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

数论是近年来发展非常迅速的一门数学学科。本研究项目研究岩泽数论的一个分支--岩泽理论。它将多项式方程的积分解和有理解与称为L函数的某些分析对象联系起来。该项目采用了新的研究方法，将朗兰兹计划和代数几何的新工具结合在一起。这些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，其普遍真理在数论中仍然是一个具有挑战性的开放问题。该研究项目旨在扩展已知满足猜想的椭圆曲线集，更具体地，研究L函数的特殊值与某些算术对象，即伽罗瓦表示的塞尔默群之间的关系。对于素数p，所研究的主要问题是岩泽猜想和p-进Bloch-Kato猜想。该项目开发了一种非普通岩泽理论的新方法：首先研究由于其普通性质而易于证明的Greenberg型岩泽主要猜想，然后利用特殊循环的显式互易律将它们与非普通岩泽理论联系起来。最终目的是证明对于所有GL(2)模形式(具有任意权并且可能在p处有分支)：当解析秩为1或0时，Birch和Swinnerton-Dyer公式的p-部分；岩泽主要猜想；中心临界L值的消失意味着相应的Selmer群具有至少1阶。本项目还旨在研究这些高阶动机的问题，特别是与酉群上的尖点形式有关的那些。

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.