General Information: This is the webpage for the Number Theory Research Seminar organized jointly by POSTECH and the PMI.
Organizers: Valentin Buciumas and Qirui Li.
Time and Place: The Number Theory Seminar will take place on Zoom, typically either on Thursday 6:00pm-7:00pm or on Friday 9am-10am. All times above and below are Korean Standard Time.
| Date | Time | Speaker | Affiliation | Title | Zoom Link |
| October 17 | 6:00pm | Yifei Zhao | University of Münster | The local Langlands correspondence for covering groups | zoom |
| Abstract: The local Langlands program for covering groups (due to M. Weissman, W.-T. Gan, F. Gao) is a conjectural parametrization of genuine smooth representations of (non-algebraic) covers of $p$-adic reductive groups by $L$-parameters. In this talk, I will explain how to construct a semisimplified version of this correspondence when the degree of the cover is coprime t o $p$. It is the shadow of a categorical spectral action for covering groups. This is joint work with T. Feng, I. Gaisin, N. Imai, and T. Koshikawa. | |||||
| October 24-27 | KMS meeting | NO SEMINAR | |||
| Abstract: | |||||
| November 1 | 9:00am | Zhiyu Zhang | Stanford University | Orbital integrals, linear algebra and Hecke algebra | zoom |
| Abstract: Orbital integrals are key numerical objects used (via fundamental lemmas) to establish Langlands functoriality for automorphic forms, e.g. endoscopic classification and base change. Explicit computations are often related to interesting linear algebra problems, e.g. given two matrices $A$, $B$ in $\operatorname{GL}_2(\mathbb{Q}_p)$, how to determine all index m such that $A^{-m}BA^{m}$ is in $\operatorname{GL}_2(\mathbb{Z}_p)$. I will report a new Jacquet-Rallis type fundamental lemma for $\operatorname{GL}_n$ to $\operatorname{GL}_n \times \operatorname{GL}_{n+1}$ (any n) on orbital integrals of Hecke functions, with explicit computations for $\operatorname{GL}_1$. Joint work with Griffin Wang. | |||||
| November 8 | 9:00am | Spencer Leslie | Boston College | Relative Langlands and endoscopy | zoom |
| Abstract: Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre-)stabilization of relative trace formulas. | |||||
| November 22 | 9:00am | Zhilin Luo | University of Chicago | Nonabelian Fourier kernels on $\operatorname{SL}_2$ and $\operatorname{GL}_2$ | zoom |
| Abstract: I will present joint work with B.C. Ngô unveiling an explicit formula for nonabelian Fourier kernels in $G=\operatorname{SL}_2$ and $\operatorname{GL}_2$ that is conjectured by A. Braverman and D. Kazhdan. I will also explain the connection with the orbital Hankel transform suggested by Ngô. If time permits, I will discuss a conjectural formula of the Lafforgue transform, a new ingredient introduced in our paper, for $G=\operatorname{GL}_n$. | |||||
| November 28 | 6:00pm | Satoshi Wakatsuki | Kanazawa University | Asymptotic behavior for traces of Hecke operators | zoom |
| Abstract: In this talk, I will discuss the asymptotic behavior of traces of Hecke operators on spaces of cusp forms. First I summarize asymptotic formulas for one variable holomorphic cusp forms and their applications. In particular, I clarify the relations between remainder terms, orbital integrals, and applications. Next, after summarizing general results, I will present our asymptotic formula and its proof for holomorphic Siegel cusp forms of general degree. Finally, we discuss future problems and their relations with orbital integrals. This is a joint work with Shingo Sugiyama and Masao Tsuzuki. | |||||
| December 6 | 10:30am | Qiao He | Columbia University | Intersection of modular correspondence with levels | zoom |
| Abstract: Modular correspondence is a classical object that contains many interesting and important arithmetic information. For example, the geometry of modular correspondence can yield the celebrating Hurwitz class formula. An arithmetic version of this was considered in the classical work of Gross-Keating, where the intersection of modular correspondence was related with certain purely analytic objects. In this talk, we will talk about a generalization of this from the hyperspecial level to vertex parahoric level. If time permits, we will also mention how to replace the modular correspondence so that we can formulate similar formulas for other GSpin Shimura varieties with vertex parahoric level. This is based on joint work with Baiqing Zhu. | |||||
| Date | Time | Speaker | Affiliation | Title | Zoom Link |
| March 28 | 5:30pm | Mikhail Borovoi | Tel-Aviv University | Galois cohomology of reductive groups over global fields | zoom |
| Abstract:
Let $F$ be a number field (say, the field of rational numbers $\mathbb{Q}$) or a $p$-adic field (say, the field of $p$-adic numbers $\mathbb{Q}_p$), or a global function field (say, the field of rational functions of one variable over a finite field $\mathbb{F}_q$).
Let $G$ be a connected reductive group over $F$ (say, $\operatorname{SO}(n)$).
One needs the first Galois cohomology set $H^1(F,G)$ for classification problems in algebraic geometry and linear algebra over $F$.
In the talk, I will give closed formulas for $H^1(F,G)$ when $F$ is as above, in terms of the algebraic fundamental group $\pi_1(G)$ introduced by the speaker in 1998.
All terms will be defined and examples will be given.
The talk is based on a joint work with Tasho Kaletha: arXiv:2303.04120. |
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| April 19 | 9:00am | Wei Zhang | MIT | $p$-adic Heights of the arithmetic diagonal cycles on unitary Shimura varieties | zoom |
| Abstract: We formulate a $p$-adic analogue of the Arithmetic Gan-Gross-Prasad Conjectures for unitary groups, relating the $p$-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a $p$-adic Rankin-Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramification are mild at inert primes, using recent progress on the arithmetic fundamental lemma and arithmetic transfer conjectures. We deduce some application to $p$-adic version of the Bloch-Kato conjecture. Joint work with Daniel Disegni. | |||||
| April 25 | 5:30pm | Chia-Fu Yu | Academia Sinica | When is a polarized abelian variety determined by its $p$-divisible group? | zoom |
| Abstract: In this talk I give a survey on some problems concerning supersingular abelian varieties and a few recent progress. In particular we shall address a few results on automorphism groups, endomorphism algebras, masses, our solution to the title, as well as results of Chai and Oort on central leaves and Newton strata. This talk is based on the joint papers with Tomoyoshi Ibukiyama, Valentijn Karemaker and Fuetaro Yobuko. | |||||
| May 2 | 5:30pm | Andreas Mihatsch | Universität Bonn | Generating series of complex multiplication cycles | zoom |
| Abstract: A classical result of Zagier states that the degrees of Heegner divisors on the modular curve form the positive Fourier coefficients of a modular form. In my talk, I will define complex multiplication cycles on the Siegel modular variety and show that their degrees have a similar modularity property. I will also explain their link with orbital integrals. This is based on joint work with Lucas Gerth, Tonghai Yang and Siddarth Sankaran. | |||||
| May 17 | 9:30am | Yanze Chen | University of Alberta | Whittaker coefficients of metaplectic Eisenstein series and multiple Dirichlet series. | zoom |
| Abstract: We investigate the Whittaker coefficients of an Eisenstein series on a global metaplectic cover of a semisimple algebraic group induced from the Borel subgroup and establish the relation with Weyl group multiple Dirichlet series. | |||||
| May 24 | 9:30am | Ben Brubaker | University of Minnesota | Solvable Lattice Models in Number Theory and Geometry | zoom |
| Abstract: We describe from first principles how lattice models may be used to represent special functions (predominantly polynomial functions in several variables) that arise naturally in the study of algebraic groups, and hence in related number theory and enumerative geometry. Examples include formulas for Whittaker functions arising in automorphic forms (the subject of the seminar on May 17) and polynomial representatives for the cohomology of flag varieties ("generalized Schubert calculus"). We will explain how lattice model connections advance our understanding of these special functions in two very different ways - in suggesting new identities and in making new connections to quantum groups and their modules. This is based on multiple joint works with numerous collaborators, including Buciumas, Bump, Gustafsson, and my current PhD student Dasher. No prior knowledge of lattice models will be assumed in the lecture. | |||||