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

摘要罗杰·豪，迪豪江耶鲁大学98 03617这个项目解决了被称为朗兰兹计划的数论领域中的几个问题。其中一个问题涉及L函数的自同构形式和特殊值的周期。研究人员也有一些剩余表示和迹公式的应用。第三部分研究了Eisenstein级数和非齐次向量空间的傅里叶系数。本世纪后半叶最重要的数学思想之一是，解析公式通常对离散信息进行编码。例如，一个人可能想要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，您可能想要计算一系列方程的解的数量，并将答案作为一个序列。数学家称这类问题为“离散”问题。微积分中出现的函数并不是“离散的”，而是“分析的”，对这些函数来说，微积分运行得很好。令人惊讶的是，正确的解析函数可以包含关于的一个或多个离散例子的答案。实际上，最早的例子是在几百年前发现的，但在过去的三十年里，这些例子已经被系统化，形成了数论的一个分支，称为朗兰兹程序。

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.