Iwasawa theory of elliptic curves and the Birch--Swinnerton-Dyer conjecture

多项式方程的有理数解的研究有着悠久的历史。椭圆曲线的情况，本质上是两个变量的三次方程，仍然令人着迷。Birch和Swinnerton-Dyer猜想（BSD）将椭圆曲线的算法与相关的Hasse-Weil l -函数联系起来。Iwasawa理论试图揭示l值的神秘算术化身。PI的目的是研究BSD猜想和变体的案例，主要基于Iwasawa理论。该项目还旨在培养研究生，并让本科生参与研究。在过去的十年里，我们见证了由Skinner和Zhang提出的关于0阶或1阶椭圆曲线的BSD猜想的关键进展。这个项目的一个目的是确定一些关键的缺失案例。例如，对于非常约化素数p， PI寻求BSD公式的p逆和p部分。该项目旨在发展椭圆曲线虚二次域上的双变量欧拉系统，以及非分裂素数下的反细胞Iwasawa理论。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。

The study of rational number solutions of polynomial equations has a rich history. The case of elliptic curves, essentially cubic equations in two variables, continues to be fascinating. The Birch and Swinnerton-Dyer conjecture (BSD) connects the arithmetic of an elliptic curve to the associated Hasse-Weil L-function. Iwasawa theory seeks to shed light on the mysterious arithmetic incarnation of L-values. The PI aims to study cases of the BSD conjecture and variants, primarily based on Iwasawa theory. The PI also aims to train graduate students, and introduce undergraduates to research. The last decade has witnessed a key progress towards the BSD conjecture for elliptic curves of rank zero or one, as initiated by Skinner and Zhang. An aim of this project is to establish some of the key missing cases. For instance, the PI seeks the p-converse and p-part of the BSD formula for primes p of non-ordinary reduction. The project aims to develop two-variable Euler systems over an imaginary quadratic field for an elliptic curve as well as anticyclotomic Iwasawa theory at non-split primes.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.