Langlands & ARithmetic GEometry Nearly Every Week
A wide-ranging seminar meeting on Mondays 3-4pm in S17-0405 (unless noted otherwise).
Please contact Dave, Ian or Si Ying if you'd like to give a talk!
Talks in 2026
February 2: Yihang Zhu (Tsinghua)
Room: S17 05-11
Title: Towards local Langlands-Kottwitz method
Abstract: The Langlands-Kottwitz method seeks to express Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of objects in local harmonic analysis which enter the Arthur-Selberg trace formula, namely orbital and twisted orbital integrals. As a local analogue, we present a formula relating the cohomology of local Shimura varieties with twisted orbital integrals. This formula bridges the point-counting formula for Shimura varieties with the point-counting formula for Igusa varieties, a manifestation of the Mantovan product relation. As an application of our local formula, we explain a new categorical-Langlands-based approach towards Rapoport's conjecture on the vanishing of certain twisted orbital integrals, which is itself a key ingredient in the global Langlands-Kottwitz method for a non-quasi-split prime. This is joint work with Rong Zhou.
March 2: Chenji Fu (Bonn)
Room: S17 05-11
Title: Depth-zero local Langlands via shtukas in mixed characteristic
Abstract: Eteve computed the Genestier-Lafforgue L-parameters for depth-zero representation for a equal-characteristic local field. I will explain how to adapt his method to the setting of Fargues-Scholze. We use certain ''smoothness'' and ''properness'' of the stack of shtuka to reduce the question to the local Hecke stack, where we can do an explicit computation using hyperbolic localizations following the work of ALWY on central sheaves in mixed characteristic.
March 16: Ian Gleason (NUS)
Room: S17 05-11
Title: On the schematic and analytic constructions of the local Langlands category
Abstract: We show a folklore conjecture stating that two categorical enhancements of the automorphic side of the local Langlands correspondence agree. The first enhancement appears in Fargues--Scholze's formulation of the categorical local Langlands correspondence, and the second one appears in Zhu's formulation. This is joint work with Hamann, Ivanov, Lourenco, Zou.
March 30: Ian Gleason (NUS)
Title: On the schematic and analytic constructions of the local Langlands category
Abstract: This is a sequel to the previous talk. In this talk I will describe the challenges tackled in order to construct the comparing functor. Depending on time constraints, I might also discuss the strategy to show that the constructed functor is an equivalence.
April 8 (Wednesday): Xiangqian Yang (Peking)
Room: S17 05-12 at 4pm
Title: On Ihara's lemma for definite unitary groups
Abstract: 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. The proof relies on the unipotent categorical local Langlands correspondence established by Zhu. We also discuss certain properties of the coherent sheaf associated to the space of automorphic forms under the categorical local Langlands functor.
April 15 (Wednesday): Tomoyuki Abe (Tokyo)
Room: S17 05-12 at 4pm
Title: Algebraicity of diagonal series
Abstract: Deligne showed that the diagonal series of an algebraic power series of n variables over a field of positive characteristic remains algebraic. He deduced this theorem using geometric method in the case n=2, and showed the general case using purely algebraic method. However, he expected a more direct proof of that fact, and gives a conjectural bound of the degree of algebraicity. The conjectural bound had been verified by Adamczewski and Bell using elementary method. We revisit this question using arithmetic D-module. This is a joint work with R. Crew.
April 29 (Wednesday): Heejong Lee (KIAS)
Room: S17 04-04 at 3pm
Title: Serre weight conjectures and modularity lifting for GSp4
Abstract: Given a Galois representation attached to a regular algebraic cuspidal automorphic representation, the Hodge--Tate weight of the Galois representation is matched with the weight of the automorphic representation. Serre weight conjectures are mod p analogue of such a correspondence, relating ramification at p of a mod p Galois representation and Serre weights of mod p algebraic automorphic forms. In this talk, I will discuss how to understand Serre weight conjectures and modularity lifting as a relationship between representation theory of finite groups of Lie type (e.g. GSp4(Fp)) and the geometry of p-adic local Galois representations. Then I will explain the proof idea in the case of GSp4. This is based on a joint work with Daniel Le and Bao V. Le Hung.
May 4: Thomas Bitoun (Shanghai)
Room: S17 04-05 at 3pm
Title: The D-module of an isolated singularity
Abstract: Let Z be the germ of a complex hypersurface isolated singularity of equation f. We consider the family of analytic D-modules generated by the powers of 1/f and relate it to the pole order filtration on the de Rham cohomology of the complement of {f=0}. This work revisits Vilonen's characterization of the intersection homology D-module.
Talks in 2025
August 25 (2pm in S17-0611): Adeel Khan (Academia Sinica)
Title: Perverse pullbacks for (-1)-shifted symplectic fibrations
Abstract: I will discuss a new type of pullback operation on perverse sheaves, which is defined for morphisms of complex algebraic stacks equipped with a certain extra structure, namely that of a so-called relative exact (-1)-shifted symplectic fibration. These perverse pullbacks are closely related to classical operations such as vanishing cycles functors and the Fourier-Sato transform. They also vastly generalize the theory of Donaldson-Thomas invariants of Calabi-Yau threefolds, and we will sketch how they lead to a proof of a conjecture of Joyce about their functoriality. Time permitting, I will attempt to briefly survey some applications of these constructions to topics like cohomological Hall algebras, curve counting on Landau-Ginzburg models, and the period sheaves for Hamiltonian G-spaces conjectured by Ben-Zvi, Sakellaridis, and Venkatesh. This is a report on joint work in progress with Tasuki Kinjo, Hyeonjun Park, and Pavel Safronov.
September 1: Dave
Title: Some computations with categorical local Langlands
Handout with examples
September 8: Dai Wenhan (NUS)
Title: Stalks of automorphic Vogan sheaves for the Steinberg parameter of GLn
Abstract: Under categorical local Langlands, certain natural coherent sheaves on Vogan stacks give rise to matching automorphic sheaves on Bun_G, which are generally hard to understand. Understanding the stalks of these automorphic sheaves at points of Bun_G would crucially provide a mechanism for the explicit matching of objects along the categorical local Langlands functor. Despite its importance, the general behavior of these stalks remains largely mysterious. In this talk we will study these sheaves for G=GLn at the Vogan stack of the Steinberg parameter. Some recent conjectures predict that at basic points, these stalks are given by generalized Steinberg representations of inner forms of G, up to degree shifts. After formulating the main conjecture, I will report some progress towards its proof in a special case on the degree-zero component of Bun_G, achieved through combinatorial computations inspired by Hansen. If time permits, I will also present an application of the result to cohomology of local Shimura varieties.
Handout
September 29: Yifei Zhao (University of Münster)
Title: On an observation of Weissman
Abstract: Around 10 years ago, Weissman constructed the local Langlands correspondence (LLC) for covers of split tori defined by
Brylinski-Deligne extensions. An intriguing phenomenon is that the LLC
map is not surjective in general. The goal of my talk is to relate
this phenomenon to the existence of "extended pure inner forms" of
covers. This is based on joint work with Luozi Shi.
October 13: Si Ying Lee (NUS)
Title: Stacks of p-isogenies with G-structure
Abstract: I will talk about constructing integral models for Hecke correspondences in both the global and local settings, using the theory of F-gauges. Using these integral models, I will explain how one can construct integral Hecke actions on coherent cohomology for compact abelian type Shimura varieties, proving a conjecture of Fakhruddin-Pilloni. This is joint work in progress with Keerthi Madapusi.
October 27: Joao Lourenco (Universite Sorbonne Paris Nord)
Title: Modular ramified Satake
Abstract: The geometric Satake equivalence of Mirkovic-Vilonen identifies the monoidal category of perverse sheaves with l-adic coefficients on the Hecke stack of a split reductive group G over a Laurent series fields with the monoidal category of representations of the Langlands dual of G. However, from the point of view of the Langlands program, it is also natural to consider ramified groups and their special parahoric models. With rational l-adic coefficients, ramified Satake was proved by Zhu and Richarz, exploiting semisimplicity of the category of representations. We extend these equivalences to integral and modular l-adic coefficients, via a careful study of semi-infinite orbit filtrations that allow us to bypass the lack of fusion, and a delicate analysis of fixed point group schemes. This is joint work with Achar, Richarz and Riche.
November 3: Joao Lourenco (Universite Sorbonne Paris Nord)
Title: Mod p sheaves on Witt flags
Abstract: Given a reductive group G over a p-adic field F with parahoric group model mathcal{G} over the ring of integers O, we can consider its associated flag variety using Witt vectors on perfect k-algebras: this is representable by a perfect ind-scheme by a deep theorem of Bhatt-Scholze. This time, we want to study its F_p-etale cohomology, but its behavior is linked with coherent cohomology due to the Artin-Schreier sequence. We are going to explain how the Cohen-Macaulayness of a perfect scheme relates to its perverse F_p-sheaves. Then, we will explain a new and uniform proof of this property for Schubert varieties in the minuscule range based on inversion of adjunction. This is joint work with Robert Cass.
November 10: Jhan-Syuan Cyu (NUS)
Title: Multiplicities of perverse sheaves in categorical local Langlands program
Abstract: Let G be a split connected reductive group over a non-archimedean local field. The categorical local Langlands conjecture (CLLC) predicts an equivalence between lisse-etale sheaves on Bun_G, the moduli stack of G-bundles on relative Fargues-Fontaine curves, and ind-coherent sheaves on Par_G, the moduli stack of l-adically continuous L-parameters. One question is to find t-structures on the automorphic side and the spectral side such that the categorical equivalence is t-exact. On the automorphic side, there is a perverse t-structure. However, the story is still mysterious on the spectral side. After localizing at certain types of L-parameters, we can say more about t-structures. In this talk, we will localize at the trivial L-parameter and compute the multiplicity of certain perverse sheaves with the help of CLLC.
December 8: Yuta Takaya (University of Tokyo)
Title: Relative Langlands duality in the categorical local Langlands correspondence
Abstract: The relation between period integrals and L-functions, studied since the work of Hecke and Tate, has recently been reformulated by Ben--Zvi-Sakellaridis-Venkatesh as relative Langlands duality. In this talk, I will present a version of this duality, the normalized period conjecture, in the framework of the categorical local Langlands correspondence a la Fargues-Scholze. I will describe computations for the Iwasawa-Tate and Hecke periods that support this conjecture, and discuss how it leads to a relation between distinguished representations and distinguished L-parameters. This talk is based on joint work with Milton Lin.
December 15: Yuta Takaya (University of Tokyo)
Room: S17-0511
Title: Special affinoids around unramified CM points of depth zero
Abstract: In the local Langlands program, both local Shimura varieties and type theory play important roles. Although the relationship between these two approaches remains largely open, starting from the work of Yoshida and Boyarchenko-Weinstein, it has been observed that certain affinoids around CM points in Lubin-Tate spaces reflect the associated type theory. In this talk, I will introduce analogous affinoids around unramified CM points in general local Shimura varieties, and explain the relation to regular depth-zero supercuspidal representations. If time permits, I will provide an outlook on the extension to positive depth.