2021 Zhuhai Algebraic Number Theory Workshop
 



We will host the 2021 Zhuhai ANT workshop online from August 24th to 27th, 2021.

The organiser and principal contact is Chao Qin.

Talks can be watched through Tencent Meeting ID: 617 6316 4303
You can also enter the meeting room by clicking here.

The workshop is open to everyone and no registration is necessary.


Programme

Aug 24th 9:00--12:00 Shenxing Zhang

On the generating fields of Kloosterman sums

14:00--17:00 Di Zhang

On the non-vanishing of theta lifts of Bianchi modular forms to Siegel modular forms

Aug 25th 9:00--12:00 Yihua Chen

Heegner Points on Modular Curves

14:00--17:00 Zeping Hao

$p$-adic $L$-functions in universal deformation families

Aug 26th 9:00--12:00 Jianing Li

Revisit Kida's formula in Iwasawa theory

14:00--17:00 Yongxiong Li 

Quadratic twists of an elliptic curve

Aug 27th 9:00--12:00 Shuai Zhai

A lower bound result for the $2$-part of the Birch-Swinnerton-Dyer exact formula

14:00--17:00 Meng-Fai Lim 

On order of vanishing of characteristics elements


Title and Abstract

 


Yihua Chen -- University of Science and Technology of China

Title: Heegner Points on Modular Curves
Abstract: In this talk, I will introduce the Heegner points on modular curves, and how to use them to construct the quadratic twist families of Mordell-Weil rank $0$ and $1$, and some result of BSD formula.



Zeping Hao -- Warwick University

Title: $p$-adic $L$-functions in universal deformation families
Abstract: In this talk, we construct a two-variable $p$-adic Rankin-Selberg $L$- function associated to a pair of modular forms, and discuss its interpolation property. The classical theory of such $p$-adic $L$-functions have been studied extensively by Hida in the late 20th century. Recently there has been substantial progress towards different perspectives of this construction. For example, David Loeffler has extended the classical setting (ordinary Hida families) to a more ”arithmetic” one by allowing the second modular form to vary in universal deformation families, at the expense of restricting tame level to be $1$. In this talk I will explain how to relax this tame level condtion, under certain simplifying assumptions imposed solely on the deformation problem.



Jianing Li -- University of Science and Technology of China

Title: Revisit Kida's formula in Iwasawa theory
Abstract: Kida's formula is about the change of $\lambda$-invariants of certain Iwasawa modules, such as $S$-ramified abelian extension, Selmer groups of elliptic curves and so on, in a Galois $p$-extension of $\mathbb{Z}_p$-fields, under the vanishing of $\mu$-invariants (in general). Here a $\mathbb{Z}_p$-field means that it is a finite extension of a cyclotomic (or a Coates-Wiles) $\mathbb{Z}_p$-extension. Some results obtained before assume that the prime $p$ is odd. We will also talk about some miscellany about the case $p = 2$.



Yongxiong Li -- Tsinghua University

Title: Quadratic twists of an elliptic curve
Abstract: In this talk, we prove the $2$-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve $X_0(14)$, using an explicit form of the Waldspurger formula. This is joint work with Junhwa Choi.



Meng-Fai Lim -- Central Normal University

Title: On order of vanishing of characteristics elements
Abstract: Let $E$ be an elliptic curve with either good ordinary or split multiplicative reduction at each prime above $p$. We study the characteristic element of the Selmer group of such an elliptic curve over a $p$-adic Lie extension. IN particular, we relate the order of vanishing of the characteristic element at certain twist to the Selmer ranks in the intermediate subextension of the said $p$-adic Lie extension.



Shuai Zhai -- Shandong University

Title: A lower bound result for the $2$-part of the Birch–Swinnerton-Dyer exact formula
Abstract: In this lecture, we will present a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over the rationals, which shows that when the $2$-part of Tamagawa factors is growing, the $2$-part of the algebraic central $L$-value is growing as well, coinciding with the Birch– Swinnerton-Dyer exact formula.



Di Zhang -- Hubei University of Arts and Science

Title: On the non-vanishing of theta lifts of Bianchi modular forms to Siegel modular forms
Abstract: In this talk, we study the theta lifting of a weight $2$ Bianchi modular form $F$ of level $\Gamma_0(n)$ with $n$ square-free to a weight $2$ holomorphic Siegel modular form. Motivated by Prasanna’s work for the Shintani lifting, we define the local Schwartz function at finite places using a quadratic Hecke character $\chi$ of square-free conductor $f$ coprime to level $n$. Then, at certain $2$ by $2$ Gram matrices related to $f$, we can express the Fourier coefficient of this theta lifting as a multiple of $L(F,\chi,1)$ by a non-zero constant. If the twisted $L$-value is known to be non-vanishing, we can deduce the non-vanishing of our theta lifting.



Shenxing Zhang -- University of Science and Technology of China

Title: On the generating fields of Kloosterman sums
Abstract: The arithmetic of exponential sums is a classical topic in number theory. In this talk, we will recall the basic properties of exponential sums, and show how to use $l$-adic method to obtain the generating fields of general twisted Kloosterman sum.




Participants

Huatong Chen
Yihua Chen
Haonan Gu
Zeping Hao
Jianing Li
Yongxiong Li
Meng-Fai Lim
Chan Ieong Kuan
Didier Lesesvre
Evgeny Mayanskiy
Chao Qin
Fengpeng Wang
Jingyi Xu
Weipeng Yang
Dongxi Ye
Miaodan Yuan
Shuai Zhai
Di Zhang
Shenxing Zhang