Analytic Properties of L-functions

L 函数的解析性质

• 批准号: 0209477
• 负责人: Andrew Wiles
• 金额: $ 35.06万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0209477&HistoricalAwards=false
• 关键词: Analytic; Properties; functions

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).

主要研究者的主要目的是通过分析方法来解决朗兰兹纲领的一些基本问题。他最感兴趣的是GL（2）上全纯形式的基变化的情况，其中涉及不可解的基变化，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），以实现这一目标。该提案的第二个目标是研究L-函数的一些基本分析问题。 这个建议最终涉及到寻找解方程的方法。自世纪以来，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。我感兴趣的情况是具有普通整数系数的方程。在过去的50年里，人们已经清楚地认识到，应该用一种完全不同的方法来研究这些一般方程，即用分析的方法（即从微积分中导出的方法），而不是用代数的方法。其思想是描述和证明解的许多性质（即使没有公式，这些性质仍然存在）。