ZUMA-NR

Schedule (2024–25 Spring)

Feb 21 (4:00–5:00pm, Friday!)

Speaker: Lei Zhang 张磊 (National University of Singapore 新加坡国立大学)
Title: Rankin–Selberg integrals and double flag varieties
The Rankin–Selberg method is a classical approach for constructing integral representations of L-functions, facilitating their meromorphic continuation and the establishment of functional equations. In this talk, we present a novel perspective that interprets Rankin–Selberg integrals within the framework of double flag varieties of finite type. We introduce the concept of strongly tempered spherical varieties of Rankin–Selberg type and, leveraging the classification by He, Nishiyama, Ochiai, and Oshima, systematically identify all Rankin–Selberg integrals associated with these varieties. This framework enables a unified treatment of local zeta integrals and the gamma factors for this family. As an application, we can derive complete L-functions corresponding to these Rankin–Selberg integrals and establish local multiplicity formulas via the local relative trace formula for certain strongly tempered spherical varieties of Gan–Gross–Prasad type over p-adic fields.

Feb 26

Speaker: Haoyu Hu 胡昊宇 (Nanjing University 南京大学)
Title: Estimate for Betti numbers of étale sheaves and applications
In this talk, we discuss a result on upper bounds of all Betti numbers of an étale sheaves on a smooth variety of positive characteristic. This bound only depends on the rank and the Abbes-Saito's logarithmic ramification conductor along the boundary. We will focus on the ideas of the proof. If time permits, we will discuss some applications. This is a joint work with J.-B. Teyssier.

Mar 5

Speaker: Bin Xu 许宾 (Sichuan University 四川大学)
Title: On the non-vanishing of central values of Rankin–Selberg L-functions
In this talk, we introduce an approach to obtaining non-vanishing results for the central values of automorphic L-functions. Specifically, we consider Rankin–Selberg L-functions associated with cuspidal automorphic representations of general linear groups, which parametrize automorphic representations of classical groups. Our method relies on the close connection between automorphic descent constructions and the Gan–Gross–Prasad conjectures. For general-rank cases, we will explain how this approach yields non-vanishing central L-values under certain assumptions. In particular, for some lower-rank cases, we can establish unconditional non-vanishing results for the central L-values of cuspidal automorphic representations twisted by certain characters.

Mar 12

Speaker: Andreas Mihatsch (Zhejiang University 浙江大学)
Title: Unitary Shimura varieties at ramified primes
A general problem in the theory of Shimura varieties is to understand the geometry mod p of suitable integral models. The purpose of my talk is to discuss the case of unitary Shimura varieties at places of the CM field that are ramified over Q. My first aim is to explain how the Eisenstein condition can be used to deal with this type of ramification. My second aim is to define an isomorphism between different kinds of local Shimura varieties in this setting. My third aim is to describe some applications to arithmetic intersection problems. This is joint work with Yu Luo and Zhiyu Zhang.

Mar 19

Speaker: Shizhang Li 李时璋 (Chinese Academy of Sciences 中国科学院)
Title: On the cohomology of rational p-adic local systems
In p-adic geometry, there are more mysterious rational p-adic local systems than in algebraic/complex geometry, for example, the analytification of P^1 is no longer simply connected due to existence of a rank 2 rational p-adic local system. The cohomology of such local systems were studied previously by Kedlaya–Liu and more recently by Anschütz–Le Bras–Mann. In this talk, we shall see a (relatively) low-tech approach to reprove some of KL/ALBM results. This is a report of an ongoing work joint with Niziol–Reinecke–Zavyalov.

Mar 26 (3:00–5:15pm, double-header!)

Speaker: Hiraku Atobe 跡部発 (Kyoto University 京都大学)
Title: Local intertwining relation for GL(n)
The local intertwining relations for classical groups are the second main local theorem in Arthur’s theory of endoscopic classification. They are identities that give precise information about the action of normalized intertwining operators on parabolically induced representations. In this talk, I will explain the local intertwining relation for twisted general linear groups, which is also required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. This talk is based on a joint work with Atsushi Ichino (Kyoto), Wee Teck Gan (Singapore), Alberto Minguez (Vienna), Tasho Kaletha (Bonn), and Sug Woo Shin (UC Berkeley).
Speaker: Sug Woo Shin (University of California, Berkeley)
Title: Local intertwining relation for classical groups
This is a continuation of Atobe’s talk, based on the same joint work with Atobe, Gan, Ichino, Kaletha, and Minguez. Here we focus on a related but different aspect of the local intertwining relation (LIR) in the context of Arthur’s inductive proof of endoscopic classification. The main theme is how to deduce the LIR and local A-packet classification for certain non-tempered representations from the tempered case via Aubert duality.

Apr 2

Speaker: Sebastian Bartling (Universität Duisburg–Essen)
Title: Rapoport–Zink spaces and close p-adic fields
I want to explain how to put Rapoport–Zink spaces and their function-field versions due to Hartl–Viehmann into a profinite family using the philosophy of close p-adic fields. If time permits, I want to sketch how one may apply this construction in order to prove the Arithmetic Fundamental Lemma in the function field case. This is work in progress with Andreas Mihatsch.

Apr 9

Speaker: Wenbin Luo 骆文斌 (Peking University 北京大学)
Title: Geometric Bogomolov conjecture for semi-abelian varieties
The geometric Bogomolov conjecture, originally formulated for abelian varieties over function fields, suggests that in a non-special closed subvariety, the points with canonical height bounded from above by a postive number (depending on the subvariety) are not Zariski dense. This was a long-standing conjecture until the proof of Xie–Yuan. We generalize and prove the conjecture for semi-abelian varieties. A new phenomenon is that a special subvariety may not have Zariski-densely many points of canonical height 0. This is a joint work with Jiawei Yu.

Apr 16

Speaker: Zicheng Qian 钱子诚 (Chinese Academy of Sciences 中国科学院)
Title: Higher Ext between locally analytic generalized Steinberg with application to higher L invariants for GL(n)
The definition/study of L invariants via locally analytic representation theory has been initiated by Breuil in the case of GL(2,Q_p). The GL(3,Q_p) case has been studied extensively by Schraen and Breuil–Ding from different aspects. Motivated by their work and Gehrmann's work on automorphic L invariants, it is clear that understanding higher Ext groups between various locally analytic generalized Steinberg is crucial to develop the theory of (higher) L invariants for GL(n). A key example of such Ext groups is Ext_G^{n-1}(1,St_G^{an}) (*) with G=PGL(n,K) and St_G^{an} being the locally analytic Steinberg representation of G. We would first present a list of results on the structure of (*) which technically depends on the interpretation of top primitive class of sl_n using relative Lie algebra cohomology w.r.t. sl_{n-1}. Then we discuss how the theory of higher L invariants for GL(n) is essentially equivalent to an isomorphism between (*) and certain Galois Ext^1 group. Finally we consider some canonical subspaces of both (*) and its Galois counterpart, and discuss how such isomorphism might be explicitly characterized.

Apr 25 (11:00–12:00 am, Friday!)

Speaker: Michael Rapoport (Universität Bonn)
Title: Integral Hecke correspondences and what I learned from studying them
I will explain more details of our approach to the problem raised in the colloquium talk. (FYI) Colloquium abstract: Shimura varieties are algebraic varieties over a number field. A characteristic feature of them is the abundance of self-correspondences, given by the so-called Hecke correspondences. In the theory of Shimura varieties the construction of integral models has seen a lot of activities over the years. However, the construction of integral models of Hecke correspondences is still in its infancy. I will explain the problem and some recent progress (joint work with Ulrich Gortz and Xuhua He).

Apr 30

Speaker: Shengkai Mao 毛盛开 (Chinese Academy of Sciences 中国科学院)
Title: Compactifications of subschemes of integral models of Hodge-type Shimura varieties
The special fiber of the integral model of a Hodge-type Shimura variety with parahoric level structure admits several natural stratifications that reflect the geometry of the associated universal abelian scheme. In this talk, I will present some new results on the compactification theory of such integral models constructed by Keerthi Madapusi, and describe the boundary behavior of various strata, including central leaves, Newton strata, Kottwitz–Rapoport strata, and Ekedahl–Kottwitz–Oort–Rapoport strata. This latter work builds on and generalizes some work by Kai-Wen Lan and Benoît Stroh.

May 14

Speaker: Yu Luo 罗渝 (University of Wisconsin–Madison)
Title: A new proof of the arithmetic Siegel–Weil formula
The arithmetic Siegel–Weil formula establishes a profound connection between intersection numbers in Shimura varieties and the Fourier coefficients of central derivatives of Eisenstein series. This result was proven by C. Li and W. Zhang in 2021 using local methods. In this talk, I will present a new proof of the formula that uses the local-global compatibility and the modularity of generating series of special divisors.

May 21

Speaker: Hongjie Yu 余红杰 (Chinese Academy of Sciences 中国科学院)
Title: Counting l-adic local systems over a curve
Given a punctured curve over a finite field, Deligne proposed counting irreducible geometric l-adic local systems with prescribed local ramification near each puncture, fixed by Frobenius pullback. This problem is closely related to determining a dimension formula for certain cuspidal automorphic forms over a function field. In this talk, I will present results in these directions and discuss some related questions arising in nearby areas.

May 28

Speaker: Marc Besson (Tsinghua University 清华大学)
Title: Smooth loci of twisted affine Schubert varieties
I will discuss some geometry of twisted affine Schubert varieties. In the untwisted case, work of Malkin, Ostrik and Vybornov, and separately work of Zhu, provide us with a good understanding of the singularities of affine Schubert varieties via level one Demazure characters. I will describe the extension of these results to the twisted case, giving us an understanding of the singularities of twisted affine Schubert varieties via twisted affine Demazure characters.

Jun 4

Speaker: Xiangqian Yang 杨向谦 (Peking University 北京大学)
Title: On Ihara's lemma for definite unitary groups
Clozel, Harris, and Taylor proposed a generalized Ihara's lemma for definite unitary groups. In this talk, we prove some cases of their conjecture under the assumption of banal coefficients. A key ingredient in the proof is a local-global compatibility result for the cohomology of Shimura varieties, formulated in terms of coherent sheaves on the moduli stack of local Langlands parameters. This result is based on the unipotent categorical local Langlands correspondence recently established by Zhu, and the construction of Igusa stacks by Daniels, van Hoften, Kim, and Zhang.

Jun 9 (4:00–5:00pm, Monday!)

Speaker: Hang Xue 薛航 (University of Arizona)
Title: Fourier–Jacobi periods on unitary groups
We prove the Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on unitary groups and an Ichino–Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Y. Liu. If time permits, I will explain some applications.