Peking Online International Number Theory Seminar (POINTS) - from 2020 to 2022

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Introduction

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University (SMS-PKU).

The seminars usually take place on Wednesday evenings (UTC +8), presumably via Zoom. By default, the talks will be given in English. The official announcements of our talks will appear on the websites of BICMR and/or SMS-PKU. The talks will also be announced on researchseminars.org.

The videos, if available, can also be found on the channel of BICMR on Bilibili.com.

Organizers

• Yiwen Ding
• Wen-Wei Li
• Ruochuan Liu
• Zhiyu Tian
• Liang Xiao
• Enlin Yang
• Xinyi Yuan

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2022

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Date/Time	Speaker	Title	Abstract & Materials
November 30, 16:00-17:00 (UTC+8)	Christophe Breuil (CNRS - Orsay)	Multivariable (phi, Gamma)-modules and modular representations of Galois and GL2	Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbf{Q}_p$, and $\pi$ a smooth representation of $\mathrm{GL}_2(K)$ on some Hecke eigenspace in the $H^1$ mod $p$ of a Shimura curve. One can associate to $\pi$ a multivariable $(\phi, O_K^*)$-module $D_A(\pi)$. I will state a conjecture which describes $D_A(\pi)$ in terms of the underlying 2-dimensional mod $p$ representation of $\mathrm{Gal}(\bar{K}/K)$. When the latter is semi-simple (sufficiently generic), I will sketch a proof of this conjecture. This is joint work with F. Herzig, Y. Hu, S. Morra and B. Schraen.
November 2, 13:30 (UTC+8)	Emanuel Scheidegger (BICMR)	Aspects of modularity for Calabi-Yau threefolds	We give an overview of some mostly conjectural aspects of modularity for Calabi-Yau threefolds. We focus on one parameter families of hypergeometric type and give computational results in terms of classical modular forms. In one case we show an explicit correspondence.
October 26, 15:00 (UTC+8)	Haoyang Guo (MPIM)	Prismatic approach to Fontaine's C_crys conjecture	Given a smooth proper scheme over a $p$-adic ring of integers, Fontaine's $C_{\mathrm{crys}}$ conjecture says that the étale cohomology of its generic fiber is isomorphic to the crystalline cohomology of its special fiber, after base changing them to the crystalline period ring. In this talk, we give a prismatic proof of the conjecture, for general coefficients, in the relative setting, and allowing ramified base rings. This is a joint work with Emanuel Reinecke.
October 19, 13:30 (UTC+8)	Xinwen Zhu (Stanford University)	The p-adic Borel hyperbolicity of A_g	A theorem of Borel says that any holomorphic map from a smooth complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the punctured disc to the arithmetic variety has no essential singularity. I will discuss some work towards a p-adic analogue of this theorem for Shimura varieties of Hodge type. Joint with Abhishek Oswal and Ananth Shankar.
September 21, 13:30 (UTC+8)	Zhiyou Wu (BICMR)	S=T for Shimura varieties	I will explain how the new $p$-adic geometry developed by Scholze can help prove the Eichler-Shimura relation for Shimura varieties of Hodge type, which has nothing to do with $p$-adic geometry a priori.
September 14, 15:00 (UTC+8)	Arthur-César Le Bras (CNRS - IRMA Strasbourg)	A Fourier transform for Banach-Colmez spaces	I will explain how to define an $\ell$-adic Fourier transform for Banach-Colmez spaces and discuss some examples. This is a joint work with Anschütz, which was motivated by the study of Fargues' geometrization conjecture for $\mathrm{GL}_n$.
August 30, 09:30-11:00 (UTC+8)	Lue Pan (Princeton University)	Regular de Rham Galois representations in the completed cohomology of modular curves	Let $p$ be a prime. I want to explain how to use the geometry of modular curves at infinite level and Hodge-Tate period map to study $p$-adic regular de Rham Galois representations appearing in the $p$-adically completed cohomology of modular curves. We will show that these Galois representations up to twists come from modular forms and give a geometric description of the locally analytic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ associated to them. These results were previously known by totally different methods.
August 24, 14:00-15:00 (UTC+8)	Masatoshi Kitagawa (Waseda University)	Uniformly bounded multiplicities in the branching problem and D-modules	In the representation theory of real reductive Lie groups, several finiteness results of lengths and multiplicities are known and fundamental. The Harish-Chandra admissibility theorem and the finiteness of the length of Verma modules and principal series representations are typical examples. More precisely, such multiplicities and lengths are bounded on some parameter sets. T. Oshima and T. Kobayashi ('13 adv. math.) gave a criterion on which branching laws have (uniformly) bounded multiplicities. In arXiv:2109.05556, I defined uniform boundedness of a family of $\mathcal{D}$-modules (and $\mathfrak{g}$-modules) to treat the boundedness properties uniformly. I will talk about its definition and applications. In particular, I will give a necessary and sufficient condition on uniform boundedness of multiplicities in the branching problem of real reductive Lie groups.
June 1, 19:30-20:30 (UTC+8)	Cheng Chen (University of Minnesota)	The local Gross-Prasad conjecture over archimedean local fields	The local Gross-Prasad conjecture is a refinement of the multiplicity one theorem for spherical pairs of Bessel type defined by a pair of special orthogonal groups. The conjecture shows that there is exactly one representation having multiplicity equal to one in each Vogan packet (with generic parameter) and it also depicts this unique representation with an epsilon character. I will introduce some recent progress for the conjecture over $\mathbb{R}$ and $\mathbb{C}$, part of the work was joint with Z. Luo. This local conjecture is a necessary ingredient for the global Gross-Prasad conjecture. Besides, the codimension-one case of the conjecture is closely related to the branching problem for special orthogonal groups.
May 18, 15:00-16:30 (UTC+8)	Jan Frahm (Aarhus Universitet)	Analytic continuation of branching laws for unitary representations	Branching problems ask for the behaviour of the restriction of an irreducible representation of a group $G$ to a subgroup $H$. In the context of smooth representations of real reductive groups, this typically leads to the study of multiplicities with which an irreducible representation of $H$ occurs as a quotient of an irreducible representation of $G$. Here, both quantitative results such as multiplicity-one theorems and qualitative results such as the Gan-Gross-Prasad conjectures are of interest. In the context of unitary representations of real reductive groups, one can go a step further and explicitly decompose an irreducible representation of $G$ into a direct integral of irreducible representations of $H$. I will explain how branching laws for unitary representations are related to those in the smooth category, and how one can use an analytic continuation procedure along a principal series parameter to obtain explicit branching laws from certain Plancherel formulas for homogeneous spaces.
May 11, 15:00 (UTC+8)	Anna Romanov (University of New South Wales)	A Soergel bimodule approach to the character theory of real groups	Admissible representations of real reductive groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80's in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of a block of the LV module using Soergel bimodules.
April 27, 15:30 (UTC+8)	Hongjie Yu (IST Austria)	Number of irreducible representations in the cuspidal automorphic spectrum	Let $G$ be a reductive group defined and split over a global function field. We are interested in the sum of multiplicities of irreducible representations containing a regular depth zero representation of $G(O)$, where $O$ is the ring of integral adeles, in the automorphic cuspidal spectrum. The sum is expressed in terms of the number of $\mathbb{F}_q$-points of Hitchin moduli spaces of groups associated to $G$. When $G=\mathrm{GL}(n)$, it implies some cases of Deligne's conjecture by Langlands correspondence.
April 20, 15:30 (UTC+8)	Haoyang Guo (Max-Planck-Institut für Mathematik)	Prismatic approach to crystalline local systems	Let $X$ be a smooth proper scheme over a $p$-adic field such that $X$ has a good reduction. Inspired by the de Rham comparison theorem in complex geometry, Grothendieck asked if there is a "mysterious functor" relating étale cohomology of the generic fiber and crystalline cohomology of the special fiber. This question was answered by work of many people, including Fontaine and Faltings. In particular, this motivates the definition of a $p$-adic Galois representation being crystalline, generalizing the étale cohomology of $X$ as above. In this talk, we will give an overview for the prismatic approach of Bhatt-Scholze on crystalline representations. Moreover, jointly with Emanuel Reinecke, we will consider the higher dimensional generalization of this approach on crystalline local systems.
April 15, 10:30-12:00 (UTC+8)	Chung Pang Mok (Soochow University)	Pseudorandom Vectors Generation Using Elliptic Curves And Applications to Wiener Processes	Using the arithmetic of elliptic curves over finite fields, we present an algorithm for the efficient generation of sequence of uniform pseudorandom vectors in high dimension with long period, that simulates sample sequence of a sequence of independent identically distributed random variables, with values in the hypercube $[0,1]^d$ with uniform distribution. As an application, we obtain, in the discrete time simulation, an efficient algorithm to simulate, uniformly distributed sample path sequence of a sequence of independent standard Wiener processes. arXiv: 2201.00357
April 13, 15:30 (UTC+8)	Ben Heuer (Universität Bonn)	Moduli spaces in p-adic non-abelian Hodge theory	In analogy to Simpson's non-abelian Hodge theory over the complex numbers, the p-adic Simpson correspondence over non-archimedean fields like $C_p$ aims to relate p-adic representations of the étale fundamental group of a smooth proper rigid space $X$ to Higgs bundles on $X$. In this talk, I will introduce p-adic moduli spaces for either side of the correspondence, and explain how these can be compared by way of a non-abelian generalisation of the Hodge-Tate sequence. This allows one to construct new geometric incarnations of the p-adic Simpson correspondence, and to interpret the choices necessary for its formulation in a geometric fashion.
April 6, 10:30-11:30 (UTC+8)	Sandeep Varma (Tata Institute of Fundamental Research)	On residues of certain intertwining operators	Let $G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $P = MN$ be a Levi decomposition of a maximal parabolic subgroup of $G$, and $\pi$ an irreducible unitary supercuspidal representation of $M(F)$. One can then consider the representation $\mathrm{Ind}_{P(F)}^{G(F)} \pi$ (normalized parabolic induction). Assume that $P$ is conjugate to an opposite by an element $w_0 \in G(F)$ that normalizes $M$, and which fixes the isomorphism class of $\pi$ (i.e., $\pi \cong \,{}^{w_0}\pi$). Then, by the work of Harish-Chandra, $\mathrm{Ind}_{P(F)}^{G(F)} \pi$ is irreducible if and only if a certain family $A(s, \pi, w_0)$ of so called intertwining operators has a pole at $s = 0$. In this case, after making certain choices, the residue of $A(s, \pi, w_0)$ at $s = 0$ can be captured by a scalar $R(\tilde \pi) \in \mathbb{C}$, which has a conjectural expression in terms of some gamma factors related to Shahidi's local coefficients, as described by Arthur's local intertwining relation. Following a program pioneered by Freydoon Shahidi, and furthered by him as well as David Goldberg, Steven Spallone, Wen-Wei Li, Li Cai, Bin Xu, Xiaoxiang Yu etc., one seeks to: (a) get explicit expressions to describe $R(\tilde \pi)$; and (b) interpret the resulting expression for $R(\tilde \pi)$ suitably, using the theory of endoscopy when applicable. So far, these questions have been studied mostly for classical (including unitary) groups, or in some simple situations. We will discuss (a) above in a non-classical and slightly "less simple" situation, in the cases where $G$ is an almost simple group whose absolute root system is of exceptional type or of type $B_n$ with $n \geq 3$ or $D_n$ with $n \geq 4$, and where $P$ is a "Heisenberg parabolic subgroup". We will then comment on what we can say of (b) above in the $G_2$, $B_3$ and $D_4$ cases. Though the reducibility results and the $R(\tilde \pi)$ values are more or less already known in these cases by the Langlands-Shahidi method and related results (e.g., the work of Henniart and Lomeli and Caihua Luo in the case of $D_4$), our investigations also suggest the existence of harmonic analytic expressions for certain gamma values, which in some cases just amount to the formal degree conjecture of Ichino, Ikeda and Hiraga, but in other cases seem slightly unwieldy and perhaps intriguing.
March 30, 10:00 (UTC+8)	Alexander Petrov (Harvard University)	On arithmetic characterization of local systems of geometric origin	I will talk about the problem of classifying local systems of geometric origin on algebraic varieties over complex numbers. Conjecture: For a smooth algebraic variety $S$ over a finitely generated field $F$, a semi-simple $\mathbb{Q}_l$-local system on $S_{\bar{F}}$ is of geometric origin if and only if it extends to a local system on $S_{F'}$ for a finite extension $F' \supset F$. My main goal will be to provide motivation for this conjecture arising from the Fontaine-Mazur conjecture, and survey known results and related problems.
March 16, 10:30-11:30 (UTC+8)	Shih-Yu Chen (Academia Sinica)	On Deligne's conjecture for critical values of tensor product L-functions and symmetric power L-functions of modular forms	In this talk, we introduce our result on the algebraicity of ratios of product of critical values of Rankin--Selberg $L$-functions and its applications. More precisely, let $\mathit{\Sigma,\Sigma'}$ (resp. $\mathit{\Pi,\Pi'}$) be cohomological tamely isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ (resp. $\mathrm{GL}_{n'}(\mathbb{A})$) such that $\mathit{\Sigma}_\infty = \mathit{\Sigma}_\infty'$ and $\mathit{\Pi}_\infty = \mathit{\Pi}_\infty'$. It is a consequence of Deligne's conjecture on critical $L$-values that the ratio $\frac{L(s, \mathit{\Sigma} \times \mathit{\Pi}) \cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi}')}{L(s,\mathit{\Sigma} \times \mathit{\Pi}')\cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi})}$ is algebraic and Galois-equivariant at critical points. We show that this assertion holds under certain parity and regularity conditions. As applications, we prove Deligne's conjecture for some tensor product $L$-functions and symmetric odd power $L$-functions for $\mathrm{GL}_2$.
March 14, 10:30-11:30 (UTC+8)	Shih-Yu Chen (Academia Sinica)	Algebraicity of critical values of automorphic L-functions: Examples and Conjectures	In this talk, we introduce some algebraicity results on the critical values of automorphic $L$-functions. The techniques in these examples are integral representation of automorphic $L$-functions, constant terms of Eisenstein series, and their cohomological interpretations. These results are compatible with Clozel's conjecture on existence of motives associated to algebraic cuspidal automorphic representations of general linear groups and Deligne's conjecture on algebraicity of critical values of motivic $L$-functions.
March 3, 10:30-12:00 (UTC+8)	Yasuhiro Terakado (NCTS)	Mass formula on the basic loci of unitary Shimura varieties	We study a mass of the group of self-quasi-isogenies of the abelian variety corresponding to a point on the basic locus in the reduction modulo p of a $\mathrm{GU}(r,s)$ Shimura variety. We give explicit formulas for the number of irreducible components of the basic locus, and for the cardinality of the zero-dimensional Ekedahl-Oort stratum, in a Shimura variety associated with a unimodular Hermitian lattice. On the way, we also give a formula for the number of connected components of a Shimura variety. This is joint work with Chia-Fu Yu.
January 13, 2022, 09:00-10:00 (UTC+8)	Liyang Yang (Princeton University)	The Jacquet-Zagier Trace Formula for GL(n)	The so-called Jacquet-Zagier trace formula was established by Jacquet and Zagier for GL(2) for two main reasons: deducing the holomorphy of adjoint L-functions and generalizing Selberg's trace formula in a different way from Arthur's truncation process. In this talk we will describe Jacquet-Zagier's trace formula in higher ranks. It plays a role in the study of holomorphic continuation of automorphic L-functions and certain Artin L-functions.
January 6, 2022, 10:00-11:00 (UTC+8)	Chung-Ru Lee (Duke University)	Endoscopic Relative Orbital Integrals on a Unitary Group	The characterization of distinguished representations is crucial for studying automorphic representations. The celebrated conjectures of Sakellaridis and Venkatesh provide such a characterization in many cases. In particular, they provide a conjectural description of the representations of a split reductive group that are distinguished by a split reductive spherical subgroup. However, there remain many mysteries when the generic stabilizer is disconnected. The comparison of relative trace formulae, initially suggested by Jacquet, has been one of the most effective ways to study distinction problems in automorphic representation theory. Stabilization is a pivotal step for the comparison of relative trace formulae. To prepare for stabilization, one needs to investigate the endoscopic relative orbital integrals. In this talk, we study the endoscopy theory for unitary groups in a relative setting where the generic stabilizer is disconnected and finite over a p-adic field. This talk aims to compute an explicit formula for endoscopic relative orbital integrals.

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2021

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Date/Time	Speaker	Title	Abstract & Materials
December 2, 2021, 08:50-09:50 (UTC+8)	Weibo Fu (Princeton University)	A derived construction of eigenvarieties	We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of p-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes associated to the locally symmetric spaces of a quasi-split reductive group $\mathbb{G}$, comparison to overconvergent cohomology, proving exactness of finite slope part functor, together with some representation-theoretic statements. As a global application, we exhibit an eigenvariety coming from data of $\mathrm{GL}_n$ over a CM field as a subeigenvariety for a quasi-split unitary group.
June 10, 2021, 10:00-11:00 (UTC+8)	Jacob Tsimerman (University of Toronto)	Abelian varieties not isogeneous to Jacobians - in arbitrary characteristic	(Joint w/ Ananth Shankar) We prove that over an arbitrary global field, for every $g>3$ there exists an abelian variety which is not isogenous to a Jacobian.
June 2, 2021, 10:00-11:00 (UTC+8)	Zicheng Qian (Toronto University)	Moduli of Fontaine-Laffaille modules and mod p local-global compatibility	We introduce a set of invariant functions on the moduli of Fontaine-Laffaille modules and prove that they separate points on the moduli in a suitable sense. Consequently, we prove the following local-global compatibility result for suitable global set up and under standard Kisin-Taylor-Wiles conditions: the Hecke eigenspace attached to a modular mod $p$ global Galois representation determines its restriction at a place unramified over $p$, if the restriction is Fontaine-Laffaille and has a generic semisimplification. The genericity assumption is mild and explicit. This is a joint work with D. Le, B.V. Le Hung, S. Morra and C. Park.
May 21, 2021, 10:00-11:00 (UTC+8)	Jinbo Ren (University of Virginia)	Some applications of Diophantine Approximation to Group theory	Transcendental Number Theory tells us an essential difference between transcendental numbers and algebraic numbers is that the former can be approximated by rational numbers "very well" but not the latter. More specifically, one has the following Fields Medal work by Roth. Given a real algebraic number $a$ of degree $\geq 3$ and any $\delta \gt 0$, there is a constant $c = c(a, \delta) \gt 0$ such that for any rational number $\eta$, we have $\lvert \eta-a \rvert \gt c H(\eta)^{-\delta}$, where $H(\eta)$ is the height of $\eta$. Moreover, we have Schmidt's Subspace theorem, a non-trivial generalization of Roth's theorem. On the other hand, we have the notion of Bounded Generation in Group Theory. An abstract group $\Gamma$ is called Boundedly Generated if there exist $g_1,g_2,\cdots, g_r\in \Gamma$ such that $\Gamma = \langle g_1\rangle \cdots \langle g_r\rangle$ where $\langle g\rangle$ is the cyclic group generated by $g$. While being a purely combinatorial property of groups, bounded generation has a number of interesting consequences and applications in different areas. For example, bounded generation has close relation with Serre's Congruence Subgroup Problem and Margulis-Zimmer conjecture. In my recent joint work with Corvaja, Rapinchuk and Zannier, we applied an "algebraic geometric" version of Roth and Schmidt's theorems, i.e. Laurent's theorem, to prove a series of results about when a group is boundedly generated. In particular, we have shown that a finitely generated anisotropic linear group over a field of characteristic zero has bounded generation if and only if it is virtually abelian, i.e. contains an abelian subgroup of finite index. In my talk, I will explain the idea of this proof and give certain open questions.
April 7, 2021, 15:00-16:00 (UTC+8)	Zhixiang Wu (Université Paris-Saclay)	Companion forms and partially classical eigenvarieties	In general, there exist $p$-adic automorphic forms of different weights with the same associated $p$-adic Galois representation. The existence of these companion forms is also predicted by Breuil's locally analytic socle conjecture in the $p$-adic local Langlands program. Under the Taylor-Wiles assumption, Breuil-Hellmann-Schraen proved the existence of all companion forms when the associated crystalline Galois representations have regular Hodge-Tate weights. In this talk, I will explain how to generalize their results to some cases when the Hodge-Tate weights are not necessarily regular. The method relies on Ding's construction of partially classical eigenvarieties and their relationships with some spaces of Galois representations.
January 21, 2021, 15:00-16:00 (UTC+8)	Hiroshi Ishimoto (Kyoto University)	A proof of Ibukiyama's conjecture on Siegel modular forms of half-integral weight and of degree 2	In 2006, Ibukiyama conjectured that there is a linear isomorphism between a space of Siegel cusp forms of degree $2$ of integral weight and that of half-integral weight. With Arthur's multiplicity formula on the odd special orthogonal group $\mathrm{SO}(5)$ and Gan-Ichino's multiplicity formula on the metaplectic group $\mathrm{Mp}(4)$, Ibukiyama's conjecture can be proven in a representation theoretic way.

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2020

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Date/Time	Speaker	Title	Abstract & Materials
December 23, 2020, 11:00-12:00 (UTC+8)	Ziquan Yang (Harvard University)	Finiteness and the Tate Conjecture in Codimension 2 for K3 Squares	Two years ago, via a refined CM lifting theory, Ito-Ito-Koshikawa proved the Tate Conjecture for squares of K3 surfaces over finite fields by reducing to Tate's theorem on the endomorphisms of abelian varieties. I will explain a different proof, which is based on a twisted version of Fourier-Mukai transforms between K3 surfaces. In particular, I do not use Tate's theorem after assuming some known properties of individual K3's. The main purpose of doing so is to illustrate Tate's insight on the connection between the Tate conjecture and the positivity results in algebraic geometry for codimension 2 cycles, through some "geometry in cohomological degree 2".
November 25, 2020, 14:30-15:30 (UTC+8)	Ariyan Javanpeykar (Johannes Gutenberg-Universität Mainz)	Hilbert's irreducibility theorem for abelian varieties	We will discuss joint work with Corvaja, Demeio, Lombardo, and Zannier in which we extend Hilbert's irreducibility theorem (for rational varieties) to the setting of abelian varieties. Roughly speaking, given an abelian variety $A$ over a number field $k$ and a ramified covering $X$ of $A$, we show that $X$ has "less" $k$-rational points than $A$.
October 21, 2020, 16:00-17:00 (UTC+8)	Jinhe Ye (MSRI)	Lovely pairs of valued fields and adic spaces	Hrushovski and Loeser used the space $\widehat{V}$ of generically stable types concentrating on $V$ to study the topology of Berkovich analytification $V^{an}$ of $V$. In this talk we will give a brief introduction to this object and present an alternative approach, based on lovely pairs of valued fields, to study various analytifications using model theory. We will provide a model-theoretic counterpart $\widetilde{V}$ of the Huber's analytification of $V$. We show that, the same as for $\widehat{V}$, the space $\widetilde{V}$ is strict pro-definable. Furthermore, we will discuss canonical liftings of the deformation retraction developed by Hrushovski and Loeser. This is a joint project with Pablo Cubides-Kovacsics and Martin Hils. Venue: Science Building 1, room 1303, Peking University (Yanyuan campus)
August 26, 2020, 10:30-11:30 (UTC+8)	Yuanqing Cai (Kyoto University)	Doubling integrals for Brylinski-Deligne extensions of classical groups	In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula. Recently, a family of global integrals that represent the tensor product $L$-functions for classical groups (joint with Friedberg, Ginzburg, and Kaplan) and the tensor product $L$-functions for covers of symplectic groups (Kaplan) was discovered. These can be viewed as generalizations of the doubling method. In this talk, we explain how to develop the doubling integrals for Brylinski-Deligne extensions of all connected classical groups. This gives a family of Eulerian global integrals for this class of non-linear extensions.
August 13, 2020, 15:00-16:00 (UTC+8)	Kei Yuen Chan (Shanghai Center for Mathematical Sciences)	Gan-Gross-Prasad conjectures for general linear groups	In this talk, I will talk about restriction problems of general linear groups over local and global fields, surrounding Gan-Gross-Prasad conjectures. In particular, I will discuss a local conjecture on predicting the branching laws of the non-tempered representations arisen from Arthur packets and my recent work on a proof of the conjecture. Along the way, I will also discuss some significant properties of restrictions such as multiplicity one, Ext-vanishing, projectivity and indecomposability.
July 22, 2020, 16:00-17:00 (UTC+8)	Esmail Arasteh Rad (Universität Münster)	Local models for moduli of global and local shtukas	Moduli spaces for global $G$-shtukas appear as function fields analogs for Shimura varieties. This can be observed for example through Langlands philosophy. They possess local counterparts which are called Rapoport-Zink spaces for local $P$-shtukas which similarly arise as function fields analogs for Rapoport-Zink spaces for $p$-divisible groups. In this talk we first recall the construction of these moduli stacks (spaces), and after providing some preliminary backgrounds, we discuss the theory of local models for them. If time permits we also discuss some of the applications.
July 8, 2020, 09:00-10:00 (UTC+8)	Xiaolei Wan (National University of Singapore)	Examples related to the Sakellaridis-Venkatesh conjecture	In this talk, I will introduce the Sakellaridis-Venkatesh conjecture on the decomposition of global period, and give examples related to this conjecture. More specifically, the cases $X = \mathrm{SO}(n-1) \backslash \mathrm{SO}(n)$ and $X = \mathrm{U}(2) \backslash \mathrm{SO}(5)$. In both cases, I will determine the Plancherel decompositions of $L^2(X_v)$, where $v$ is a local place. Then I will prove the local relative character identity. In the global setting, I will give the factorization of the global period of $X = \mathrm{SO}(n-1) \backslash \mathrm{SO}(n)$, where the local functional comes from the local Plancherel decomposition. The example $X = \mathrm{U}(2) \backslash \mathrm{SO}(5)$ is slightly beyond the SV conjecture but we still have a decomposition of the global period as the sum of two factorizable elements.
July 1, 2020, 16:00-17:00 (UTC+8)	Jun Su (Cambridge University)	Arithmetic group cohomology: coefficients and automorphy	Cohomology of arithmetic subgroups, with coefficients being algebraic representations of the corresponding reductive group, has played an important role in the construction of Langlands correspondence. Traditionally the first step to access these objects is to view them as cohomology of (locally constant) sheaves on locally symmetric spaces and hence connect them with spaces of functions. However, sometimes infinite dimensional coefficients also naturally arise, e.g. when you try to attach elliptic curves to weight 2 eigenforms on $\mathrm{GL}_2$ / an imaginary cubic field, and the sheaf theoretic viewpoint might no longer be fruitful. In this talk we'll explain a different but very simple understanding of the connection between arithmetic group cohomology (with finite dimensional coefficients) and function spaces, and discuss the application of this idea to infinite dimensional coefficients.
June 24, 2020, 09:30-10:30 (UTC+8)	Andrea Dotto (University of Chicago)	Mod p Bernstein centres of p-adic groups	The centre of the category of smooth mod $p$ representations of a $p$-adic reductive group does not distinguish the blocks of finite length representations, in contrast with Bernstein's theory in characteristic zero. Motivated by this observation and the known connections between the Bernstein centre and the local Langlands correspondence in families, we consider the case of $\mathrm{GL}_2(\mathbb{Q}_p)$ and we prove that its category of representations extends to a stack on the Zariski site of a simple geometric object: a chain $X$ of projective lines, whose points are in bijection with Paskunas's blocks. Taking the centre over each open subset we obtain a sheaf of rings on $X$, and we expect the resulting space to be closely related to the Emerton-Gee stack for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$. Joint work in progress with Matthew Emerton and Toby Gee.
June 17, 2020, 15:00-16:00 (UTC+8)	Miyu Suzuki (Kanazawa University)	Prehomogeneous zeta functions and toric periods for inner forms of GL(2)	I will explain a new application of prehomogeneous zeta functions to non-vanishing of periods of automorphic forms. The zeta functions we use were first introduced by F. Sato and a general theory is developed by the recent work of Wen-Wei Li. They can be used to show non-vanishing of infinitely many toric periods of cuspidal representations of inner forms of $\mathrm{GL}(2)$. If time permits, I will mention future works based on the local theory of Wen-Wei Li. This is a joint work with Satoshi Wakatsuki.
June 10, 2020, 18:00-19:00 (UTC+8)	Xiaoyu Zhang (Universität Duisburg-Essen)	p-adic family of modular forms on GSpin Shimura varieties	The theory of $p$-adic interpolation of modular forms on the upper half plane started with Serre for Eisenstein series and then was developed by Hida for ordinary cuspidal modular forms. This theory plays an important role in the construction of $p$-adic $L$-functions, modularity theorems, etc. In this talk, I will generalize this theory to modular forms on $\mathrm{GSpin}$ Shimura varieties. In such cases, the ordinary locus may be empty and we need to work with the $\mu$-ordinary locus. Then we follow Hida's idea to construct $p$-adic families of modular forms and give the control theorem on the dimension of the space of such $p$-adic families.
June 3, 2020, 09:30-10:30 (UTC+8)	Linhui Shen (Michigan State University)	Quantum geometry of moduli spaces of local systems	Let $G$ be a split semi-simple algebraic group over $\mathbb{Q}$. We introduce a natural cluster structure on moduli spaces of $G$-local systems over surfaces with marked points. As a consequence, the moduli spaces of $G$-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the $q$-deformed Toda systems, quantum groups, and the modular functor conjecture for such representations. This talk will mainly be based on joint work with A.B. Goncharov.
May 27, 2020, 09:30-10:30 (UTC+8)	Xiyuan Wang (Johns Hopkins University)	The Tate conjecture for a concrete family of elliptic surfaces	We prove the Tate conjecture for a concrete family of elliptic surfaces. This is a joint work with Lian Duan. In this talk, I will begin with an general introduction to the Tate conjecture and the Fontaine-Mazur conjecture. Then I will focus on the Tate conjecture for a family of elliptic surfaces introduced by Geemen and Top, and try to explain the motivation and elementary idea behind the proof.
May 20, 2020, 13:00-14:00 (UTC+8)	Huy Dang (University of Virginia)	Hurwitz trees and deformations of Artin-Schreier covers	In this talk, we introduce the notion of Hurwitz tree for an Artin-Schreier deformation (deformation of $\mathbf{Z}/p$-covers in characteristic $p>0$). It is a combinatorial-differential object that is endowed with essential degeneration data, measured by Kato's refined Swan conductors, of the deformation. We then show how the existence of a deformation between two covers with different branching data (e.g., different number of branch points) equates to the presence of a Hurwitz tree with behaviors determined by the branching data. One application of this result is to prove that the moduli space of Artin-Schreier covers of fixed genus $g$ is connected when $g$ is sufficiently large. If time permits, we will discuss a generalization of the Hurwitz tree technique to all cyclic covers and beyond.
May 13, 2020, 18:00-19:00 (UTC+8)	James Newton (King's College London)	Symmetric power functoriality for modular forms	Langlands functoriality predicts the transfer of automorphic representations along maps of L-groups. In particular, the symmetric power representation $\mathrm{Symm}^{n-1}$ of $\mathrm{GL}(2)$ should give rise to a lifting from automorphic representations of $\mathrm{GL}(2)$ to automorphic representations of $\mathrm{GL}(n)$. I will discuss joint work with Jack Thorne, in which we prove the existence of all symmetric power lifts for many cuspidal Hecke eigenforms (for example, those of square-free level).
May 6, 2020, 17:00-18:00 (UTC+8)	Timo Richarz (TU Darmstadt)	Applications of Néron blowups to integral models of moduli stacks of shtukas	I will explain how Néron blowups as defined in Arnaud's talk encode level structures on moduli stacks of bundles. This can be used to define integral models of moduli stacks of shtukas with level structures. In the case of parahoric group schemes, these models are analogues of the integral models of Shimura varieties defined by Kisin-Pappas. If time permits, I indicate some problems arising outside the case of parahoric level structures.
May 6, 2020, 16:00-17:00 (UTC+8)	Arnaud Mayeux (BICMR)	Dilatations and Néron blowups (with Timo Richarz and Matthieu Romagny)	In this talk, I will introduce general dilatations for schemes, extending Bosch-Lütkebohmert-Raynaud dilatations. In the case of group schemes, dilatations are called Néron blowups. Dilatations and Néron blowups enjoy many properties that I will state. We will see later in this seminar some applications of this theory (level structures, moduli of shtukas).