Mathematical Sciences: Eisenstein Series on the Metaplectic Group, Special Values of Automorphic L-Functions and Functional Equations

该奖项支持斯坦福大学 Daniel Bump 教授在自同构方面的研究。 Bump 博士计划中的大部分研究是与加州大学圣克鲁斯分校的 S. Friedberg 和布朗大学的 J. Hoffstein 合作的延续。他们通过将 Novodvorsky 引入的积分卷积应用于这些爱森斯坦级数，利用了某些爱森斯坦级数的自同构 L 函数在元波群上出现的二次扭曲。这产生了具有已知极点的狄利克雷级数，从而产生了一种新型的非零定理。他们已经获得了一个结果，当与 Kolyvagin 最近的工作相结合时，该结果可应用于椭圆曲线上的 Birch-Swinnerton-Dyer 猜想。 非欧几里得平面几何始于 19 世纪初，最初是作为一种数学好奇心，但到了该世纪末，数学家们意识到许多具有根本重要性的物体在其基本性质上都是非欧几里得的。 对非欧几里得平面几何的详细研究催生了现代数学的几个分支，其中模和自同构形式的研究是最活跃的之一。 该领域主要关注有关整数的问题，但在几何和分析的使用中，它保留了与其历史根源的联系。

This award supports the research in Automorphic Forms of Professor Daniel Bump of Stanford University. Much of Dr. Bump's planned research is a continuation of joint work with S. Fried- berg of the University of California at Santa Cruz and J. Hoff- stein of Brown University. They exploit the occurrence of quadratic twists of automorphic L-functions of certain Eisenstein series on the metaplectic group, by applying to these Eisenstein series an integral convolution introduced by Novodvorsky. This results in a Dirichlet series with known poles, leading to nonvanishing theorems of a new type. They have already obtained a result which, when combined with recent work of Kolyvagin, has applications to the Birch-Swinnerton-Dyer conjectures on elliptic curves. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of Modular and Automorphic Forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of Geometry and Analysis, it retains connection to its historical roots.