Euler Systems, p-adic Deformations, and the Birch-Swinnerton-Dyer Conjecture

欧拉系统、p-adic 变形和 Birch-Swinnerton-Dyer 猜想

基本信息

批准号：
1946136
负责人：
Francesc Castella
金额：
$ 8.81万
依托单位：
University of California-Santa Barbara
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1946136&HistoricalAwards=false
关键词：
Euler Systems adic Deformations Birch

项目摘要

This project is concerned with research in number theory. A central focus in this area of mathematics is understanding the mechanism whereby local information can be packaged to get access to the global information of interest, such as the solutions to polynomial equations. Certain analytic objects, the so-called L-functions, are expected to encode such mechanism. The celebrated conjecture of Birch and Swinnerton-Dyer (one of the millenium prize problems) revolves around this theme, as does Dirichlet's class number formula from the nineteenth century. This project aims to enhance our understanding of the Birch and Swinnerton-Dyer conjecture and of closely related problems. Progress in these directions may have an impact on areas, such as cryptography, exploiting the complexity of the arithmetic of elliptic curves.Euler systems build a bridge between certain arithmetic objects and their analytic counterparts (L-functions), hence providing very powerful tools for tackling problems on the passage from local to global. The project aims to exploit the Euler systems for tensor products and triple products of modular forms, and especially their variation in p-adic families, to obtain new results on fundamental open problems in number theory, such as Greenberg's conjecture on the generic order of vanishing of L-functions in Hida families, the p-part of the Birch and Swinnerton-Dyer formula in ranks 0 and 1, and the construction of explicit classes in Selmer groups of elliptic curves of rank 2, curves that lie just beyond our current understanding of the Birch and Swinnerton-Dyer conjecture.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.

该项目与数量理论的研究有关。在这一数学领域的主要重点是了解可以包装本地信息以访问全球感兴趣信息的机制，例如对多项式方程的解决方案。某些分析对象（所谓的L功能）有望编码这种机制。 Birch和Swinnerton-Dyer（千年奖问题之一）的著名猜想围绕着这个主题，Dirichlet的班级编号公式也是19世纪的。该项目旨在增强我们对桦木和Swinnerton-Dyer猜想以及紧密相关的问题的理解。这些方向上的进展可能会影响诸如密码学之类的区域，利用椭圆曲线算术的复杂性。EulerSystems在某些算术对象及其分析对应物（L功能）之间建立了一个桥梁，因此为从本地通往全球解决方面的问题提供了非常有力的工具。 The project aims to exploit the Euler systems for tensor products and triple products of modular forms, and especially their variation in p-adic families, to obtain new results on fundamental open problems in number theory, such as Greenberg's conjecture on the generic order of vanishing of L-functions in Hida families, the p-part of the Birch and Swinnerton-Dyer formula in ranks 0 and 1, and the construction of explicit classes in Selmer groups在等级2的椭圆形曲线中，曲线超出了我们目前对桦木和Swinnerton-Dyer猜想的理解。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响来评估的评估值得支持的。