Mathematical Sciences: Eisenstein Series on the Metaplectic Group

这项拟议的研究是与Daniel Bump博士和Jeffrey Hoffstein博士已经开始的联合工作的继续，他们建立了某些L函数的非零结果，这些函数产生于自同构形理论。沿着这些直线的部分结果已经允许Friedberg博士和他的合作者得出结论：当定义在有理数上的椭圆曲线的L函数在值1处非零时，椭圆曲线的q-有理点群和泰特-沙法雷维奇群都是有限的。在代数几何中，“曲线”可以在平面上用两个变量的单一多项式来定义。当此多项式中至少有一个单项式具有三次之和，且没有一项单项式具有更高的次数时；当所得到的曲线不与自身相交时，该曲线称为“椭圆”。这些曲线看似如此简单，但却承载着惊人的深层结构，并一直是令人震惊的难度猜想的基础。在这些猜想中，有Birch和Swinnerton-Dyer在1960年代的S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对Birch猜想和Swinnerton-Dyer猜想的首批案例进行验证的重要部分。

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.