Mathematical Sciences: Eisenstein Series on the Metaplectic Group

数学科学：爱森斯坦Metaplectic群系列

• 批准号: 8821762
• 负责人: Solomon Friedberg
• 金额: $ 10.43万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8821762&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Eisenstein; Series; Metaplectic

The proposed research is a continuation of joint work already begun with Dr. Daniel Bump and Dr. Jeffrey Hoffstein, which established nonvanishing results for certain L-functions arising in the theory of automorphic forms. Partial results along these lines have already allowed Dr. Friedberg and his collaborators to conclude that when the L-function of an elliptic curve defined over the rational numbers is nonzero at the value 1, both the group of Q-rational points and the Tate-Shafarevich group of the elliptic curve are finite. In algebraic geometry, a "curve" is what may be defined in the plane by a single polynomial equation in two variables. When at least one of the monomials in this polynomial has total degree three, and no monomials have higher degree; and when the resulting curve does not cross itself, the curve is called "elliptic". These curves, seemingly so simple, carry nonetheless an astonishing amount of deep structure, and have been the basis of farreaching conjectures of astonishing difficulty. Among these conjectures are those made in the 1960's by Birch and Swinnerton- Dyer, connecting values of certain functions to the structure of elliptic curves. Dr. Friedberg's work is an essential part of the recent verification of the first cases of the Birch and Swinnerton-Dyer Conjectures.

拟议的研究是联合工作的继续 已经开始由丹尼尔·邦普博士和杰弗里·霍夫斯坦博士， 它建立了某些L-函数的非零结果 产生于自守形式理论。部分结果沿着 这些线路已经让弗里德伯格博士和他的 合作者得出结论，当椭圆函数的L-函数 定义在有理数上的曲线在值 1，Q-有理点群和Tate-Shafarevich 椭圆曲线上的所有点都是有限的。 在代数几何学中，“曲线”可以定义为 用一个二元多项式方程来表示平面。当 在这个多项式中至少有一个单项式具有总次数 三是不求高，不求高，不求高。 结果曲线不与自身相交，则该曲线称为 “椭圆”。这些看似简单的曲线， 数量惊人的深层结构， 具有惊人难度的深远的挑战。其中 这些照片是20世纪60年代由伯奇和斯温纳顿制作的- Dyer，将某些函数的值与 椭圆曲线弗里德伯格博士的工作是一个重要组成部分， 最近核实的第一个案件的桦树和 Swinnerton-Dyer猜想