丁一文（北京国际数学中心）

Time：2020年9月24日

Tencent Meeting ID:106 321 103

Abstract:Using Emerton’s parabolic ordinary part functor and p-adic local Langlands correspondence for GL_2(Q_p), we construct some families of p-adic automorphic representations, that we call GL_2(Q_p)-ordinary families. We will give some basic properties of these families and will report an “R=T” result on these families.

Talk5：

2020年10月15日 张翀（南京大学）

题目：p进约化群表示的分支律

摘要：分支律关心群表示限制在子群后的性质。此报告主要谈论p进约化群复表示的分支律，将综述相关工具和一些进展。

Talk6：

2020年10月22日 胡永泉（晨兴数学中心）

Title: Gelfand-Kirillov dimensions in p-adic Langlands program

Abstract: The Gelfand-Kirillov dimension is an important notion in the study of non-commutative noetherian rings. In this talk, I will first explain this notion for mod p and p-adic representations of p-adic groups, and then explain a control theorem of the Gelfand-Kirillov dimension for mod p representations of GL_2 and its application in the p-adic Langlands program. This is a joint work with Breuil, Herzig, Morra, Schraen, and with Wang.

Talk7：

2020年10月29日 阳恩林（北京大学）Tencent Meeting ID: 106 321 103

Title: Localized characteristic classes for constructible etale sheaves

Abstract: In this talk, I will first recall the theory of singular support of constructible etale sheaves after Beilinson, and then define a localized version of characteristic class, which generalize Abbes- Saito’s definition. I will also show a relationship between the localized characteristic class and the relative characteristic class. This is joint work with Yigeng Zhao in progress.

Talk8：

2020年11月5号上午10点-11点，袁新意（北京大学）Tencent Meeting ID: 106 321 103

Title: Introduction to the Colmez conjecture

Abstract: In this talk, I will introduce the Colmez conjecture, and its averaged version proved by Yuan-Zhang and Andreatta-Goran-Howard-Madapusi Pera. Some related terms in the talk are CM abelian varieties, Faltings height, Shimura varieties, and Artin L-functions.

Talk9：

2020年11月19日上午10点半--11点半，胡勇（南方科技大学）Tencent Meeting ID: 106 321 103

Title: Chow groups of quadrics

Abstract: In the 1990's, Karpenko made a systematic study of Chow groups of smooth projective quadrics in characteristic different from 2, by using Swan's work on the K-theory of quadrics. His results have soon found important applications in computation of unramified cohomology of quadrics, and have played a key role in Izhboldin's remarkable theorem that the u-invariant of a field can be 9. This talk is a survey of these results, as well as some key ingredients in the proofs. If time permits, I will talk about some recent observations in the characteristic 2 case, in a joint work with Ahmed Laghribi and Peng Sun.

Talk10：

2020年11月26日上午10点--11点，刘若川教授（北京大学）Tencent Meeting ID: 106 321 103

Title: Introduction to p-adic local Langlands program

Abstract: I will give a brief introduction to p-adic local Langlands program and survey some recent progress.

Talk11：

2020年12月3日上午10点-11点， 杨同海教授（美国威斯康星大学） Tencent Meeting ID: 106 321 103

Title: On a conjecture of Gross and Zagier on algebraicity

Abstract: The automorphic Green function $G_s(z_1, z_2)$ for $\SL_2(\Z)$, also called the resolvent kernel function for $\Gamma$, plays an important role in both analytic and algebra number theory, e.g. in the Gross-Zagier formula and Gross-Kohnen-Zagier formula. It is transcendental in nature, even its CM values are transcendental. It is quite interesting to have the following conjectural algebraicity property. For a weakly holomorphic modular form $f(\tau)=\sum_{m} c_f(m) q^m$ of weight $-2j$ ($j \ge 0$), consider the linear combination $$G_{1+j, f}(z_1, z_2) = \sum_{m >0} c_f(-m) m^j G_{1+j}^m(z_1, z_2)$$where $G_s^m(z_1, z_2) is the Hecke correspondence of $G_s(z_1, z_2)$ under the Hecke operator $T_m$ on the first (or second) variable. Gross-Zagier conjectured in 1980s that for any two CM points $z_i$ of discriminants $d_i$$$(d_1 d_2)^{j/2} G_{j+1, f} (z_1, z_2) = \frac{w_{d_1}w_{d_2}}{4}\cdot \log|\alpha|$$for some algebraic number $\alpha$, where $w_i$ is the number of units in $O_{d_i}$. In this talk, I will describe some progress on this conjecture. If time permits, I will also explain how one method to attack this conjecture also produces an analogue of the Gross-Kohnen-Zagier theorem in Kuga varieties.

Talk12：

2020年12月10日上午10点-11点， 祝辉林副教授（厦门大学） Tencent Meeting ID: 106 321 103

Title：On Some Diophantine Equations

Abstract: In this talk we give a survey about some Diophantine equations, including generalized Fermat equation and purely ternary exponential Diophantine equation. We will provide some opinions about some Diophantine equations that we are interested in.