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

PGL(2) 的强次凸性和最优大筛不等式

基本信息

批准号：
EP/W009838/1
负责人：
Ian Petrow
金额：
$ 45.98万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW009838%2F1
关键词：
Strong subconvexity optimal large sieve

项目摘要

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.

这个项目致力于发展自同构形式(某些高度对称的波形)在数论(研究整数的性质)中的应用。自同构形式在被称为朗兰兹计划的广泛的思想和猜想网络中扮演着基本构件的角色，该计划将数论、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。自同构形应用于数论最重要的方法之一是通过L函数理论，它是自同构形的化身。L函数的典型例子是Riemann Zeta函数。我们之所以关心Riemann Zeta函数，是因为它是由Berhnard Riemann在19世纪50年代发现的，用于控制素数在数行上的分布。黎曼在1859年的回忆录中提到，泽塔(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“黎曼假说”。今天，它是数学中最大的悬而未决的猜想之一。这项研究的目的是在不使用任何未经证实的猜测的情况下，建立黎曼假说的几个结果。我们要开发的一个应用程序是给出L函数在其对称性中心点的估计。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。次凸性问题有许多数论应用，例如，用只取整数值的多变量二次方程表示大整数n的次数的近似公式。L函数的次凸估计的另一个应用是解决数学物理中的一个问题，即量子唯一遍历性。这项建议中的一个项目试图证明L函数的非常强的次凸界(‘Weyl’强度)，该函数附属于从2x2矩阵群到标度的任何自同构形式。推广的黎曼假设的第二个应用是算术级数(即具有固定公差的整数序列)中的素数的个数。利用L函数，可以给出小于给定界的素数除以互质数q后剩余a的近似公式。推广的黎曼假设对这种近似的误差的大小给出了很强的控制。尽管我们不能(在没有证明GRH的情况下)说误差总是如此可控，但我们可以说存在极大误差的Q的数量是非常小的。这是20世纪60年代的经典结果，被称为Bombieri-Vinogradov定理。证明Bombieri-Vinogradov定理的主要技术输入是一个被称为大筛不等式的不等式。这个不等式所说的是，在精确的定量意义上，所有的乘法谐波都近似彼此正交。Bombieri-Vinogradov定理只是大筛不等式的大量应用之一--它是数论中极其灵活和普遍的工具。这项研究的第二个主要目的是证明2x2矩阵群上的自同构型的一个大筛子不等式，这是解析数论中的一个著名的突出问题。Weyl强度次凸性和大筛子不等式这两个目标直到最近才进入我们当前工具的显著范围，这是由于Pi和Young最近在2019年所做的工作，以及将成为项目合作伙伴的胡在2020年开发的一个新的迹公式。这些项目非常及时，处于数论研究的前沿。