- Conference Summary
- Participants
- Schedule
- Title & Abstract

Arithmetic and Topology

Dec 16,2022 - Dec 17,2022

This online conference is hosted by the Department of Mathematics and the Shenzhen International Center for Mathematics at SUSTech. We intend to bring experts in both Number Theory and Topology to share recent advances in these related fields, and to foster interdisciplinary communication and collaboration.

Stavros Garoufalidis,Shenzhen International Center for Mathematics

Hui Gao, Southern University of Science and Technology

Yifei Zhu, Southern University of Science and Technology

- Weiyan Chen, Tsinghua University
- Xing Gu, Westlake University
- Yiqin He, Peking University
- Yongquan Hu, Chinese Academy of Sciences
- Zhen Huan, Huazhong University of Science and Technology
- Zicheng Qian, Chinese Academy of Sciences
- Guozhen Wang, Fudan University
- Haoran Wang, Tsinghua University
- Seokbeom Yoon, Southern University of Science and Technology
- Ningchuan Zhang, University of Pennsylvania
- Bin Zhao, Capital Normal University

All talks will be China Standard Time (GMT+8)

8:50–9:00 am

**Opening remarks**

9:00–9:50 am

**Guozhen Wang**

10:00–10:50 am

**Weiyan Chen**

11:00–11:50 am

**Zhen Huan**

2:00–2:50 pm

**Yongquan Hu**

3:00–3:50 pm

**Haoran Wang**

4:00–4:50 pm

**Bin Zhao**

10:00–10:50 am

**Ningchuan Zhang**

11:00–11:50 am

**Xing Gu**

2:00–2:50 pm

**Seokbeom Yoon**

3:00–3:50 pm

**Zicheng Qian**

4:00–4:50 pm

**Yiqin He**

**Guozhen Wang**

**(TBA)**

**Weiyan Chen**

**(TBA)**

**Twisted Real quasi-elliptic cohomology**

**Zhen Huan**

Quasi-elliptic cohomology is closely related to Tate *K*-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories. It can be interpreted by orbifold loop spaces and expressed in terms of equivariant *K*-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate *K*-theory of symmetric groups modulo a certain transfer ideal.

In this talk we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct Real analogues of the string power operation of quasi-elliptic cohomology. We also explore the relation of the theory to the Tate curve. This is joint work with Matthew Young.

**Multivariable ( φ, O_{K}^{x})-modules and mod p representations of GL_{2}**

**Yongquan Hu**

Let *p* be a prime number, *K* a finite unramified extension of ℚ_{p}, and *π* a smooth representation of GL_{2}(*K*) on some Hecke eigenspace in mod *p* cohomology of Shimura curves. One can associate to *π* a multivariable (*φ*, *O*_{K}^{x})-module. The aim of this talk is to explain the construction and some recent results around it. This is joint work with C. Breuil, F. Herzig, S. Morra and B. Schraen.

**On some mod p representations of quaternion algebra over ℚ_{p}**

**Haoran Wang**

Scholze proposed a mod *p* Jacquet–Langlands correspondence for GL(*n*, *K*), where *K* is a finite extension of ℚ_{p}. I will discuss some results about Scholze's functors in the case of GL(2, ℚ_{p}). This is a joint work with Yongquan Hu.

**Slopes of modular form and ghost conjecture**

**Bin Zhao**

In 2016, Bergdall and Pollack raised a conjecture towards the computation of the *p*- adic slopes of Hecke cuspidal eigenforms whose associated *p*-adic Galois representations satisfy the assumption that their mod *p* reductions become reducible when restricted to the *p*-decomposition group. In this talk, I will report a joint work with Ruochuan Liu, Nha Truong and Liang Xiao to prove this conjecture under mild assumptions. I will first give the statement of this conjecture and explain the intuition behind its formulation. I will then explain some key strategies in our proof. If time permits, I will mention some arithmetic consequences of this conjecture.

**A Quillen–Lichtenbaum Conjecture for Dirichlet L-functions**

**Ningchuan Zhang**

The original version of the Quillen–Lichtenbaum Conjecture, proved by Voevodsky and Rost, connects special values of the Dedekind zeta function of a number field with its algebraic *K*-groups. In this talk, I will discuss a generalization of this conjecture to Dirichlet *L*-functions. The key idea is to twist algebraic *K*-theory spectra of number fields with equivariant Moore spectra associated to Dirichlet characters. Rationally, we obtain a Quillen–Borel type theorem for Artin *L*-functions. This is joint work in progress with Elden Elmanto.

**Xing Gu**

**(TBA)**

**Seokbeom Yoon**

**(TBA)**

**On Breuil–Schraen ℒ-invariants for GL _{n}**

**Zicheng Qian**

The study of Breuil–Schraen ℒ-invariants is motivated by that of Steinberg case of *p*-adic Langlands correspondence. Many results are known for GL(2, ℚ_{p}) and GL(3, ℚ_{p}) due to work of Breuil, Ding and Schraen. In this talk, we sketch a few recent progress towards general GL_{n}.

**Parabolic simple ℒ-invariants and local-global compatibility**

**Yiqin He**

Let *L* be a finite extension of ℚ_{p} and *ρ*_{L} be a potentially semistable noncrystalline *p*-adic representation of Gal_{L} such that the associated *F*-semisimple Weil–Deligne representation is absolutely indecomposable. Via a study of Breuil's parabolic simple ℒ-invariants, we attach to *ρ*_{L} a locally ℚ_{p}-analytic representation Π(*ρ*_{L}) of GL_{n}(*L*), which carries the exact information of the Fontaine–Mazur parabolic simple ℒ-invariants of *ρ*_{L}. When *ρ*_{L} comes from a patched automorphic representation of *G*(𝔸_{F+}) (for a unitary group *G* over a totally real field *F*^{+} which is compact at infinite places and GL_{n} at *p*-adic places), we prove under mild hypothesis that Π(*ρ*_{L}) is a subrepresentation of the associated Hecke-isotypic subspace of the Banach spaces of (patched) *p*-adic automophic forms on *G*(𝔸_{F+}).