New Bounds for Automorphic L-Functions

主要研究者打算证明自守L-函数的新的次凸边界或改进现有的次凸边界。这样的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导出。作为回报，他们提供了解决几个深丢番图问题的解决方案的关键等分布现象。无条件地证明这些边界也揭示了广义黎曼假设，因为它们是它的结果。建议的焦点是GL（2）x GL（1）和GL（2）x GL（2）类型的族。导致这些家庭的subconvex界限的技术到目前为止已被限制到有理数或全纯形式的领域。主要研究者将尝试将这些技巧推广到全真实的数域和非全纯形式。由于它们的基本特征，整数是许多理论和实践结构的来源，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。自守形式和它们的L-函数是使整数的隐藏对称性可见的数学对象之一。人们已经观察到，但没有严格证明在一个单一的情况下，每一个自守L-函数的零点分布在一个非常特殊的方式。这个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.