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

欧拉系统、岩泽理论和椭圆曲线算术

• 批准号: 2401321
• 负责人: Francesc Castella
• 金额: $ 22.31万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401321&HistoricalAwards=false
• 关键词: Euler; Systems; Iwasawa; Theory; Arithmetic

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.

椭圆曲线是一类多项式方程（两个变量的三次方程），人们对其进行了数百年的研究，但许多基本问题仍然悬而未决。例如，目前还没有经过验证的算法来确定给定的椭圆曲线是否具有有限或无限多个有理解。在过去的一个世纪里，数学家们推测这些问题的答案可以从复变量的某些函数中提取，即椭圆曲线的 L 函数。欧拉系统和岩泽理论是迄今为止用于研究这些以及算术与分析之间的相关推测联系的两个最强大的工具。该奖项将通过开发欧拉系统和岩泽理论的新结果和技术来增进我们对椭圆曲线算法的理解。该奖项还将支持多项指导、培训、传播和外展活动。更具体地说，PI 及其合作者要进行的研究将主要集中在解决方案将显着增进我们对 Birch 核心问题的理解的问题上。和斯温纳顿-戴尔猜想以及分析等级 1 情况下的相关问题，并为分析等级 2 及更高等级的更神秘的情况提供了新的线索。在 1 级中，他们将在完全实数域上定义的椭圆曲线的情况下证明著名的 Gross-Zagier 和 Kolyvagin 定理的第一个 p 逆矩阵。在第二级，他们将继续研究几年前由 Darmon-Rotger 引入的广义 Kato 类，在超奇异情况下建立新的非零结果。他们还将研究与 Mazur-Rubin 符号猜想相关的 2 阶 Q 上椭圆曲线的 Selmer 基的系统 p-adic 构造。对于任意阶的椭圆曲线，他们将建立相关的欧拉系统和科利瓦金系统的各种非平凡结果，正如科利瓦金和马祖尔-塔特首先猜想的那样。该奖项反映了 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。