Elliptic Curves and Cohomological Automorphic Forms over CM Fields

CM 域上的椭圆曲线和上同调自同构

• 批准号: 1902155
• 负责人: Shiang Tang
• 金额: $ 15万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902155&HistoricalAwards=false
• 关键词: Elliptic; Curves; Cohomological; Automorphic; Forms

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.

对对称性的研究遍及数学。在数论中，有趣的对称性存在于代数数之间，这些数是整数系数多项式的根。这些对称性支撑了算术、几何和分析之间的一种令人惊讶的联系，即朗兰兹互易性。由此产生的桥梁之间建立看似不同的领域带来了强大的分析和代数工具承担算术问题。这个项目的目的是建立新的情况下朗兰兹互惠，并将由此产生的工具应用到数论的问题。证明伽罗瓦表示的自同构是现代代数数论的一个重要主题，是目前唯一已知的技术，建立了算术L-函数的许多几何性质。这个项目的一部分目的是建立自同构的许多椭圆曲线在虚二次域，或更一般的CM数域。这个项目的第二部分旨在完善我们的知识，本地全球的兼容性在朗兰兹计划CM领域。这种更好的兼容性将被应用到本项目第三部分的相邻塞尔默组的研究中，建立布洛赫-加藤和佩林-里乌的案例，以及对Venkatesh计划的应用。该奖项反映了NSF的法定使命，并被认为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进行评估来支持。