The Subconvexity Problem

素数在数学中具有基本的重要性，因为每个数字都可以唯一地写成素数的乘积。素数的一个神秘特征是，关于它们如何分布的统计数据类似于随机数字序列所产生的统计数据。尽管事实上一个数字是否是素数并不是随机的。l函数是一种数学对象，它可以让我们从数论中推测出质数和其他对象的统计特征。本课题研究了关于l -函数的一个重要猜想，即所谓的次凸性问题，它表明这些l -函数所取的值比预期的要小。如果是真的，这将是解锁L函数包含的统计信息的重要一步。本奖项的目的是在高阶l函数的次凸性问题上取得进展，并在尽可能广泛的自同构形式下证明次凸性。该项目将重点关注U(n+1) x U(n)和GL(n+1) x GL(n)组。PI将使用算术放大，表示理论和来自微局部分析的思想来解决问题。微型局部升降机将发挥关键作用。所使用的方法在本质上并不取决于小组的等级。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Prime numbers are of fundamental importance in mathematics, as every number can be uniquely written as a product of primes. A mysterious feature of prime numbers is that the statistics of how they are distributed is similar to what a random sequence of numbers would produce. This is despite the fact that whether a number is prime or not is in no respect random. L-functions are mathematical objects that conjecturally allow us to understand this statistical feature of primes and other objects from the theory of numbers. This project investigates an important conjecture about L-functions, known as the subconvexity problem, which states that the values taken by these L-functions are smaller than expected. If true, it would be an important step in unlocking the statistical information that L- functions contain.The aim of this award is to make progress on the subconvexity problem for L-functions in higher rank, and to prove subconvexity for as wide a range of automorphic forms as possible. The project will focus on the groups U(n+1) x U(n) and GL(n+1) x GL(n). The PI will approach the problem using arithmetic amplification, representation theory, and ideas from microlocal analysis. A key role will be played by microlocal lifts. The methods used will not depend on the rank of the group in an essential way.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.