Mathematical Sciences: Automorphic Forms on GL(r) and the Metaplectic Groups

本研究将集中于自守形式理论 关于元群的 它将试图表明，L- 与n阶Dirichlet特征标相关的函数出现 作为n重覆盖上Eisenstein级数的Fourier系数 GL（n）。 这个结果对解析数有应用 理论和制定适当的概括， Waldspurger定理 它将进一步试图表明， n重自守形式的Euler积 SL（r）的覆盖是具有以下形式的Rankin-Selberg卷积： 在GL（n）的n重覆盖上的theta函数。 它也将 试图证明爱森斯坦的傅里叶系数 GL（r+1）的n重覆盖上的级数是Rankin-Selberg n重覆盖上的θ函数卷积 GL（n-1）。 此外，凹凸将研究自守形式和L- 在特殊群G2上的函数。 最后工作将完成 上非分歧Whittaker函数的高阶卷积 从广义巴恩斯引理的观点出发， 广义超几何级数 本文的研究属于自守形式的一个分支 数论的，其中数论函数被编码 复杂的分析函数， 分析来承担数论问题。 这 这一想法在许多方面得到了推广，并已被证明是一种 许多领域的基本工具。 多变量推广 是这项研究的重点，Bump是这方面的领导者。 方向 这项研究的结果将被证明是非常 令人兴奋.

This research will focus on the theory of automorphic forms on the metaplectic groups. It will attempt to show that L- functions associated with Dirichlet characters of order n occur as Fourier coefficients of Eisenstein series on the n-fold covers of GL(n). This result would have applications to analytic number theory and to the formulation of the proper generalization of Waldspurger's theorem. It will further attempt to show that the Euler product associated with an automorphic form on the n-fold cover of SL(r) is the Rankin-Selberg convolution of the form with a theta function on the n-fold cover of GL(n). Also it will attempt to show that the Fourier coefficients of an Eisenstein series on the n-fold cover of GL(r+1) are Rankin-Selberg convolutions of the form with theta functions on the n-fold cover of GL(n-1). Also Bump will investigate automorphic forms and L- functions on the exceptional group G2. Finally work will be done on the higher convolutions of nonramified Whittaker functions on GL(r,R) from the point of view of generalized Barnes' lemmas and generalized hypergeometric series. This research is in the area of automorphic forms, a branch of number theory wherein number theoretic functions are encoded into complex analytic functions allowing the deep tools of analysis to come to bear on the number theory problems. This idea is generalized in many ways and has proved to be a fundamental tool in many areas. Multivariable generalizations are the focus of this research and Bump is a leader in this direction. The results of this research will prove to be very exciting.