New Bounds for Automorphic L-Functions

主要研究人员打算证明自同构L函数的新的次凸界或改进已有的次凸界。这些界限反映了它们的源对象的算术性质，因为它们不能从简单的分析原理中得出。作为回报，它们提供了解决解决均匀分布现象的几个深层次丢番图问题的关键。无条件地证明这些界限也有助于理解广义黎曼假设，因为它们是广义黎曼假设的结果。提案的重点是GL(2)x GL(1)和GL(2)x GL(2)类型族。到目前为止，导致这些族的次凸界的技术仅限于有理数或全纯形式的领域。主要研究人员将尝试将这些技巧扩展到全实数域和非全纯形式。该建议属于整数理论。由于整数的基本性质，整数是许多理论和实践构造的源头，包括通过互联网进行安全通信或通过噪声信道进行有效通信的算法。自同构型及其L函数是使整数的隐藏对称性可见的数学对象之一。已经观察到每个自同构L函数的零点是以一种非常特殊的方式分布的，但没有一个实例得到了严格的证明。这种观察是广义黎曼假设，它暗示了整数的许多其他未知的性质。对于各种重要问题，关于L函数大小的一个较弱的假设就足够了。首席研究员打算证明这一较弱假说的新实例。

The principal investigator intends to prove new subconvex bounds for automorphic L-functions or improve on existing subconvex bounds. Such bounds reflect the arithmetic nature of their source objects as they cannot be derived from simple analytic principles. In return, they provide the key to the solution of several deep diophantine problems addressing equidistribution phenomena. Proving these bounds unconditionally also sheds light on the Generalized Riemann Hypothesis as they are consequences of it. In the focus of the proposal are families of GL(2) x GL(1) and GL(2) x GL(2) type. The techniques leading to subconvex bounds for these families have so far been restricted to the field of rational numbers or holomorphic forms. The principal investigator will try to extend these techniques to totally real number fields and non-holomorphic forms.This proposal belongs to the theory of integers. Because of their fundamental character, integers are at the source of many theoretical and practical constructions including algorithms for secure communication through the internet or efficient communication through a noisy channel. Automorphic forms and their L-functions are among the mathematical objects that make the hidden symmetries of integers visible. It has been observed, but not proved rigorously in a single instance, that the zeros of every automorphic L-function are distributed in a very special way. This observation is the Generalized Riemann Hypothesis which implies many otherwise unknown properties of the integers. For various important questions a weaker hypothesis, concerning the size of L-functions, suffices. The principal investigator intends to prove new instances of this weaker hypothesis.