# 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.

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

(TBA)

## Past talks

### 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