Arithmetic Aspects of Special Values of L-Functions

L 函数特殊值的算术方面

基本信息

批准号：
2001409
负责人：
Ashay Burungale
金额：
$ 11万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2022-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001409&HistoricalAwards=false
关键词：
Arithmetic Aspects Special Values Functions

项目摘要

Understanding the structure of the set of rational points on an elliptic curve (essentially a cubic equation) has been an aim in number theory for over a century. It has connections to open problems (such as the congruent number problem) buried in antiquity. The Birch and Swinnerton-Dyer (BSD) conjecture predicts a mystifying relation among the rational points and the associated L-function. The intriguing L-functions appear in various areas of mathematics and also in physics. Even though analytic in appearance, the L-functions reveal striking affinity to arithmetic.The PI seeks to study certain arithmetic aspects of special values of L-functions. The first part of the project aims to establish a p-adic criterion for an elliptic curve to have analytic rank (the vanishing order of the associated L-function at its center) one. It will build on the recent progress towards the BSD conjecture (due to Skinner, Zhang and others) and aims to remove some of the key hypotheses. The other part of the project aims to study mod p non-vanishing of special values of L-functions in vertical and horizontal families. It will build on the progress during the last two decades (due to Hida, Michel--Venkatesh and others).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

一个多世纪以来，理解椭圆曲线（本质上是三次方程）上有理点集的结构一直是数论的目标。它与古代埋藏的开放问题（例如全等数问题）有联系。 Birch 和 Swinnerton-Dyer (BSD) 猜想预测了有理点和相关 L 函数之间的神秘关系。有趣的 L 函数出现在数学和物理学的各个领域。尽管表面上是分析性的，但 L 函数显示出与算术的惊人亲和力。PI 旨在研究 L 函数特殊值的某些算术方面。该项目的第一部分旨在为椭圆曲线建立一个 p 进准则，使其具有解析等级（相关 L 函数在其中心的消失阶）。它将建立在 BSD 猜想的最新进展（由于 Skinner、Zhang 等人）的基础上，旨在消除一些关键假设。该项目的另一部分旨在研究垂直和水平族中 L 函数特殊值的 mod p 不消失。它将建立在过去二十年的进步基础上（归功于 Hida、Michel-Venkatesh 等人）。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。