Explicit Approaches to the Birch and Swinnerton-Dyer Conjecture

Birch和Swinnerton-Dyer猜想是数论中最核心的未解问题之一。 拟议的项目旨在从两个角度研究这一猜想。 首先，PI打算继续他的计划，在许多特定情况下使用岩泽理论，欧拉系统，L函数和伽罗瓦上同调的显式计算来明确验证整个猜想;这项工作同时激励新算法的发展和现有定理的改进。 其次，PI计划对科利瓦金的Heegner点的欧拉系统的性质进行理论和计算研究，希望将科利瓦金在20世纪90年代早期未完成的几个想法发扬光大。椭圆曲线在现代数论和算术几何中发挥着核心作用。 例如，安德鲁·怀尔斯证明了费马大定理，他证明了格哈德·弗雷附加到费马主张的一个反例上的椭圆曲线将附加到一个模形式上，而这个模形式不存在。 我们对椭圆曲线的理解是广泛的，但许多问题仍然没有解决。 也许最核心的未解决的问题是伯奇和斯温纳顿-戴尔猜想，它涉及椭圆曲线的许多算术不变量。 该项目的目标是提供大量的新数据，技术和思想相关的攻击伯奇和Swinnerton-Dyer猜想。

ABSTRACT for award DMS-0653968 of SteinThe Birch and Swinnerton-Dyer conjecture is one of the most central unsolved problems in number theory. The proposed project aims to study this conjecture from two perspectives. First, the PI intends to continue his program to explicitly verify the full conjecture in many specific cases using explicit computations with Iwasawa theory, Euler systems, L-functions, and Galois cohomology; this work simultaneously motivates the development of new algorithms and the refinement of existing theorems. Second, the PI plans to carry out a theoretical and computational study of properties of Kolyvagin's Euler system of Heegner points, with the hope of carrying forward several of the ideas Kolyvagin left unfinished in the early 1990s.Elliptic curves play a central role in modern number theory and arithmetic geometry. For example, Andrew Wiles proved Fermat's Last Theorem by showing that the elliptic curve attached by Gerhard Frey to a counterexample to Fermat's claim would be attached to a modular form, and that this modular form cannot exist. Our understanding of the world of elliptic curves is extensive, but many questions remain unresolved. Perhaps the most central unsolved problem is the Birch and Swinnerton-Dyer conjecture, which relates many of the arithmetic invariants of an elliptic curve. The goals of this project are to provide substantial new data, techniques, and ideas relevant to attacks on the Birch and Swinnerton-Dyer conjecture.