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: 1001252
• 负责人: Alex Kontorovich
• 金额: $ 12万
• 依托单位: Brown University
• 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: 2010-07-15 至 2010-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001252&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.

该合作提案涉及开发传统上称为库兹涅佐夫迹公式的基本恒等式的新更高阶版本，该公式将某个微分算子的频谱与该算子作用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项建立渐近或强界。第一个应用是在各种更高阶的情况下获得某些 L 函数的稀疏族的对称类型。预计会有更广泛的进一步应用。 此外，还将研究以下研究问题：将执行对新类多重狄利克雷级数的搜索；将研究更高级别的上尖峰表示；仿射线性筛法将与双线性形式方法相结合，以展示包含无穷多个素数的细轨道；自守形式、表示和 L 函数的理论是现代数论的中心主题，并提供了代数几何、表示论等不同数学领域之间的联系、概率、组合学和数学物理。因此，对上述对象的理解的进展通常会对其他领域产生重大影响。例如，保护互联网和手机无线通信安全的加密算法通常严重依赖于素数的深层属性。该提案还包括对从事这些不断发展的数学领域工作的本科生、研究生和博士后进行指导的重要教育和传播部分，希望将传统上代表性不足的群体带入该领域。