New Bounds for Automorphic L-Functions

本课题旨在证明自同构l函数的新的次凸界或改进现有的次凸界。这种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推导出来。作为回报，它们提供了解决几个涉及平均分配现象的深层丢芬图问题的关键。无条件地证明这些边界也有助于阐明广义黎曼假设，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)类型的家族。迄今为止，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。首席研究员将尝试将这些技术扩展到全实数域和非全纯形式。这个建议属于整数理论。由于整数的基本特性，它是许多理论和实践结构的源泉，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。自同构形式及其l -函数是使整数的隐藏对称性可见的数学对象之一。每个自同构l函数的零点都以一种非常特殊的方式分布，这已经被观察到，但没有在单个实例中得到严格证明。这个观察就是广义黎曼假设，它暗示了整数的许多其他未知性质。对于许多重要的问题，一个较弱的假设，关于l函数的大小，就足够了。首席研究员打算证明这个较弱的假设的新实例。

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.