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.

傅立叶分析表明，可以定期的函数（例如，根据时间重复其以前值的时间固定时间的时间数量）可以实现为三角函数的总和。 具有更复杂，非共同，周期性的功能是现代数字理论的核心。 Langlands的基本视野预测了与算术有关的此类功能与对称性之间的联系，即来自多项式方程的根。 该提案与新的伪装中的周期性功能有关，当时这些功能不是在组上定义的，而是在组的有限封面上定义。这将算术和分析以新的方式融合。 此外，要研究的技术可能与数学物理学的构造有联系。该项目的主要研究对象是覆盖组的自态形式。 首席研究人员将首先关注Eisenstein系列，这些系列是通过平均过程获得的。 来自自动形式的许多标准工具都没有延续（例如，惠特克功能通常不是唯一的），而是令人惊讶的图片融合表示理论（规范基础，米尔科维\'c-vilonen循环），而数字理论则出现。 主要研究人员将研究这些及其数量理论应用。他还将研究元素艾森斯坦系列的残留物，“较高的theta系列”。 要理解这些对象附加的一能轨道，在不同组上建立这种系列之间的关系，并在全球和本地结构中使用它们，还有很多事情要做。第三个项目涉及独特的功能和Iwahori Hecke代数。 它为独特功能的新结构提供了潜力。