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

这个项目是关于数论的研究。这一数学领域的中心焦点是理解如何将局部信息打包以访问感兴趣的全局信息（例如多项式方程的解）的机制。某些解析对象，即所谓的l -函数，有望编码这种机制。著名的伯奇和斯温纳顿-戴尔猜想（千禧年奖问题之一）围绕着这个主题，就像19世纪的狄利克雷类数公式一样。本项目旨在增进我们对Birch和Swinnerton-Dyer猜想以及密切相关问题的理解。在这些方向上的进展可能会对诸如密码学、利用椭圆曲线算法的复杂性等领域产生影响。欧拉系统在某些算术对象和它们的解析对象（l -函数）之间建立了一座桥梁，因此为解决从局部到全局的问题提供了非常强大的工具。本项目旨在利用模形式的张量积和三重积的欧拉系统，特别是它们在p进族中的变化，获得数论中基本开放问题的新结果，如格林伯格关于l -函数在Hida族中消失的一般阶的猜想，Birch和Swinnerton-Dyer公式在0阶和1阶中的p部分，以及2阶椭圆曲线Selmer群中的显式类的构造。这些曲线超出了我们目前对Birch和Swinnerton-Dyer猜想的理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.