Residual Representations, Relative Trace Formulas, Fourier Coefficients of Eisenstein Series

残差表示、相对迹公式、爱森斯坦级数的傅立叶系数

• 批准号: 9803617
• 负责人: Dihua Jiang
• 金额: --
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803617&HistoricalAwards=false
• 关键词: Residual; Representations; Relative; Trace; Formulas

ABSTRACT Roger Howe, Dihau Jiang Yale University 98 03617 This project tackles several problems in an area of number theory known as the Langlands Program. One problem concerns the periods of automorphic forms and special values of L-functions. The investigators also have some applications of residual representations and trace formula. The third part of the study concerns Fourier coefficients of Eisenstein series and prehomogeneous vector spaces. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples about. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, these examples have been systematized into a branch of number theory called the Langlands program.

摘要 Roger Howe，Dihau Jiang耶鲁大学98 03617 这个项目解决了数理论领域中的几个问题，称为朗兰兹纲领。其中一个问题涉及自守形式的周期和L-函数的特殊值。研究者们也对剩余表示和迹公式有一些应用。 研究的第三部分涉及爱森斯坦级数和预齐次向量空间的傅立叶系数。 世纪后半叶最重要的数学思想之一是，解析公式经常编码离散信息。例如，一个人可能想计算一个特定方程的解的数量，但发现很难找到解。另一方面，你可能想计算一个方程序列的解的数量，并将答案作为一个序列。数学家称这类问题为“离散”问题。出现在微积分中的函数，以及微积分如此有效的函数，不是“离散的”，而是“分析的”。“令人惊讶的是，正确的解析函数可以包含离散例子的答案。 实际上，第一个例子是在几百年前发现的，但在过去的30年里，这些例子已经被系统化，成为数论的一个分支，称为朗兰兹纲领。