Collaborative Research: Automorphic Forms, Representations and L-functions

合作研究：自守形式、表示和 L 函数

• 批准号: 1064214
• 负责人: Alex Kontorovich
• 金额: $ 12万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-06 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1064214&HistoricalAwards=false
• 关键词: Collaborative; Research; Automorphic; Forms; Representations

This collaborative proposal is concerned with developing a new higher rank version of a fundamental identity known classically as the Kuznetsov trace formula, which relates the spectrum of a certain differential operator to the geometry of the space on which the operator acts. The aim is to establish either asymptotics or strong bounds for all the different terms appearing in the formula. A first application is to obtain the symmetry types of certain thin families of L-functions in various higher rank situations. A broad range of further applications are expected. Additionally, the following research problems will be investigated: a search for a new class of Multiple Dirichlet Series will be executed; supercuspidal representations in higher rank will be studied; the Affine Linear Sieve will be combined with bilinear forms methods to exhibit thin orbits containing an infinitude of primes; and finally, effective infinite-volume counting problems will be attacked.The theory of automorphic forms, representations, and L-functions is a central theme in modern number theory, and has provided links between such diverse areas of mathematics as algebraic geometry, representation theory, probability, combinatorics, and mathematical physics. Thus progress in the understanding of the aforementioned objects often has a significant impact in other fields. For example, cryptographic algorithms which secure wireless communication for the internet and cellular phones often rely heavily on deep properties of prime numbers. The proposal also includes a significant educational and dissemination component in the mentoring of undergraduate, graduate students, and postdocs working in these evolving parts of mathematics, with the hope of bringing traditionally under-represented goups into the field.

该协作提案与开发一个新的高级版本的基本身份的新高级版本是经典地称为Kuznetsov Trace公式，该公式将某些差异操作员的光谱与操作员行为的空间的几何形状联系起来。目的是为公式中出现的所有不同术语建立渐近学或强度。第一个应用是在各种较高等级情况下获得某些L功能的某些薄项属家庭的对称类型。预计会有广泛的进一步应用。 此外，将研究以下研究问题：将执行对新的多个Dirichlet系列的搜索；将研究更高等级的超级质表示。仿射线性筛将与双线性形式的方法结合使用，以表现出含有无限质量的薄轨道；最后，将攻击有效的无限体积计数问题。自动形式，表征和L功能的理论是现代数字理论中的一个核心主题，并且在代数几何（例如代数几何），表示理论等数学领域之间提供了联系。 ，概率，组合学和数学物理学。因此，对上述对象的理解进展通常会对其他领域产生重大影响。例如，保护互联网和蜂窝电话的无线通信的加密算法通常在很大程度上依赖于质数的深度属性。该提案还包括在这些不断发展的数学部分中工作的本科生，研究生和博士后的指导，并希望将传统上代表性不足的GOUP带入该领域。