Subconvexity Bounds of L-Functions

“这个奖项是根据2009年美国复苏和再投资法案(公法第111-5条)资助的。”这项提议旨在打破L$的凸性界限--对高级别群的函数。主要研究者在突破GL(3)上自对偶L函数的凸性界方面已经取得了第一个进展。文中从多个方面讨论了高阶群上L函数突破凸性界的有关问题。提出了具体的猜想，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公开问题之一-n次L函数的林德洛夫假设密切相关。次凸界是通向Lindeloff假设的第一步。该建议涉及到构造在某些几何空间上的某些数理论函数(具有一个复参数的无限和)的大小的研究。确定这些函数的界限很重要，不仅因为它的理论价值，而且因为它的广泛应用。例如，一个更强的量子唯一遍历性猜想是一个应用，它告诉我们某些测度在某些几何空间上的收敛以及这些测度的收敛速度。这是数学物理中的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.