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Elliptic Curves, p-adic Deformations, and Iwasawa Theory

椭圆曲线、p 进变形和岩泽理论

基本信息

批准号：
2101458
负责人：
Francesc Castella
金额：
$ 18.01万
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：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101458&HistoricalAwards=false
关键词：
Elliptic Curves adic Deformations Iwasawa

项目摘要

A central problem in number theory is understanding the (integer, or rational) solutions to polynomial equations with integer coefficients. Elliptic curves are a class of such equations that has been studied since antiquity, yet for which many questions still remain open. Elliptic curves are a central object of study in this research. Since the fundamental work of Gross-Zagier and Kolyvagin, powerful tools from algebraic geometry and representation theory have guided research in this direction. The PI aims to leverage some recent developments in those areas to further our understanding of the arithmetic of elliptic curves and related problems.More specifically, the research is divided into two related parts. The projects in the first part focus on the construction of linearly independent Selmer classes for higher rank elliptic curves. Darmon and Rotger have proposed very precise conjectures in the rank 2 setting, and these projects are expected to yield new results in this direction. The second part focuses on problems in Iwasawa theory, with applications towards the Birch and Swinnerton-Dyer conjecture, a non-vanishing conjecture by Greenberg in the case of root number -1, and the conjectured local indecomposability of the Galois representations associated to ordinary non-CM forms.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.

数论中的一个中心问题是理解具有整数系数的多项式方程的（整数或有理数）解。椭圆曲线是一类这样的方程，自古以来就被研究过，但仍有许多问题有待解决。椭圆曲线是本研究的中心研究对象。自Gross-Zagier和Kolyvagin的基础工作以来，代数几何和表示理论的强大工具引导了这一方向的研究。PI旨在利用这些领域的一些最新发展来进一步理解椭圆曲线的算法和相关问题。更具体地说，研究分为两个相关的部分。第一部分的课题主要是构造高阶椭圆曲线的线性无关Selmer类。Darmon和Rotger在排名2的情况下提出了非常精确的推测，这些项目有望在这个方向上产生新的结果。第二部分着重于Iwasawa理论中的问题，应用于Birch和Swinnerton-Dyer猜想，格林伯格在根数-1情况下的非消失猜想，以及与普通非cm形式相关的伽罗瓦表示的推测局部不可分解性。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。