Explicit Approaches to the Birch and Swinnerton-Dyer Conjecture

关于斯坦奖DMS-0653968，白桦和斯温纳顿-戴尔猜想是数论中最核心的悬而未决的问题之一。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函数和伽罗华上同调的显式计算来显式地验证完整的猜想；这项工作同时推动了新算法的发展和现有定理的完善。其次，PI计划对柯利威因的海格纳点的欧拉系的性质进行理论和计算研究，以期发扬柯利文在20世纪90年代初留下的几个未完成的思想。椭圆曲线在现代数论和算术几何中起着核心作用。例如，安德鲁·怀尔斯证明了费马最后定理，证明了格哈德·弗雷在费马主张的反例中附加的椭圆曲线将附在模形式上，并且这种模形式不存在。我们对椭圆曲线世界的理解是广泛的，但许多问题仍然没有解决。也许最核心的悬而未决的问题是Birch和Swinnerton-Dyer猜想，它与椭圆曲线的许多算术不变量有关。这个项目的目标是提供大量新的数据、技术和想法，与攻击Birch和Swinnerton-Dyer猜想有关。

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.