Euler Systems, Iwasawa Theory, and the Arithmetic of Elliptic Curves

椭圆曲线是一类多项式方程(两个变量的三次多项式)，已经研究了几个世纪，但许多基本问题仍然悬而未决。例如，目前还没有成熟的算法来确定给定的椭圆曲线是否有有限或无限多个有理数解。在过去的一个世纪里，数学家们猜测，这些问题的答案可以从复变量的某些函数中提取出来，即椭圆曲线的L函数。欧拉系统和岩川理论是迄今为止可用于研究这些以及算术和分析之间的相关猜想联系的两个最强大的工具。这一奖项将通过发展欧拉系统和岩泽理论的新结果和新技术，促进我们对椭圆曲线算法的理解。该奖项还将支持几个指导、培训、传播和推广活动。更具体地说，PI和他的合作者所进行的研究将主要集中在这些问题上，这些问题的解决方案将显著促进我们对Birch和Swinnerton-Dyer猜想核心问题以及分析排名为1的相关问题的理解，并为分析排名2和更高的更神秘的案例提供新的线索。在秩1中，他们将证明在定义在全实域上的椭圆曲线的情况下，与Gross-Zagier和Kolyvan in的著名定理第一个p-逆。在第二阶，他们将继续研究几年前由Darmon-Rotger引入的广义Kato类，在超奇异情形下建立新的非零化结果。他们还将研究与Mazur-Rubin的符号猜想有关的q上2阶椭圆曲线的Selmer基的一个系统的p-adi构造。对于任意等级的椭圆曲线，他们将建立相关的欧拉系统和Kolyvan in系统的各种非平凡结果，正如Kolyvan in和Mazur-Tate首先猜测的那样。这一奖项反映了NSF的法定使命，并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，被认为值得支持。

Elliptic curves are a class of polynomial equations (of degree three in two variables) that have been studied for centuries, yet for which many basic questions remain open. For instance, at present there is no proven algorithm to decide whether or not a given elliptic curve has finite or infinitely many rational solutions. Over the past century, mathematicians conjectured that an answer to these questions could be extracted from certain functions of a complex variable, namely the L-function of the elliptic curve. Euler systems and Iwasawa theory are two of the most powerful tools available to date for the study of these and related conjectured links between arithmetic and analysis. This award will advance our understanding of the arithmetic of elliptic curves by developing new results and techniques in Euler systems and Iwasawa theory. The award will also support several mentoring, training, dissemination, and outreach activities.More specifically, the research to be pursued by the PI and his collaborators will largely focus on problems whose solutions will significantly advance our understanding of issues at the core of the Birch and Swinnerton-Dyer conjecture and related questions in situations of analytic rank 1, and shed new light on the much more mysterious cases of analytic rank 2 and higher. In rank 1, they will prove the first p-converse to the celebrated theorem of Gross-Zagier and Kolyvagin in the case of elliptic curves defined over totally real fields. In rank 2, they will continue their investigations of the generalized Kato classes introduced a few years ago by Darmon-Rotger, establishing new nonvanishing results in the supersingular case. They will also study a systematic p-adic construction of Selmer bases for elliptic curves over Q of rank 2 in connection with the sign conjecture of Mazur-Rubin. For elliptic curves of arbitrary rank, they will establish various non-triviality results of associated Euler systems and Kolyvagin systems, as first conjectured by Kolyvagin and Mazur-Tate.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.