MODULAR FORMS AND ANALYTIC NUMBER THEORY

本计画拟探讨几组位于模形式理论与解析数论交叉点的问题，其中一组问题是关于奇异模的真实的二次类比。经典奇异模是模j-函数在虚二次无理数处的特殊值，在类域理论和半整权弱全纯模形式理论中具有重要意义。真实的二次类似物通过j函数的某些循环积分来定义。建议理解它们与真实的二次域的可能关系，以及它们的"迹“的渐近行为，这些迹作为一种新的模拟模形式的傅立叶系数出现。也要调查的是新的连接之间的周期函数的模积分和某些正交多项式。 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Jacobi多项式的扰动，推广了Atkin多项式。另一个研究课题是利用高阶自守形式研究多项式同余的根的分布。数论是数学中最古老的部分之一，今天仍然享有显着的进步。模形式理论在数论中占据中心地位，并且已经被证明是数论和数学其他部分中新思想的源泉。 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。该方案的一个重要组成部分是博士后、研究生和本科生三个层次的研究与教育的融合，提出了几个适合本科生的研究问题，这些问题也将涉及PI、研究生和博士后的参与。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。其他计划中的活动旨在提升数学教师的研究经验，他们将在研究前阶段教育学生。这些都是旨在成本效益的方式，有利于未来的研究在美国的数学。

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.