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的Daniel Bump教授的自同构形式的研究。Bump博士计划的大部分研究是与加州大学圣克鲁兹分校的S.Fry-berg和布朗大学的J.Hoff-Stein合作的继续。通过将Novodvorsky引入的积分卷积应用于某些Eisenstein级数，他们利用了自同构的Eisenstein级数的二次扭曲在亚普勒群上的出现。这导致了具有已知极点的Dirichlet级数，从而得到了一种新类型的非零定理。他们已经得到了一个结果，当与Kolyvan in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十九世纪初，最初是一种数学奇闻，但到那个世纪末，数学家们已经意识到许多具有根本重要性的物体本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已经形成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。这个领域主要涉及关于整数的问题，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.