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.

了解椭圆曲线上一组理性点的结构（本质上是立方方程）已成为一个多世纪以来数字理论的目标。它与埋在上古时的开放问题（例如一致的数字问题）有联系。桦木和swinnerton-dyer（BSD）猜想预测了理性点和相关的L功能之间的神秘关系。有趣的L功能出现在数学的各个领域以及物理学领域。即使外观进行分析，L功能也揭示了与算术的惊人亲和力。PI试图研究L功能特殊值的某些算术方面。该项目的第一部分旨在建立椭圆曲线的P-ADIC标准，以使分析等级（其中心相关L功能的消失顺序）。它将基于最近向BSD猜想（由于Skinner，Zhang和其他人）的进展，并旨在消除一些关键的假设。该项目的另一部分旨在研究垂直和水平家庭中L功能的特殊值的模式。它将基于过去二十年的进步（由于HIDA，Michel-venkatesh等）。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。