New Directions in the Theory of Automorphic Forms

自守形式理论的新方向

• 批准号: 1701638
• 负责人: William Duke
• 金额: $ 41万
• 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: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701638&HistoricalAwards=false
• 关键词: New; Directions; Theory; Automorphic; Forms

This research project concerns number theory, the oldest branch of mathematics. More specifically, it focuses on the study of automorphic forms. Automorphic forms are a very special class of functions that form an important bridge connecting the discrete objects of algebraic number theory and the continuous objects of analytic number theory. Automorphic forms are used as tools to study number theoretic functions, e.g. by measuring their rates of growth, discovering formulas for them, or proving relations that they satisfy. Classically, special values of automorphic forms provided the solution to Kronecker's problem for extensions of the rational numbers that do not lie in the real line. More recently, automorphic forms played a pivotal role in the proof of Fermat's Last Theorem. This project aims to further explore the connection of automorphic forms to other mathematical structures.The main object of the proposed research is to further understand and exploit the relationship between automorphic forms and quadratic number fields. This relationship is exceedingly rich and unites the study of diverse objects such as Heegner points, closed geodesics, the hyperbolic Laplacian, Kloosterman sums, L-functions and class fields. The theory for imaginary quadratic fields is in general better developed and simpler, especially in relation to class field theory. The investigators will concentrate mostly on the real quadratic case. In one direction, they will study some new geometric invariants associated to real quadratic fields that were introduced recently. These invariants are certain surfaces that are bounded by modular closed geodesics. The investigators will study the distribution of the areas of the surfaces, especially as this relates to ideal classes and also investigate some new geometric problems about the closed geodesics. They plan to express various invariants for real quadratic fields (such as surface integrals of modular functions) in terms of the Fourier coefficients of automorphic forms. This naturally leads to problems involving sums of Kloosterman sums and to extensions of recent work on uniform estimates for such sums.

这个研究项目涉及数论，这是数学中最古老的分支。更具体地说，它侧重于自同构形式的研究。自同构形式是一类非常特殊的函数，它是连接代数数论的离散对象和解析数论的连续对象的重要桥梁。自同构形式被用作研究数论函数的工具，例如，通过测量它们的增长率，发现它们的公式，或证明它们满足的关系。经典地，自同构形式的特殊值为不在实数线上的有理数的扩展提供了Kronecker问题的解。最近，自同构形式在费马大定理的证明中发挥了关键作用。本项目旨在进一步探索自同构形式与其他数学结构的联系。本研究的主要目的是进一步理解和开发自同构形式与二次元数域之间的关系。这种关系非常丰富，并将各种对象的研究结合起来，如Heegner点、封闭测地线、双曲拉普拉斯、Kloosterman和、l函数和类域。一般来说，虚二次场的理论发展得更好，也更简单，特别是在类场论方面。调查人员将主要集中于真正的二次型情况。一方面，他们将研究最近引入的与实数二次域相关的一些新的几何不变量。这些不变量是由模封闭测地线限定的曲面。研究人员将研究曲面面积的分布，特别是与理想类有关的分布，并研究关于封闭测地线的一些新的几何问题。他们计划用自同构形式的傅里叶系数来表示实二次域的各种不变量（例如模函数的表面积分）。这自然导致了涉及Kloosterman和和的问题，以及最近关于这种和的统一估计的工作的扩展。

项目成果

Markov spectra for modular billiards

模块化台球的马尔可夫谱

• DOI: 10.1007/s00208-018-1781-x
• 发表时间: 2019
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Andersen, Nickolas;Duke, William
• 通讯作者: Duke, William

Kronecker’s first limit formula, revisited

重新审视克罗内克的第一个极限公式

• DOI: 10.1007/s40687-018-0138-0
• 发表时间: 2018
• 期刊: Research in the Mathematical Sciences
• 影响因子: 1.2
• 作者: Duke, W.;Imamoḡlu, Ö.;Tóth, Á.
• 通讯作者: Tóth, Á.