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

## 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

Please click on the Article Index or "Next" button to see the talks.

### Multivariable (phi, Gamma)-modules and modular representations of Galois and GL2

**Speaker:** Christophe Breuil (CNRS - Orsay)

**Time:** 16:00-17:00, November 30 (UTC+8)

**Abstract:** 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.

### Aspects of modularity for Calabi-Yau threefolds

**Speaker:** Emanuel Scheidegger (BICMR)

**Time:** 13:30, November 2 (UTC+8)

**Abstract:** 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.

### Prismatic approach to Fontaine's C_crys conjecture

**Speaker:** Haoyang Guo (MPIM)

**Time:** 15:00, October 26 (UTC+8)

**Abstract:** 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.

### The p-adic Borel hyperbolicity of A_g

**Speaker:** Xinwen Zhu (Stanford University)

**Time:** 13:30, October 19 (UTC+8)

**Abstract:** 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.

### S=T for Shimura varieties

**Speaker:** Zhiyou Wu (BICMR)

**Time:** 13:30, September 21 (UTC+8)

**Abstract:** 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.

### A Fourier transform for Banach-Colmez spaces

**Speaker:** Arthur-César Le Bras (CNRS - IRMA Strasbourg)

**Time:** 15:00, September 14 (UTC+8)

**Abstract:** 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 \(\mathrm{GL}_n\).

*Regular de Rham Galois representations in the completed cohomology of modular curves*

**Speaker:** Lue Pan (Princeton University)

**Time:** 09:30-11:00, August 30 (UTC+8)

**Abstract:** 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.

*Uniformly bounded multiplicities in the branching problem and D-modules*

**Speaker:** Masatoshi Kitagawa (Waseda University)

**Time:** 14:00-15:00, August 24 (UTC+8)

**Abstract:** 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. Slides

*The local Gross-Prasad conjecture over archimedean local fields*

**Speaker:** Cheng Chen (University of Minnesota)

**Time:** 19:30-20:30, June 1 (UTC+8)

**Abstract:** 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. Slides Video

*Analytic continuation of branching laws for unitary representations*

**Speaker:** Jan Frahm (Aarhus Universitet)

**Time:** 15:00-16:30, May 18 (UTC+8)

**Abstract:** 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. Slides Video

*A Soergel bimodule approach to the character theory of real groups*

**Speaker:** Anna Romanov (University of New South Wales)

**Time:** 15:00, May 11 (UTC+8)

**Abstract:** 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.

*Number of irreducible representations in the cuspidal automorphic spectrum*

**Speaker:** Hongjie Yu (IST Austria)

**Time:** 15:30 April 27 (UTC+8)

**Abstract:** 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 \( G=GL(n) \), it implies some cases of Deligne's conjecture by Langlands correspondence. Video

*Prismatic approach to crystalline local systems*

**Speaker:** Haoyang Guo (Max-Planck-Institut für Mathematik)

**Time:** 15:30 April 20 (UTC+8)

**Abstract:** 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.

*Pseudorandom Vectors Generation Using Elliptic Curves And Applications to Wiener Processes*

**Speaker:** Chung Pang Mok (Soochow University)

**Time:** 10:30-12:00 April 15 (UTC+8)

**Abstract:** 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. Video

**arXiv:** 2201.00357

*Moduli spaces in p-adic non-abelian Hodge theory*

**Speaker:** Ben Heuer (Universität Bonn)

**Time:** 15:30 April 13 (UTC+8)

**Abstract:** 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.

*On residues of certain intertwining operators*

**Speaker:** Sandeep Varma (Tata Institute of Fundamental Research)

**Time:** 10:30-11:30 April 6 (UTC+8)

**Abstract:** 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. Slides Video

*On arithmetic characterization of local systems of geometric origin*

**Speaker:** Alexander Petrov (Harvard University)

**Time:** 10:00 March 30 (UTC+8)

**Abstract:** 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. Video

*On Deligne's conjecture for critical values of tensor product L-functions and symmetric power L-functions of modular forms*

**Speaker:** Shih-Yu Chen (Academia Sinica)

**Time:** 10:30-11:30 March 16 (UTC+8)

**Abstract:** 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 \( \mathrm{GL}_2 \). Video

*Algebraicity of critical values of automorphic L-functions: Examples and Conjectures*

**Speaker:** Shih-Yu Chen (Academia Sinica)

**Time:** 10:30-11:30 March 14 (UTC+8)

**Abstract:** 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. Video

*Mass formula on the basic loci of unitary Shimura varieties*

**Speaker:** Yasuhiro Terakado (NCTS)

**Time:** 10:30-12:00 March 3 (UTC+8)

**Abstract:** 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. Video

*The Jacquet-Zagier Trace Formula for GL(n)*

**Speaker:** Liyang Yang (Princeton University)

**Time:** 09:00-10:00 January 13, 2022 (UTC+8)

**Abstract:** 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'strace formula in higher ranks. It plays a role in the study of holomorphic continuation of automorphic L-functions and certain Artin L-functions. Slides Video

*Endoscopic Relative Orbital Integrals on a Unitary Group*

**Speaker:** Chung-Ru Lee (Duke University)

**Time:** 10:00-11:00 January 6, 2022 (UTC+8)

**Abstract:** 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. Slides Video

*A derived construction of eigenvarieties*

**Speaker:** Weibo Fu (Princeton University)

**Time:** 08:50 - 09:50 December 2, 2021 (UTC+8)

**Abstract:** 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 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.

*Abelian varieties not isogeneous to Jacobians - in arbitrary characteristic*

**Speaker:** Jacob Tsimerman (University of Toronto)

**Time:** 10:00 - 11:00 June 10, 2021 (UTC+8)

**Abstract:** (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.

*Moduli of Fontaine-Laffaille modules and mod p local-global compatibility*

**Speaker:** Zicheng Qian (Toronto University)

**Time:** 10:00 - 11:00 June 2, 2021 (UTC+8)

**Abstract:** 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-lobal 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.

*Some applications of Diophantine Approximation to Group theory*

**Speaker:** Jinbo Ren (University of Virginia)

**Time:** 10:00 - 11:00, May 21, 2021 (UTC+8)

**Abstract:** 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 \(\delta > 0\), there is a constant \(c = c(a, \delta) > 0\) such that for any rational number \(\eta\), we have \(|\eta-a|>c H(\eta)^{-\delta}\), 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.

*Companion forms and partially classical eigenvarieties*

**Speaker:** Zhixiang Wu (Université Paris-Saclay)

**Time:** 15:00-16:00, April 7, 2021 (UTC+8)

**Abstract:** 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.

*A proof of Ibukiyama's conjecture on Siegel modular forms of half-integral weight and of degree 2*

**Speaker:** Hiroshi Ishimoto (Kyoto University)

**Time:** 15:00-16:00, January 21, 2021 (UTC+8)

**Abstract:** 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. Slides Video

*Finiteness and the Tate Conjecture in Codimension 2 for K3 Squares*

**Speaker:** Ziquan Yang (Harvard University)

**Time:** 11:00-12:00, December 23, 2020 (UTC+8)

**Abstract:** 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".

*Hilbert's irreducibility theorem for abelian varieties*

**Speaker:** Ariyan Javanpeykar (Johannes Gutenberg-Universität Mainz)

**Time:** 14:30-15:30, November 25, 2020 (UTC+8)

**Abstract:** 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 \(A\). Video

*Lovely pairs of valued fields and adic spaces*

**Speaker:** Jinhe Ye (MSRI)

**Time:** 16:00-17:00, October 21, 2020 (UTC+8)

**Abstract:** 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. Slides

**Venue:** Science Building 1, room 1303, Peking University (Yanyuan campus)

*Doubling integrals for Brylinski-Deligne extensions of classical groups*

**Speaker:** Yuanqing Cai (Kyoto University)

**Time:** 10:30-11:30 August 26, 2020 (UTC +8)

**Abstract:** 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. Slides Video

*Gan-Gross-Prasad conjectures for general linear groups*

**Speaker:** Kei Yuen Chan (Shanghai Center for Mathematical Sciences)

**Time:** 15:00-16:00 August 13, 2020 (UTC +8)

**Abstract:** 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. Slides Video

*Local models for moduli of global and local shtukas*

**Speaker:** Esmail Arasteh Rad (Universität Münster)

**Time:** 16:00-17:00 July 22, 2020 (UTC +8)

**Abstract:** 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. Slides

*Examples related to the Sakellaridis-Venkatesh conjecture*

**Speaker:** Xiaolei Wan (National University of Singapore)

**Time:** 09:00-10:00 July 8, 2020 (UTC +8)

**Abstract:** 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. Slides

*Arithmetic group cohomology: coefficients and automorphy*

**Speaker:** Jun Su (Cambridge University)

**Time:** 16:00-17:00 July 1, 2020 (UTC +8)

**Abstract:** 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.

*Mod p Bernstein centres of p-adic groups*

**Speaker:** Andrea Dotto (University of Chicago)

**Time:** 09:30-10:30 June 24, 2020 (UTC +8)

Abstract: 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.

*Prehomogeneous zeta functions and toric periods for inner forms of GL(2)*

**Speaker:** Miyu Suzuki (Kanazawa University)

**Time:** 15:00-16:00 June 17, 2020 (UTC +8)

**Abstract:** 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. Slides Video

*p-adic family of modular forms on GSpin Shimura varieties*

**Speaker:** Xiaoyu Zhang (Universität Duisburg-Essen)

**Time:** 18:00-19:00 June 10, 2020 (UTC +8)

**Abstract:** 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. Slides

*Quantum geometry of moduli spaces of local systems*

**Speaker:** Linhui Shen (Michigan State University)

**Time:** 09:30-10:30 June 3, 2020 (UTC +8)

**Abstract:** 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. Slides

*The Tate conjecture for a concrete family of elliptic surfaces*

**Speaker:** Xiyuan Wang (Johns Hopkins University)

**Time:** 09:30-10:30 May 27, 2020 (UTC +8)

**Abstract:** 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.

*Hurwitz trees and deformations of Artin-Schreier covers*

**Speaker:** Huy Dang (University of Virginia)

**Time:** 13:00-14:00 May 20, 2020 (UTC +8)

**Abstract:** 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.

*Symmetric power functoriality for modular forms*

**Speaker:** James Newton (King's College London)

**Time:** 18:00-19:00 May 13, 2020 (UTC +8)

**Abstract:** 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).

*Applications of Néron blowups to integral models of moduli stacks of shtukas*

**Speaker:** Timo Richarz (TU Darmstadt)

**Time:** 17:00-18:00 May 6, 2020 (UTC +8)

**Abstract:** 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.

*Dilatations and Néron blowups (with Timo Richarz and Matthieu Romagny)*

**Speaker:** Arnaud Mayeux (BICMR)

**Time:** 16:00-17:00 May 6, 2020 (UTC +8)

**Abstract:** 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). Slides