Analytic Properties of L-functions

主要研究人员的主要目的是通过分析方法来解决朗兰兹计划的一些基本问题。他最感兴趣的是GL(2)上的全纯形式的基变化的情况，其中涉及不可溶基变化，但所提出的方法应该适用于更一般的情况。目前的意图是允许(在适当的情况下)使用黎曼假设来实现这一目标。该建议的第二个目的是研究L函数的一些基本分析问题。这一建议最终涉及到寻找解方程的方法。自十九世纪以来，人们就知道某些方程有公式(如二次方程和三次方程)，但大多数没有。我感兴趣的情况是具有普通整数系数的方程。在过去的五十年里，很明显，应该有一种完全不同的方式来研究这些一般方程，使用来自分析的方法(即来自微积分的方法)，而不是使用来自代数的方法。这个想法是为了描述和证明解的许多性质(这些性质仍然存在，即使可能没有公式)。

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).