# Peking Online International Number Theory Seminar (POINTS) - 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 slides or videos, if available, can be found on the page of *Past talks*. You can also have a look of the channel of BICMR on Bilibili.com.

If you are willing to give a talk, please feel free to contact any one of the organizers below. Their E-mail addresses can be easily found on the websites of BICMR or SMS-PKU.

## Organizers

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

## Upcoming talks

To be scheduled.

## Past talks

*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