Elliptic Curves and Cohomological Automorphic Forms over CM Fields

对称性的研究遍及数学领域。在数论中，代数数之间存在着有趣的对称性，代数数是具有整数系数的多项式的根。这些对称性巩固了算术、几何和分析之间令人惊讶的联系，这种联系被称为朗兰兹互惠。由此产生的在看似不同的领域之间建立的桥梁带来了强大的分析和代数工具来处理算术问题。这个项目的目的是建立新的情形，并将所得到的工具应用于数论中的问题。证明伽罗瓦表示的自同构是现代代数数论中的一个重要主题，也是目前已知的唯一建立算术L函数的许多猜想性质的方法。这个项目的一部分目的是在虚二次域上建立许多椭圆曲线的自同构，或者更一般地在CM数域上建立自同构。这个项目的第二部分旨在提炼我们在朗兰兹计划中关于CM领域的局部-全球兼容性的知识。这种更好的兼容性随后将应用于本项目第三部分中的伴随Selmer群的研究，建立Bloch-Kato和Perrin-Riou猜想的案例，以及对Venkatesh计划的申请。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 symmetry pervades mathematics. In number theory, interesting symmetries exist among algebraic numbers, numbers which are roots of polynomials with integer coefficients. These symmetries underpin a surprising connection, known as Langlands reciprocity, between arithmetic, geometry, and analysis. The resulting bridges constructed between seemingly disparate areas bring powerful analytic and algebraic tools to bear on arithmetic questions. This project aims to establish new cases Langlands reciprocity, and to apply the resulting tools to questions in number theory.Proving automorphy of Galois representations is an important theme in modern algebraic number theory, and is currently the only known technique that establishes many conjectural properties of arithmetic L-functions. Part of this project aims to establish automorphy of many elliptic curves over imaginary quadratic fields, or more generally CM number fields. The second part of this project aims to refine our knowledge of local-global compatibility in the Langlands program over CM fields. This finer compatibility will then be applied to the study of adjoint Selmer groups in the third part of this project, establishing cases of conjectures of Bloch-Kato and Perrin-Riou, as well as having applications to a program of Venkatesh.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.