Strong subconvexity and an optimal large sieve inequality for PGL(2)

这个项目旨在发展自守形式（某些高度对称的波形）在数论（研究整数的性质）中的应用。自守形式在一个广泛的思想和结构网络中扮演着基本的构建块的角色，这个网络被称为朗兰兹纲领，它将数论、代数几何、动力学、分析和数学物理等不同的数学领域联系在一起。自守形式可以应用于数论的最重要的方法之一是通过L-函数的理论，它充当自守形式的化身。L函数的典型例子是黎曼zeta函数。我们关心黎曼zeta函数，因为它是在19世纪50年代由Bernard Riemann发现的，用于控制素数在数线上的分布。黎曼在他1859年的回忆录中提到，zeta（s）的所有非平凡根很可能都位于Re（s）=1/2这条线上，但他无法证明这一点。这个猜想现在被称为“黎曼猜想”。它是今天数学中最伟大的未解难题之一。本提案中的研究旨在建立黎曼假设的几个结论，而不使用任何未经证明的假设。一个应用程序，我们的目标是开发的L-函数在其对称性的中心点的估计。这是一个非常活跃和令人兴奋的研究领域，被称为次凸性问题。次凸性问题在数论中有许多应用，例如用一个只取整数值的多元二次方程来表示一个大整数n的近似公式。L-函数的次凸估计的另一个应用是数学物理中被称为量子唯一遍历性的问题。该提案中的一个项目试图证明附加到任何自守形式的L-函数的非常强的次凸边界（“Weyl”强度），这些自守形式产生于2x2矩阵的组，直到缩放。广义黎曼假设的第二个应用是算术级数中素数的数量（即具有固定公差的整数序列）。利用L-函数，可以给出小于给定界的素数除以某个互质数q后的余数a的个数的近似公式。广义黎曼假设对这种近似中的误差大小提供了非常强的控制。尽管我们不能（在没有证明GRH的情况下）说误差总是如此受控，但我们可以说存在异常大误差的q的数量非常小。这是20世纪60年代的一个经典结果，被称为贝塞里-维诺格拉多夫定理。证明贝塞里-维诺格拉多夫定理的主要技术输入是一个被称为大筛不等式的不等式。这个不等式的意思是，所有乘性谐波在精确的定量意义上彼此近似正交。贝塞里-维诺格拉多夫定理只是大筛不等式的众多应用之一，它是数论中极其灵活和普遍的工具。本研究的第二个主要目标是证明2x2矩阵群上自守形式的大筛不等式，这是解析数论中一个著名的突出问题。由于PI和Young在2019年的最新工作以及Hu在2020年开发的新迹公式，Weyl强度次凸和大筛不等式这两个目标最近才进入我们现有工具的惊人范围。这些项目非常及时，处于数论研究的前沿。

This project seeks to develop applications of automorphic forms (certain highly symmetric wavesforms) to number theory (the study of the properties of whole numbers). Automorphic forms play the role of fundamental building blocks in a wide-ranging web of ideas and conjectures known as the Langlands program, which ties together such diverse areas of mathematics as number theory, algebraic geometry, dynamics, analysis, and mathematical physics. One of the most important ways that automorphic forms can apply to number theory is via the theory of L-functions, which act as avatars of automorphic forms. The prototypical example of an L-function is the Riemann zeta function. We care about the Riemann zeta function because it was found in 1850s by Berhnard Riemann to control the distribution of prime numbers on the number line. Riemann mentioned in his 1859 memoir that it is very probable that all nontrivial roots of zeta(s) lie on the line Re(s)=1/2, but he was unable to prove this. This conjecture is now known as the "Riemann Hypothesis". It is today one of the greatest unsolved conjectures in mathematics. The research in this proposal aims to establish several of the consequences of the Riemann Hypothesis without using any unproven conjectures. One application that we aim to develop is to give estimates for L-functions at the center point of their symmetry. This is a highly active and exciting area of reserach known as the subconvexity problem. There are many number theoretic applications of the subconvexity problem, for instance approximate formulas for the number of representations of a large integer n by a quadratic equation in several variables taking only integer values. Another application of subconvex estimates for L-functions is to a question in mathematical physics known as quantum unique ergodicity. One project in this proposal seeks to prove very strong subconvex bounds (of 'Weyl' strength) for L-functions attached to any automorphic form arising from the group of 2x2 matrices up to scaling.A second application of the generalised Riemann hypothesis is to the number of primes in an arithmetic progression (i.e. a sequence of whole numbers having a fixed common difference). Using L-functions, one can give an approximate formula for the number of primes less than a given bound which are of remainder a after dividing by some coprime number q. The generalised Riemann hypothesis gives a very strong control on the size of the error made in this approximation. Even though we cannot (without proving the GRH) say that the error is always so controlled, we can say that the number of q's for which there is an exeptionally large error is exceedingly small. This is a by now classical result from the 1960s known as the Bombieri-Vinogradov theorem.The main technical input to proving the Bombieri-Vinogradov theorem is an inequality known as the large sieve inequality. What this inequality says is that all multiplicative harmonics are approximately orthogonal to each other, in a precise quantitative sense. The Bombieri-Vinogradov theorem is only one of a large number of applications of the large sieve inequality - it is extremely flexible and ubiquitous tool in number theory. The second major goal of this reserach grant is to prove a large sieve inequality for automorphic forms on the group of 2x2 matrices up to scaling, a well-known outstanding problem in analytic number theory. The two goals of Weyl-strength subconvexity and the large sieve inequality have only very recently come within striking range of our current tools due to recent work of the PI and Young in 2019 and a new trace formula developed by Hu in 2020, who will be a project partner. The projects are highly timely and at the cutting edge of research in number theory.