Semiclassical Analysis, Amplification, and Subconvexity

L-函数是一个复变量的函数，其研究是现代数论的核心。 它们的引入是为了研究有关素数分布的深层问题。 今天，我们知道L函数在特殊点的值与数学的许多领域有关，从整数方程到数学物理。 当这些特殊值很小时，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。 本研究项目试图通过量子物理学的方法来证明这一点。主要研究者将使用半经典分析的技术来解决L函数的次凸性问题。 这两个领域之间的联系是由周期积分公式，如兰金-塞尔伯格，Waldspurger，和一野池田。 PI希望算术流形的几何结构将使其更容易研究振荡积分和等分布问题，将出现在这项工作中。 PI还旨在证明自守形式的混沌性结果，通过证明它们的Lp范数小于一般波包的Lp范数。

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.