Analytic Properties of L-functions

首席研究员的主要目标是利用分析方法解决朗兰兹程序中的一些基本问题。他最感兴趣的是GL(2)上全纯形式的碱变化，其中涉及不溶性碱变化，但所提出的方法应该更普遍地工作。目前的意图是允许使用黎曼假设（在适当的设置）来实现这一目标。本文的第二个目标是研究l函数的一些基本解析问题。这个建议的最终目的是寻找解方程的方法。自19世纪以来，人们已经知道某些方程有公式（如二次方程和三次方程），但大多数方程没有公式。我感兴趣的是具有普通整数系数的方程。在过去的五十年中，很明显，应该有一种完全不同的方法来研究这些一般方程，即使用分析方法（即从微积分中推导出来的方法），而不是使用代数方法。其思想是描述和证明解的许多性质（这些性质仍然存在，即使它们可能没有公式）。

The principal investigator intends primarily to pursue the goal of attacking some of the fundamental problems of the Langlands program by means of analytic methods. He is most interested in the case of base change for holomorphic forms on GL(2) where a non-soluble base change is involved, but the proposed methods should work much more generally. At present the intention is to permit the use of the Riemann Hypothesis (in an appropriate setting) to achieve this goal. A second goal of the proposal is to study some of the fundamental analytic questions on L-functions. This proposal is ultimately concerned with finding methods to solve equations. Since the nineteenth century it has been known that certain equations have formulas (like the quadratic equation and the cubic equation) but that most do not. The cases that interest me are the equations with ordinary integer coefficients. It has become apparent in the last fifty years that there should be a completely different way of studying these general equations by using methods from analysis (i.e. derived from calculus) rather than by using methods from algebra. The idea is to describe and prove many properties of the solutions (which still exist even though there may be no formulas for them).