Applications of Double Dirichlet Series to Automorphic Forms and Number Theory

调查员继续他的调查应用程序的双重狄利克雷级数的研究自守L-级数和数论。 他和各种合作者一直在发展理论的双重狄利克雷级数几年。 它在许多情况下提供了Rankin-Selberg方法的简单替代。 例如，它提供了GL（2）上一般自守形式的对称平方L-级数的整体的一个非常简短的证明。 研究人员希望尽可能地扩展这种技术，以了解更高的对称幂L-函数和标准L-函数的高阶扭曲的集体行为。 作为一个额外的输入源，试图猜测潜在的应用程序的双重狄利克雷级数方法，调查研究梅林变换的某些概括的metaplectic形式。 它们是由其他非全等子群构造的。 这些形式的理论是相应的丰富，并应有助于进一步的应用的双重Dirichlet级数。L-级数的研究一直是数学发展的一个重要组成部分，在过去的100年。 例如，某些L-级数提供了导致最近证明费马最后定理的链条中的重要环节。 二重狄利克雷级数的研究导致了对L-级数的更好的理解，这反过来又导致了对一些非常深入的数学领域的进一步了解。 目前还无法预测这项研究的最终影响，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.