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

这个项目是关于数论的研究.在这个数学领域的一个中心焦点是理解的机制，使本地信息可以打包，以获得感兴趣的全球信息，如解决方案多项式方程。某些分析对象，所谓的L函数，被期望编码这样的机制。著名的猜想伯奇和斯温纳顿戴尔（一个千禧年奖的问题）围绕这个主题，因为没有狄利克雷的类数公式从19世纪。这个项目旨在提高我们对Birch和Swinnerton-Dyer猜想以及相关问题的理解。这些方向的进展可能会对一些领域产生影响，如密码学，利用椭圆曲线算术的复杂性。欧拉系统在某些算术对象和它们的解析对应对象（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。该项目旨在利用模形式的张量积和三重积的Euler系统，特别是它们在p-adic族中的变化，以获得数论中基本开放问题的新结果，例如格林伯格关于希达族中L函数消失的一般阶的猜想，Birch和Swinnerton-Dyer公式在秩0和1中的p部分，以及在塞尔默组的秩为2的椭圆曲线中构造显式类，这些曲线超出了我们目前对伯奇和斯温纳顿-戴尔猜想的理解。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.