Applications of Double Dirichlet Series to Automorphic Forms and Number Theory

研究者继续研究二重狄利克雷级数在自同构l -级数和数论研究中的应用。他和许多合作者几年来一直在发展双狄利克雷级数理论。在许多情况下，它为Rankin-Selberg方法提供了一个简单的替代。例如，它提供了GL(2)上一般自同构形式的对称平方l级数的完整的一个非常简短的证明。研究者希望尽可能地扩展这种技术，以理解高对称幂l函数和标准l函数的高阶扭曲的集体行为。作为尝试猜测双重狄利克雷级数方法的潜在应用的额外输入来源，研究者研究了某些广义形式的Mellin变换。这些是由其他非同余子群构造的。这些形式的理论相应丰富，应有助于进一步深入了解二重狄利克雷级数的应用。l级数的研究是近百年来数学发展的重要组成部分。例如，某些l级数提供了最近证明费马大定理的链条中的重要环节。对二重狄利克雷级数的研究使我们对l级数有了更好的理解，从而使我们对数学的一些非常深刻的领域有了进一步的了解。目前还无法预测这项调查的最终影响，但潜在的应用领域包括密码学。

The investigator continues his investigation of applications of double Dirichlet series to the study of automorphic L-series and number theory. He and various collaborators have been developing the theory of double Dirichlet series for several years. It offers a simple replacement for the Rankin-Selberg method in many instances. For example, it provides a very short proof of the entirety of the symmetric square L-series of a generic automorphic form on GL(2). The investigator hopes to extend this technique as far as possible to understand higher symmetric power L-functions and the collective behavior of higher order twists of standard L-functions. As an additional source of input for trying to guess potential applications of the double Dirichlet series method, the investigator studies the Mellin transforms of certain generalizations of metaplectic forms. These are constructed from other non congruence subgroups. The theory of these forms is correspondingly rich and should contribute further insights into applications of double Dirichlet series.The study of L-series has been an essential part of the development of mathematics over the last 100 years. For example, certain L-series provided vital links in the chain that led to the recent proof of Fermat's last theorem. The study of double Dirichlet series is leading to an improved understanding of L-series which in turn leads to further knowledge of some very deep areas of mathematics. It is impossible to predict the ultimate impact of this investigation, but potential areas of application include cryptography.