Subconvexity Bounds of L-Functions

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”该建议旨在打破高阶群上$L$-函数的凸性界限。在突破GL(3)上的自对偶l函数的凸界方面，主要研究者已经取得了第一个进展。本文讨论了高秩群上l -函数的凸界破缺的各个方面的问题。提出了具体的推测，讨论了可能遇到的困难，提出了新的技术和工具。这些项目与数论中最基本的开放问题之一密切相关- n次l -函数的Lindeloff假设。次凸边界是林德洛夫假设的第一步。本文研究了构造在几何空间上的若干理论函数（具有一个复参数的无穷和）的大小。限定这类函数的重要性不仅在于它的理论价值，还在于它的广泛应用。例如，一个应用将是一个更强的量子唯一遍历性猜想，它告诉我们某些测度在某些几何空间上的收敛性以及这些测度的收敛率。这是数学物理中一个非常重要的问题。其他的应用将在未来被发现。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."This proposal is aimed at breaking the convexity bounds for $L$-functions on high rank groups. The principal investigator has already made the first progress on breaking the convexity bounds for self dual L-functions on GL(3). The problems related to breaking the convexity bounds for L-functions on high rank groups in various aspects are discussed in the proposal. Concrete conjectures are made, possible difficulties are discussed, new techniques and tools are mentioned. These projects are closely related to one of the most fundamental open problems in number theory - the Lindeloff hypothesis for degree n L-functions. The subconvexity bounds are the first steps toward the Lindeloff hypothesis.The proposal is concerned with the study of the size of certain number theoretical functions (infinite sums with one complex parameter) which are constructed on some geometric spaces. Bounding such functions are important not only because of its theoretical value but also because of its vast applications. For example, one application would be a stronger quantum unique ergodicity conjecture which tells us the convergence of certain measures on certain geometric spaces as well as the rate of convergence of these measures. This is a very important problem in mathematical physics. Other applications will be found in the future.