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.

该项目涉及数论研究。该数学领域的一个中心焦点是理解可以打包局部信息以获取感兴趣的全局信息（例如多项式方程的解）的机制。某些分析对象，即所谓的 L 函数，预计会编码这种机制。伯奇和斯温纳顿-戴尔的著名猜想（千禧奖问题之一）围绕着这个主题，十九世纪的狄利克雷的类数公式也是如此。该项目旨在增强我们对伯奇和斯温纳顿-戴尔猜想以及密切相关问题的理解。这些方向的进展可能会对密码学、利用椭圆曲线算术的复杂性等领域产生影响。欧拉系统在某些算术对象与其分析对象（L 函数）之间架起了一座桥梁，从而为以下领域提供了非常强大的工具：解决从局部到全球的过渡问题。该项目旨在利用欧拉系统的张量积和模形式的三重积，特别是它们在 p-adic 族中的变体，以获得数论中基本开放问题的新结果，例如格林伯格关于消失的通用顺序的猜想Hida 族中的 L 函数、0 级和 1 级 Birch 和 Swinnerton-Dyer 公式的 p 部分，以及 2 级椭圆曲线 Selmer 群中显式类的构造，曲线超出了我们目前对 Birch 和 Swinnerton-Dyer 猜想的理解。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。