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 Daniel Bump 博士和 Jeffrey Hoffstein 博士已经开始的联合工作的延续，该研究为自守形式理论中出现的某些 L 函数建立了非零结果。沿着这些思路的部分结果已经让 Friedberg 博士和他的合作者得出结论，当在有理数上定义的椭圆曲线的 L 函数在值 1 处不为零时，Q 有理点群和 Tate-椭圆曲线的 Shafarevich 群是有限的。 在代数几何中，“曲线”是可以通过两个变量的单个多项式方程在平面中定义的东西。当该多项式中至少有一个单项式的总次数为三，并且没有单项式具有更高的次数时；当所得曲线不与自身相交时，该曲线称为“椭圆”。这些曲线看似简单，却蕴含着惊人数量的深层结构，并且是难度惊人的影响深远的猜想的基础。这些猜想包括 Birch 和 Swinnerton-Dyer 在 1960 年代提出的猜想，将某些函数的值与椭圆曲线的结构联系起来。弗里德伯格博士的工作是最近验证第一批伯奇猜想和斯温纳顿-戴尔猜想的重要组成部分。