The Birch--Swinnerton-Dyer conjecture: beyond dimension 1

数论的核心问题之一是描述多项式方程的有理解。线性方程和二次方程的情况是基本的，但三次方程(所谓的“椭圆曲线”)的情况要深刻得多。椭圆曲线的研究是数论的一个主要分支，与许多其他学科有联系，包括拓扑学、几何学和密码学。对于椭圆曲线，曲线上的有理点集形成一个阿贝尔群，该群被认为是有限生成的。我们将曲线的“阶”定义为该群的阶；椭圆曲线理论中的中心问题是如何确定这个阶。Birch和Swinnerton-Dyer猜想是数学中最著名的公开问题之一，它给出了一个关于这一阶的猜想公式，它与一个被称为L函数的辅助对象有关。这个项目的目标是利用被称为欧拉系统的数学工具，在椭圆曲线上的Birch-Swinnerton-Dyer猜想及其推广到高维对象(阿贝尔簇)方面取得新的进展。

One of the central problems of number theory is to describe the rational solutions of polynomial equations. The case of linear and quadratic equations is elementary, but the case of cubic equations (so-called "elliptic curves") is vastly deeper. The study of elliptic curves is a major strand of number theory, with connections to many other disciplines including topology, geometry, and cryptology. For an elliptic curve, the set of rational points on the curve forms an abelian group, which is known to be finitely generated. We define the "rank" of the curve to be the rank of this group; the central problem in the theory of elliptic curves is how to determine this rank. The Birch and Swinnerton-Dyer conjecture, one of the most famous open problems in mathematics, gives a conjectural formula for this rank, relating it to an auxiliary object known as an L-function.The goal of this project is to make new progress on the Birch--Swinnerton-Dyer conjecture for elliptic curves, and its generalisations to higher-dimensional objects ('abelian varieties'), using mathematical tools known as Euler systems.