# Peking Online International Number Theory Seminar (POINTS) - from 2020 to 2021

## 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.

*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