Schedule (2025–26 Fall)
Sep 3
- Speaker: Yihang Zhu 朱艺航 (Tsinghua University 清华大学)
- Title: Towards local Langlands–Kottwitz method
- The Langlands–Kottwitz method seeks to express Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of objects in local Harmonic analysis, namely orbital and twisted orbital integrals. We present a local analogue of that, relating the cohomology of some local Shimura varieties with twisted orbital integrals. The discrete summation over Kottwitz triples in the global case is replaced by an integral over semi-simple conjugacy classes. As application, we explain a new point of view towards Rapoport's conjecture on the vanishing of certain twisted orbital integrals, which is a key ingredient in the global Langlands–Kottwitz method for a non-quasi-split prime. This is joint work with Rong Zhou.
Sep 17
- Speaker: Alfio Fabio La Rosa (Zhejiang University 浙江大学)
- Title: Arithmetic applications of the trace formula
- A conjecture of K. Buzzard and T. Gee posits that certain properties of the infinitesimal character of the Archimedean component of an automorphic representation translate into arithmetic properties of its non-Archimedean component. In particular, it is conjectured that C-algebraic automorphic representations are C-arithmetic. All the known cases rely on the existence of geometric models for the representation in question, but many C-algebraic representations do not admit such realisations. The main purpose of this talk is to propose to investigate the conjecture through the trace formula: I will present a non-geometric proof that cuspidal automorphic representations which are regular discrete series at infinity are C-arithmetic. I will conclude by discussing how the conjecture can be reduced to the cuspidal case.
Sep 24
- Speaker: Pol van Hoften (Zhejiang University 浙江大学)
- Title: P-adic Fourier theory in families
- Classical Fourier theory describes measures on a locally compact abelian group in terms of functions on its Pontryagin dual. In this talk, I will explain an analogous theory for (families of) p-divisible rigid analytic groups and their duals that recovers the Amice transform when applied to the open unit disk considered as multiplicative group. This is joint work with Andrew Graham and Sean Howe.
Oct 15
- Speaker: Ruotao Yang 杨若涛 (Chinese Academy of Sciences 中国科学院)
- Title: The application of factorization method in the study of Iwahori Gaiotto conjecture
- The Gaiotto conjecture says that the category of representations of the quantum supergroup can be realized as a category of certain sheaves on the affine Grassmannian. It is the quantum version of a particular case of the relative Langlands conjecture by Ben-Zvi–Sakellaridis–Venkatesh. In this talk, we will mainly focus on the tamely ramified version of the Gaiotto conjecture, and see the application of the factorization method in its study. The talk is organized as follows: first we will review the background and explain the statement of the conjecture; then we will roughly sketch the strategy of the proof using the factorization method; if time permits, we will explain the application of the factorization method in the general relative Langlands program. The talk is based on ongoing works with Michael Finkelberg and Roman Travkin.
Oct 22
- Speaker: Teruhisa Koshikawa 越川皓永 (Kyoto University 京都大学)
- Title: Cuspidal coherent sheaves
- I will explain some aspect of cuspidal coherent sheaves and relevant stuffs in the categorical local Langlands program.
Oct 29 (5:00–6:00pm!)
- Speaker: Masao Oi 大井雅雄 (National Taiwan University 臺灣大學)
- Title: Types and positive-depth Deligne–Lusztig induction
- In this talk, I would like to discuss a comparison of two kinds of representations of p-adic reductive groups arising from different origins. One is Yu's algebraic construction, further developed by Kim-Yu and Kaletha. The other is a geometric construction recently established by Chan–Ivanov, which generalizes the classical Deligne–Lusztig construction to the positive-depth setting. Our basic strategy is to compare the trace characters of those two representations. A central idea in our argument is to introduce an analogue of the classical Green function (in the context of Deligne–Lusztig theory) for both the algebraic and geometric representations, which enables us to get a character formula completely parallel to the classical Deligne–Lusztig character formula. This talk is based on my joint work with Charlotte Chan (University of Michigan).
Nov 5
- Speaker: Shih-Yu Chen 陳昰宇 (National Tsing Hua University 臺灣清華大學)
- Title: On the Blasius conjecture for the twisted standard L-functions
- The behavior of periods of motives under twisting by Artin motives was studied in depth by Blasius. Building on Deligne's conjecture concerning the critical values of motivic L-functions, Blasius proposed a refined conjecture on the algebraicity of the critical values of twisted motivic L-functions, as well as its automorphic analogue in the setting of standard L-functions of algebraic cuspidal automorphic representations twisted by Artin representations. In this talk, we present results establishing new cases of the automorphic analogue under suitable regularity assumptions.
Nov 19
- Speaker: Si Ying Lee (National University of Singapore 新加坡国立大学)
- Title: p-isogenies with G structure
- I will talk about defining the notion of p-isogenies using the theory of F-gauges, and how this allows us to construct integral models of Hecke correspondences. I will also discuss some expected consequences of this, such as constructions of Rapoport–Zink spaces, and a general framework on understanding integral Hecke actions on Shimura varieties. This is joint work in progress with Keerthi Madapusi.
Nov 26 (3:00–4:00pm!)
- Speaker: Elias Caeiro (École Normale Supérieure)
- Title: (Stark-)Heegner points and class number problems
- The classical class number one problem, conjectured by Gauss and solved by Heegner, Baker and Stark, states that there are only nine imaginary quadratic fields of class number one. In this talk, I will explain how Heegner points and the Gross–Zagier formula may be used to approach geometrically the class number h problem for a fixed h. Assuming Darmon’s conjectural real multiplication theory, this method can be adapted to determine all real quadratic fields of Richaud–Degert type which have class number one, giving a conditional improvement on the work of Biró and Lapkova. This is joint work with Henri Darmon and Jingxuan Geng.
Dec 3
- Speaker: Peihang Wu 吴沛航 (Peking University 北京大学)
- Title: Canonical extensions of p-adic shtukas on compactifications
- In this talk, we discuss the extension of p-adic shtukas on (integral models of) toroidal compactifications of Shimura varieties. We will explain the theory from a more general context and explain how to construct such extensions for toroidal compactifications. We will also discuss its applications for proving that many strata of special fibers are well positioned and for studying the uniqueness and functoriality of compactifications. This is based on a joint work in preparation with Shengkai Mao.
Dec 10
- Speaker: Jiawei An 安嘉伟 (Chinese Academy of Sciences 中国科学院)
- Title: Slopes of L-invariants for modular forms
- In this talk, I will discuss a distribution conjecture (Bergdall–Pollack) concerning the slopes (i.e., the p-adic valuations) of L-invariants attached to p-new eigenforms, which is analogous to the Gouvêa distribution conjecture for the slopes of modular forms. I will also explain how the recent breakthrough – the "local ghost theorem" of Liu, Truong, Xiao, and Zhao – can be used to compute these slopes and to prove the conjecture under certain generic local conditions on the associated Galois representations.
Dec 17
- Speaker: Fangu Chen 陈梵谷 (University of California, Berkeley)
- Title: A generalization of Elkies' theorem on infinitely many supersingular primes
- In 1987, Elkies proved that every elliptic curve defined over Q has infinitely many supersingular primes. In this talk, I will present an extension of this result to certain abelian fourfolds in Mumford's families and more generally, to certain families of Kuga–Satake abelian varieties constructed from K3-type Hodge structures with real multiplication. I will review Elkies' proof and explain how his strategy of intersecting with CM cycles can be adapted to our setting. I will also discuss some of the techniques in our proof to study the local properties of the CM cycles.
Dec 25 (4:00–5:00pm, Thursday!)
- Speaker: Yuanyang Jiang 江元旸 (Université Paris-Saclay)
- Title: Locally analytic completed cohomology of Hilbert modular varieties
- We generalize a result of Lue Pan on locally analytic completed cohomology of modular curves to the case of Hilbert modular varieties. As an application, we prove that for parallel weight Hecke classes appearing in the completed cohomology of Hilbert modular varieties, de Rhamness of the associated Galois representation will imply classicality. One central problem is to understand certain "partial de Rham cohomology", which we will prove to be classical by developing a version of Jacquet–Langlands transfer of D-modules using theory of analytic de Rham stacks.
Dec 31 (a pre-talk is scheduled at 3pm!)
- Speaker: Tony Feng (University of California, Berkeley)
- Title: Higher theta functions over function fields
- The theory of theta functions and arithmetic theta functions is used classically to access the special values and special derivatives of automorphic L-functions. In joint work with Zhiwei Yun and Wei Zhang, we have constructed a theory of higher theta functions over function fields, which we expect to give access to higher derivatives of automorphic L-functions in terms of arithmetic geometry. This has led us, also partly jointly with Adeel Khan, to unexpected and fruitful connections between theta functions, derived algebraic geometry and motivic homotopy theory.