Analysis on arithmetic quotients

20世纪数论的主要发展之一是罗伯特·朗兰兹的 关于自同构形式的工作，其最终目标是理解L函数，它是 解析数论自早期由勒琼·迪尔奇莱特发明以来 19世纪。朗兰兹在这一领域的第一个重要成果是他构造了爱森斯坦级数， 这暗示了一族新的L函数族的亚纯延拓。他对爱森斯坦的研究 自从他第一次这样做以来，系列在近45年的时间里没有被微妙地简化过，我希望两者都能 简化和延伸，从一个新的角度来看待它。 朗兰兹的技巧只给出了亚纯延拓，但也给出了其他几种延拓，主要是Freydoon 普渡大学的沙希迪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函数得到了某些新的估计 数值常数导致了拉马努扬猜想的类似。从某种意义上说，这些 技术已经走到了死胡同，因为所有相关的有限维约化列表 团体已经筋疲力尽了。尽管沙希迪公布了负面证据，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.