Elliptic Curves, p-adic Deformations, and Iwasawa Theory

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）解。椭圆曲线是一类这样的方程，自古以来就被研究，但其中许多问题仍然是开放的。椭圆曲线是本研究的中心研究对象。自从格罗斯-扎吉尔和科利瓦金的基础工作以来，代数几何和表示论的强大工具一直在指导这一方向的研究。本研究的目的是利用这些领域的一些最新进展来加深我们对椭圆曲线算法及其相关问题的理解。在第一部分的项目集中在构造线性无关的塞尔默类的高秩椭圆曲线。Darmon和Rotger在秩2的设置中提出了非常精确的模型，这些项目有望在这个方向上产生新的结果。第二部分着重于岩泽理论中的问题，并应用于Birch和Swinnerton-Dyer猜想，Greenberg在根数为-1的情况下提出的一个非零猜想，和有限局部不可分解的伽罗瓦表示与普通的非-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.