Artihmetic Geometry: Iwasawa Theory, the Bloch-Kato Conjecture, and the Birch and Swinnerton-Dyer Conjecture

算术几何：岩泽理论、布洛赫-加藤猜想、伯奇和斯温纳顿-戴尔猜想

• 批准号: 1600636
• 负责人: Xin Wan
• 金额: $ 13.85万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600636&HistoricalAwards=false
• 关键词: Artihmetic; Geometry; Iwasawa; Theory; Bloch

Number theory is a subject in mathematics that has been developing very rapidly in recent years. This research project studies Iwasawa theory, a branch of number theory. It relates integral and rational solutions of polynomial equations to certain analytic objects known as L-functions. The project exploits novel approaches to this study, which combine new tools from the Langlands program and algebraic geometry. The new approaches have been used to prove that a majority of elliptic curves satisfy the Birch and Swinnerton-Dyer conjecture, whose general truth is still a challenging open question in number theory. This research project aims to expand the set of elliptic curves known to satisfy the conjecture.More concretely, this project studies the relationships between special values of L-functions and certain arithmetic objects, namely the Selmer groups of Galois representations. For a prime number p the main problems under study are the Iwasawa main conjectures and p-adic Bloch-Kato conjectures. The project develops a new approach towards non-ordinary Iwasawa theory: first to study Greenberg type Iwasawa main conjectures that are accessible to proof due to their ordinary nature, and then to relate them to non-ordinary Iwasawa theory using explicit reciprocity laws of special cycles. The ultimate goal is to prove, for all GL(2) modular forms (of any weight and possibly with ramification at p): the p-part of the Birch and Swinnerton-Dyer formula in the case when analytic rank is 1 or 0; the Iwasawa main conjecture; and that the vanishing of central critical L-value implies the corresponding Selmer group has rank at least 1. The project also aims to study these problems for higher rank motives, especially those associated to cusp forms on unitary groups.

数论是近年来发展非常迅速的数学学科。本研究课题研究数论的分支岩泽理论。它将多项式方程的积分和有理解与称为L-函数的某些分析对象联系起来。该项目利用新的方法进行这项研究，其中联合收割机新的工具，从朗兰兹计划和代数几何。新的方法已经被用来证明大多数椭圆曲线满足Birch和Swinnerton-Dyer猜想，其普遍真理仍然是数论中一个具有挑战性的开放问题。 本研究课题的目的是扩展满足猜想的椭圆曲线的集合，具体而言，研究L-函数的特殊值与伽罗瓦表示的塞尔默群等算术对象之间的关系。对于一个素数p的主要问题正在研究中的Iwasawa主要图和p-adic Bloch-Kato图。该项目开发了一种新的方法来研究非普通岩泽理论：首先研究格林伯格型岩泽主图，由于它们的普通性质而易于证明，然后使用特殊循环的显式互易定律将它们与非普通岩泽理论联系起来。最终目标是证明，对于所有GL（2）模形式（任意权且可能在p处有分支）：当解析秩为1或0时Birch和Swinnerton-Dyer公式的p-部分;岩泽主猜想;以及中心临界L-值的消失意味着相应的塞尔默群的秩至少为1。该项目还旨在研究这些问题的高阶动机，特别是那些与酉群上的尖点形式。