### Morningside Seminar on Number Theory

##### 2021-09-14 10:00-12:00

Speaker: Dr.  Zhiyu Zhang (MIT)

Time: 10:00-12:00 September 14, 2021

Place: MCM110 & Zoom (Zoom ID: 4120194771   Password: mcm1234)

Title: Maximal parahoric arithemtic transfers and modularity of the arithmetic theta series

Abstract: The modularity of classical theta series of a positive definite i.e sign (n,0) quadratic lattice is well-known and has many applications. For nice (i.e parahoric level) hermitian lattices L of sign (n-1,1), the arithmetic theta series of L is Kudla's generating series of special cycles \Theta_L:=\sum_m Z(m) q^m on the parahoric integral model M of unitary Shimura varieties for L. In this talk, for all nice L we show the conjectured modularity of \Theta_L in arithmetic Chow group when intersecting with any 1-cycle C in M, under certain assumptions e.g M is proper and the CM extension F/F_0 is unramified. Moreover, we show a double modularity result relating "local Fourier transforms" of \Theta_L, Z(m) and M.

To attack the AGGP conjecture for M which generalizes the Gross-Zagier formula for Shimura curves to higher dimensions, we need the arithmetic fundamental lemma (AFL) at good places and arithmetic transfer conjecutre (ATC) at bad places. We will give a natural formulation of ATCs for all maximal parahoric levels at unramified bad places, via a small resolution of the singularity of M x M (with a moduli interpretation) using the (refined) Balloon-Ground stratification on the special fiber. The blow up may be thought as an arihtmetic analog of the Atiyah flop. As an application of the double modularity, we prove the ATCs over Q_p for p>2 via truncated globalizations and the double induction method, inspired by the fundamental proof of AFL for p>=n by W. Zhang in 2019. In particular, we establish the AFL for all p>2.

Speaker: Prof. Hao Chen (Jinan University)

Time: 16:00-17:30  September 13, 2021

Place: MCM610 & Zoom (Zoom ID: 4120194771   Password: mcm1234)

Title: Hamming 纠错编码和Insertion Deletion纠错编码

Abstract:  在经典的Hamming纠错编码理论中，各种构造和上界结果已经比较完整。Insertion deletion纠错编码自1965年提出以来，一些基本问题和上界进展很慢，Haeupler-Shahrasbi在2017年提出的算法性构造，Insertion deletion编码理论取得突破性进展。本报告介绍Hamming纠错编码的一些经典结果和密码学应用，并且介绍Insertion deletion编码的Haeupler-Shahrasbi突破性结果，Insertion deletion编码一些新的上界，达到上界的最优Insertion deletion编码的算法性构造结果。

Speaker: Dr. Yousheng Shi (University of Wisconsin-Madison)

Time: 14:00-15:00  July 22, 2021(Beijing time)

Place: MCM110 & Zoom (Zoom ID: 4120194771   Password: mcm1234)

Title: Kudla Rapoport conjecture over the ramified primes

Abstract: Kudla-Rapoport conjecture compares the intersection number of special cycles on unitary Rapoport-Zink spaces associated to a quadratic extension of local fields with local densities of Hermitian forms. Over the good primes the conjecture was solved by Li and Zhang recently. Over the bad primes the conjecture needs modification.  In this talk, I will review the global motivation of the conjecture, introduce a precise version of it over the ramified primes and then talk about verification of the conjecture in some cases. This is based on the joint work with Qiao He and Tonghai Yang.

Speaker: Dr. Jinbo Ren (University of Virginia)

Time: 9:00-10:00, June 10, 2021 (Beijing time)
Place: ZOOM (Zoom ID: 412 019 4771  Password: mcm1234)
Title: Some applications of Diophantine Approximation in Group theory
Abstract: Transcendental Number Theory tells us algebraic numbers cannot be approximated by rational numbers “very well”.  More specifically, one has the following Fields Medal work by Roth. Given an algebraic number $a$ of degree $\geq 3$ and any $\delta>2$, 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 Subspace theorem by Schmidt and the $p$-adic version by Schlickewei, which are non-trivial generalizations 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 $\g_1,g_2,\dots, g_r\in \Gamma$ 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 the Margulis-Zimmer conjecture.
In my recent joint work with Corvaja, Rapinchuk and Zannier, we applied an “algebraic geometric” version of Subspace Theorem, 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 this talk, I will explain the idea of the proof and give certain open problems.

Speaker: Dr. Yanshuai Qin (UC Berkeley)
Time: 16:30-17:30, June 11, 2021 (Beijing time)
Place: MCM110 & ZOOM (Zoom ID: 412 019 4771  Password: mcm1234)
Title: A relation between Brauer groups and Tate-Shafarevich groups for high dimensional fibrations
Abstract: Let $\mathcal{X} \rightarrow C$ be a dominant morphism between smooth geometrically connected varieties over a finitely generated field such that the generic fiber $X/K$ is smooth, projective and geometrically connected. We prove a relation between the Tate-Shafarevich group of $Pic^0_{X/K}$ and the geometric Brauer groups of $\mathcal{X}$, $X$ and $C$, generalizing a theorem of Artin and Grothendieck for fibered surfaces to arbitrary relative dimensions.