Elliptic Curves and Cohomological Automorphic Forms over CM Fields

对称性的研究遍及数学。在数论中，代数数之间存在着有趣的对称性，代数数是整数系数多项式的根。这些对称性在算术、几何和分析之间建立了一种令人惊讶的联系，即朗兰兹互易性。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解决算术问题。本项目旨在建立朗兰互惠的新案例，并将结果工具应用于数论问题。证明伽罗瓦表示的自同构是现代代数数论中的一个重要课题，也是目前已知的唯一能够建立算术l函数许多猜想性质的方法。这个项目的一部分目的是建立虚二次域上的许多椭圆曲线的自同构，或者更一般的CM数域。该项目的第二部分旨在完善我们在CM领域的Langlands程序中的局部-全局兼容性知识。这种更精细的相容性将在本项目的第三部分应用于伴随Selmer群的研究，建立Bloch-Kato和Perrin-Riou猜想的案例，并应用于Venkatesh程序。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。

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.