Semiclassical Analysis, Amplification, and Subconvexity

L函数是一个复变量的函数，它的研究是现代数论的核心。引入它们是为了研究有关素数分布的深入问题。今天，我们知道L函数在特殊点处的值与许多数学领域有关，从求解整数方程到数学物理。当这些特定值很小时，它告诉我们某些高度结构化的数学对象实际上是随机行为的。这个研究项目试图用量子物理的方法来证明这一点。主要的研究人员将使用半经典分析的方法来研究L函数的次凸性问题。这两个区域之间的联系由诸如Rankin-Selberg、Waldspurger和Ichino-Ikeda的周期积分公式提供。PI希望算术流形的几何结构可以更容易地研究在这项工作中将出现的振荡积分和均匀分布问题。PI还通过证明自同构形式的Lp范数比一般波包的Lp范数小，从而证明了自同构形式的混沌结果。

L-functions are functions of one complex variable whose study is central to modern number theory. They were introduced in order to study deep questions about the distribution of prime numbers. Today, we know that the values of L-functions at special points are linked to many areas of mathematics, from solving equations in integers to mathematical physics. When these special values are small, it tells us that certain highly structured mathematical objects are in fact behaving randomly. This research project project attempts to prove that this happens by using methods from quantum physics.The principal investigator will use techniques from semiclassical analysis to approach the subconvexity problem for L-functions. The link between these two areas is provided by period integral formulae such as those of Rankin-Selberg, Waldspurger, and Ichino-Ikeda. The PI hopes that the geometric structure of arithmetic manifolds will make it easier to study the oscillatory integrals and equidistribution problems that will arise in this work. The PI also aims to prove chaoticity results for automorphic forms, by showing that their Lp norms are smaller than those of general wave packets.