Analytic theory of automorphic forms

自守形式的解析理论

• 批准号: 1401008
• 负责人: Matthew Young
• 金额: $ 13.27万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401008&HistoricalAwards=false
• 关键词: Analytic; theory; automorphic; forms

The prime numbers have fascinated people for millennia because of their orderly definition but unpredictable behavior. It is only recently, in the past few decades, that primes have proved useful in cryptography, where it is often important to generate large primes. For instance, one can program a computer to find the first prime with 100 digits. Knowing how long this takes in general amounts to studying the distribution of the prime numbers. This distribution is intimately connected to properties of the Riemann zeta function, the most basic example of a so-called L-function. The PI will study many of the statistical properties of the Riemann zeta function and other more general L-functions. This project will study analytic properties of L-functions and automorphic forms. For instance, the PI plans to study the quantum unique ergodicity conjecture in some new contexts such as in higher rank and along an arbitrary curve. The PI has had prior success for the Eisenstein series restricted to a particular geodesic in the upper half plane. These problems fit into the general setting of how eigenfunctions of the Laplacian behave as the eigenvalue grows large, a topic of great interest in geometry, mathematical physics, etc. However, in the arithmetical setting these problems are related to quantitative analytic properties of L-functions.

素数吸引了人们几千年，因为它们的有序定义，但不可预测的行为。直到最近，在过去的几十年里，素数才被证明在密码学中是有用的，在密码学中，生成大素数通常很重要。例如，我们可以给计算机编程，让它用100位数找到第一个素数。知道这个过程一般需要多长时间就等于研究了素数的分布。这种分布与黎曼zeta函数的性质密切相关，这是所谓L函数的最基本例子。PI将研究黎曼zeta函数和其他更一般的L函数的许多统计性质。本专题将研究L-函数和自守形式的解析性质。例如，PI计划在更高秩和沿着任意曲线等新的背景下研究量子唯一遍历猜想。PI先前成功地将爱森斯坦级数限制在上半平面的特定测地线上。这些问题适合的一般设置如何本征函数的拉普拉斯行为的本征值增长大，一个主题的极大兴趣几何，数学物理等，但是，在算术设置这些问题是有关定量分析性质的L-函数。

项目成果

A generalized cubic moment and the Petersson formula for newforms

广义立方矩和新形式的 Petersson 公式

• DOI: 10.1007/s00208-018-1745-1
• 发表时间: 2018
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Petrow, Ian;Young, Matthew P.
• 通讯作者: Young, Matthew P.