Collaborative Research: Automorphic Forms, Representations and L-functions

这项合作提案涉及开发一个新的更高等级的基本恒等式版本，经典地称为库兹涅佐夫迹公式，该公式将某个微分算子的谱与该算子作用的空间的几何形状联系起来。其目的是为公式中出现的所有不同项建立渐近或强界。第一个应用是得到某些薄族L函数族在各种高阶情形下的对称性。预计会有更广泛的进一步应用。此外，还将研究以下问题：寻找一类新的多重狄利克雷级数；研究更高等级的超尖球面表示；仿射线性筛法与双线性形式方法相结合，展示包含无穷多个素数的薄轨道；最后，将涉及有效的无限体积计数问题。自同构形式、表示和L函数理论是现代数论的一个中心主题，它将代数几何、表示论、概率论、组合学和数学物理等不同数学领域联系起来。因此，在理解上述目标方面取得的进展往往对其他领域产生重大影响。例如，保护互联网和移动电话无线通信安全的密码算法通常严重依赖于素数的深层性质。该提案还包括对从事这些不断发展的数学领域的本科生、研究生和博士后进行指导的重要教育和传播部分，希望将传统上代表不足的群体带入这一领域。

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.