MODULAR FORMS AND ANALYTIC NUMBER THEORY

模形式和解析数论

• 批准号: 1001527
• 负责人: William Duke
• 金额: $ 27万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001527&HistoricalAwards=false
• 关键词: MODULAR; FORMS; ANALYTIC; NUMBER; THEORY

This project proposes to investigate several sets of problems that lie in the intersection of the theory of modular forms and analytic number theory.One set of problems concerns real quadratic analogues of singular moduli. Classical singular moduli are special values of the modular j-function at imaginary quadratic irrationalities and have well-known importance for class field theory and the theory of half-integral weight weakly holomorphic modular forms. The real quadratic analogues are defined through certain cycle integrals of the j-function. It is proposed to understand their possible relations to real quadratic fields as well as the asymptotic behavior of their ``traces'', which occur as Fourier coefficients of a new kind of mock modular form. Also to be investigated are new connections between the period functions of modular integrals and certain orthogonal polynomials. One problem here is to apply Riemann-Hilbert analysis to obtain strong asymptotics for the orthogonal polynomials associated to rational period functions. These polynomials are perturbations of Jacobi polynomials and generalize Atkin's polynomials. Another research topic is to study the distribution of roots of polynomial congruences using higher rank automorphic forms.Number theory, which is one of the oldest parts of mathematics, continues to enjoy remarkable advances today. The theory of modular forms occupies a central position within number theory and has proven to be a wellspring of new ideas both within number theory and in other parts of mathematics. The research proposed here is intended to contribute in a meaningful way to the theory of modular forms by developing new connections to analytic number theory as well as to other parts of analysis. An important component of this proposal is the integration of research and education at the postdoctoral, graduate and undergraduate levels.Several research problems that are suitable for undergraduates are proposed, and these will also involve the participation of the PI, graduate students, and postdocs. The aim is to help introduce the undergraduates to research and to develop the mentoring skills of the graduate students and postdocs. Other planned activities are designed to elevate the research experience of mathematics teachers who will be educating students at pre-research stages. These are intended to be cost-effective ways to benefit future research in mathematics in the United States.

本项目拟研究模形式理论与解析数论交叉的几组问题。一组问题涉及奇异模的实数二次类。经典奇异模是模j函数在虚二次无理数处的特殊值，在类场论和半积分权弱全纯模形式理论中具有众所周知的重要性。通过j函数的某些循环积分来定义实二次类。提出了它们与实二次域的可能关系，以及它们的“迹”的渐近行为，这些“迹”表现为一种新的模拟模形式的傅里叶系数。本文还研究了模积分的周期函数与某些正交多项式之间的新联系。这里的一个问题是应用黎曼-希尔伯特分析来获得与有理周期函数相关的正交多项式的强渐近性。这些多项式是雅可比多项式的扰动，是对阿特金多项式的推广。另一个研究课题是利用高阶自同构形式研究多项式同余根的分布。数论是数学中最古老的部分之一，今天继续享有显著的进步。模形式理论在数论中占有中心地位，并已被证明是数论和其他数学领域新思想的源泉。这里提出的研究旨在通过发展与解析数论以及分析的其他部分的新联系，以有意义的方式为模形式理论做出贡献。该提案的一个重要组成部分是博士后、研究生和本科水平的研究和教育的整合。提出了几个适合本科生的研究问题，这些问题也将涉及项目负责人、研究生和博士后的参与。其目的是帮助本科生进行研究，并培养研究生和博士后的指导技能。其他计划的活动旨在提高数学教师的研究经验，他们将在研究前阶段教育学生。这些都是经济有效的方法，有利于美国未来的数学研究。