Topics in Automorphic Forms

自守形式主题

• 批准号: 1500977
• 负责人: Solomon Friedberg
• 金额: $ 20.16万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500977&HistoricalAwards=false
• 关键词: Topics; Automorphic; Forms

Fourier analysis implies that functions that are periodic -- for example, quantities depending on time that repeat their previous value a fixed amount of time later -- can be realized as sums of trigonometric functions. Functions with more complicated, non-commutative, periodicities are at the heart of modern number theory. A fundamental vision of Langlands predicts connections between such functions and symmetries related to arithmetic, that is, coming from roots of polynomial equations. This proposal is concerned with periodic functions in a new guise, when the functions are defined not on a group but on a finite cover of a group. This blends arithmetic and analysis in a new way. Moreover, the techniques to be studied may have connections to constructions in mathematical physics.The main objects of study in this project are automorphic forms on covering groups. The principal investigator will focus first on Eisenstein series, which are obtained by an averaging process. Many of the standard tools from automorphic forms do not carry over (e.g. Whittaker functionals are typically not unique), but a surprising picture blending representation theory (canonical bases, Mirkovi\'c-Vilonen cycles) and number theory is emerging. The principal investigator will investigate these and and their number-theoretic applications. He also will study the residues of metaplectic Eisenstein series, "higher theta series." There is much to be done to understand the unipotent orbits attached to these objects, to develop relations between such series on different groups, and to use them both globally and in local constructions. A third project concerns unique functionals and Iwahori Hecke algebras. It offers the potential for new constructions of unique functionals.

傅立叶分析意味着周期性函数（例如，取决于时间的量在固定的时间后重复其先前值）可以实现为三角函数的和。 具有更复杂、非交换性、周期性的函数是现代数论的核心。 朗兰兹的基本愿景预测了这些函数和与算术相关的对称性之间的联系，即来自多项式方程的根。 该提案涉及一种新形式的周期函数，即函数不是在群上定义的，而是在群的有限覆盖上定义的。这以一种新的方式融合了算术和分析。 此外，要研究的技术可能与数学物理中的构造有关。该项目的主要研究对象是覆盖群上的自同构形式。 首席研究员将首先关注通过平均过程获得的爱森斯坦级数。 许多来自自守形式的标准工具并没有延续下去（例如惠特克泛函通常不是唯一的），但令人惊讶的混合表示理论（规范基、米尔科维-维洛宁循环）和数论的图景正在出现。 首席研究员将研究这些及其数论应用。他还将研究爱森斯坦系列的残余物，“更高的θ系列”。 要了解这些物体的单能轨道，发展不同群体的此类系列之间的关系，并在全球和当地的建设中使用它们，还有很多工作要做。第三个项目涉及独特泛函和 Iwahori Hecke 代数。 它为独特功能的新结构提供了潜力。