The 27th Autumn Workshop on Number Theory 

• Jungtaek Hong: Global Geometrization of Local Smooth Integral Models in the Hitchin Fibration for $\mathrm{GL}_n$: The Bass Case

Abstract: Smoothening is a new method, proposed by Cho, Kang, and Lee, to study the local orbital integrals for $\mathrm{GL}_n$ (where $n=3$ or in the Bass case). The method relies on stratifying the scheme of matrices with fixed characteristic polynomial over a complete DVR into a disjoint union of smooth strata, according to a datum called a local type. In this talk,I will outline the geometrization of this local smoothening method within the framework of the global Hitchin fibration for $\mathrm{GL}_n$ in the case regarding Bass orders , by defining the global type and providing the analogous stratification. I will also explain the geometric meaning of the stratification, and provide a closed formula for the $\ell$-adic cohomology of the compactified Jacobian of a spectral curve over a finite field with double singularities whose local rings are integral domains. 

• Yeansu Kim: Aubert Duals of Strongly Positive Representations of Metaplectic Groups

Abstract: The Aubert involution plays an important role in the representation theory of p-adic groups and the study of their unitary duals. I will briefly review this duality and present joint work with Gyujin Oh on the Aubert duals of strongly positive representations of metaplectic groups over non-Archimedean local fields of characteristic different from two. Our results extend known descriptions for classical groups to the metaplectic setting and provide analogous descriptions for odd general spin groups.

• Heejong Lee: Serre weight conjectures and modularity lifting for GSp4

Abstract: Serre weight conjectures predict congruences between automorphic forms in terms of mod p Galois representations. During the first half of the 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. SL2(Fp), GSp4(Fp)) and the geometry of p-adic local Galois representations. This part will focus on examples of modular forms and associated Galois representations, keeping the exposition elementary. During the second half of the talk, I will explain the proof of Serre weight conjectures for GSp4 and how to deduce modularity lifting by combining Serre weight conjectures with Breuil-Mezard conjecture (a la Feng-Le Hung-Lin) and certain weight shifting mod p differential operators on GSp4 Shimura varieties (a la Ortiz). This is based on a joint work with Daniel Le and Bao V. Le Hung.

• Yuchan Lee: Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$: The $L$-function of a Cubic Order and Its Functional Equation

Abstract:  Beyond Endoscopy, proposed by Langlands, is a strategy toward establishing the general principle of functoriality. A key problem in this program is to isolate the contribution of the trivial representation from the regular elliptic part of the Arthur-Selberg trace formula. For $\mathrm{GL}_2(\mathbb{Q})$, Altuğ carried out such an isolation using the approximate functional equation and Poisson summation.  More recently, Deng and Espinosa developed an analogous approach for $\mathrm{GL}_3(\mathbb{Q})$, in which the analytic properties of an $L$-function attached to a cubic order play a central role.  

In this talk, I will introduce this $L$-function and explain how it arises in the Beyond Endoscopy framework for $\mathrm{GL}_3$. I will then present an independent proof of its functional equation and discuss its consequences for the isolation of the trivial representation.

• Colin Jia Sheng Loh: Period integrals of distinguished polarised strongly tempered hyperspherical varieties 

Abstract:  One of the central problems in the theory of automorphic forms is the construction of integral representations of automorphic L-functions. The framework of relative Langlands duality, proposed by Ben-Zvi, Sakellaridis and Venkatesh, provides a conceptual approach to relating such integral representations to hyperspherical varieties. Recent work of Mao, Wan and Zhang gives a complete list of strongly tempered hyperspherical varieties and proposes new period integrals associated with them. In this talk, I will present new period integrals arising from distinguished polarised strongly tempered hyperspherical varieties, discuss the L-functions represented by these integrals, and explain their relations with some previously studied Rankin-Selberg integrals.

• Andrei Seymour-Howell: Explicit trace formulae for automorphic forms over number fields

Abstract: Arthur's work on the trace formula is one of the pivotal results in the study of automorphic forms of the past few decades. Whilst his formula is valid for very general groups, deriving explicit versions of the trace formula in some settings can yield more precise results, such as finding the second order terms in the Weyl law and numerical computations.  

In this talk, I shall discuss the background and results of two separate but related projects. Firstly, in joint work with Seiji Kuga and Satoshi Wakatsuki, we derive a trace formula for Hilbert modular forms over real quadratic number fields and derive results related to generalised Hurwitz class numbers. Secondly, in separate ongoing work with Andrew Knightly and Min Lee, we derive an explicit trace formula of the Hecke operator of Maass cusp forms over arbitrary number field that are unramified at all finite places.

• Nobuki Takeda: Harder’s Conjecture and Hermitian Automorphic Forms

Abstract: Harder’s conjecture predicts congruences between elliptic modular forms and degree-two Siegel modular forms. In this talk, I will explain an approach to this conjecture using Hermitian modular forms on U(2,2). We first obtain, from the divisibility of a critical L-value, a congruence between a Hermitian Klingen-Eisenstein lift and a Hermitian cusp form. We then study the Galois representation associated with this cusp form and show, under explicit arithmetic assumptions, that it comes from a degree-two Siegel cusp form satisfying the congruence predicted by Harder. This gives new cases of Harder’s conjecture, including cases for both even and odd k. This talk is based on joint work with Hidenori Katsurada.

•  Joven Jia Xuan Teo: Closure ordering of nilpotent orbits in p-adic unitary groups

Abstract: The nilpotent orbits of a classical group defined over a local field of characteristic zero admit a partition-type classification. In the Archimedean case, it is known that the closure ordering of nilpotent orbits can be described combinatorially in terms of these signed Young diagrams. In this talk, we discuss how a similar description can be obtained for unitary groups in the nonarchimedean case.

• Satoshi Wakatsuki: On the absolute convergence of the Arthur trace formula

Abstract: In this talk, I will survey work of Finis, Lapid, and Mueller on the absolute convergence of the Arthur trace formula.

• Yiyang Wang: D4/G2: some new results

Abstract: Among spherical pairs for simple groups, there are (F4, Spin9) and (Spin8, G2). The relative functoriality for the first is related to the exceptional theta correspondence for PGL(2) x F4 in (adjoint) E7,  while the latter is related to the  exceptional theta correspondence for SL(2)^3/mu_2 x Spin8 in (adjoint) E7. It turns out that the relative aspects of  the PGL(2) x F4 theta correspondence have been studied thoroughly by  Karasiewicz-Savin (local), Le-Wang (local) and Y. Shan (global) in recent years, but the theta correspondence for SL(2)^3/mu_2 x Spin8 has been left untouched for some time.  I will try to explain some of my recent work on this model, focusing more on local results (which seem to be more interesting) than global results.