Topics in Arithmetic Geometry: Moduli Varieties, L-functions, Arakelov Theory and Their Interactions and Applications

算术几何主题：模簇、L 函数、Arakelov 理论及其相互作用和应用

• 批准号: 1700883
• 负责人: Shou-wu Zhang
• 金额: $ 19.32万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700883&HistoricalAwards=false
• 关键词: Topics; Arithmetic; Geometry; Moduli; Varieties

This project lies in a subfield of mathematics known as arithmetic geometry. Many of the questions in the project are motivated by the philosophy that information about algebraic equations over the integers can be obtained by geometric methods. Solution to the problems under study in this project will have substantial impact on research in cryptography, theoretical physics, and quantum computing.The investigator will study moduli varieties, L- functions, Arakelov theory and their interactions. This work will have applications to solution to the Birch and Swinnerton-Dyer conjecture, its generalization by Beilinson and Bloch, as well as related problems. . In particular, research will be done on 1) the arithmetic Gan-Gross-Prasad conjecture; 2) heights of Drinfeld-Heegner cycles with arbitrary length; 3) generalization of the Gross-Zagier formula to higher dimensions.

该项目属于数学的一个子领域，称为算术几何。该项目中的许多问题都是由以下理念引发的：整数代数方程的信息可以通过几何方法获得。该项目所研究问题的解决将对密码学、理论物理和量子计算的研究产生重大影响。研究人员将研究模簇、L-函数、阿拉克洛夫理论及其相互作用。这项工作将应用于解决 Birch 和 Swinnerton-Dyer 猜想、Beilinson 和 Bloch 对其的推广以及相关问题。 。具体研究：1）算术Gan-Gross-Prasad猜想； 2）任意长度的Drinfeld-Heegner循环的高度； 3）将Gross-Zagier公式推广到更高维度。

项目成果

On the averaged Colmez conjecture

关于平均科尔梅兹猜想

• DOI: 10.4007/annals.2018.187.2.4
• 发表时间: 2018-01
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Yuan, Xinyi;Zhang, Shou
• 通讯作者: Zhang, Shou

Linear forms, algebraic cycles, and derivatives of L-series

L 级数的线性形式、代数循环和导数

• DOI: 10.1007/s11425-019-1589-7
• 发表时间: 2019-10-15
• 期刊: Science China Mathematics
• 作者: Shou

A p-adic Waldspurger formula

p 进 Waldspurger 公式

• DOI: 10.1215/00127094-2017-0045
• 发表时间: 2018-03
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Liu, Yifeng;Zhang, Shouwu;Zhang, Wei
• 通讯作者: Zhang, Wei