Analysis on arithmetic quotients

20世纪数论的一个主要发展是罗伯特·朗兰兹的 工作的自守形式，其最终目标是理解L-函数，一个主要的工具， 解析数论，自从他们第一次发明的勒琼Dircihlet在早期 世纪。 朗兰兹的第一个重要成果在这一领域是他的建设爱森斯坦系列， 这意味着一族新的L-函数的亚纯延拓。 他对爱森斯坦的研究 自从他第一次做这个系列以来，在将近45年的时间里，这个系列并没有被巧妙地简化，我希望这两个 简化和扩展它，从一个新的角度来看它。 朗兰兹的技术只给出了亚纯延拓，但其他几个，主要是Freydoon 普渡大学的Shahidi，用爱森斯坦级数证明了更强的性质， 包括一个函数方程，并应用这些L-函数，以获得新的估计某些 数值常数导致一个类似的拉马努金猜想。 从某种意义上说， 技术已经走到了死胡同，因为所有相关的有限维还原列表 团体已经筋疲力尽。 尽管Shahidi公布了负面证据，但我认为， 无限维Kac-Moody群是值得探索的。

One of the major developments in number theory during the 20th century was Robert Langlands' work on automorphic forms, with its ultimate goal of understanding L-functions, a primary tool in analytic number theory ever since they were first invented by Lejeune Dircihlet in the early 19th century. Langlands' first important results in this field was his construction of Eisenstein series, which implied the meromorphic continuation of a new family of L-functions. His work on Eisenstein series has not been subtantially simplified in the nearly 45 years since he first did it, and I hope to both simplify and extend it, looking at it from a new perspective. Langlands' technique gave only meromorphic continuation, but several others, primarily Freydoon Shahidi of Purdue University, have used Eisenstein series to prove stronger properties, including a functional equation, and applied these L-functions to obtain new estimates of certain numerical constants leading to an analogue of Ramanujan's conjecture. In one sense these techniques have come to a dead end, since the list of all relevant finite-dimensional reductive groups has been exhausted. In spite of negative evidence published by Shahidi, I believe the infinite-dimensional Kac-Moody groups are worth exploring.