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

Birch--Swinnerton-Dyer 猜想：超越 1 维

• 批准号: EP/V046853/1
• 负责人: David Loeffler
• 金额: $ 12.87万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV046853%2F1
• 关键词: Birch; Swinnerton; Dyer; conjecture; beyond

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.

数论的中心问题之一是描述多项式方程的有理解。线性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，曲线上的有理点的集合形成一个阿贝尔群，已知该阿贝尔群是双生成的。我们定义曲线的“秩”为这个群的秩;椭圆曲线理论的中心问题是如何确定这个秩。Birch和Swinnerton-Dyer猜想是数学中最著名的公开问题之一，它给出了这个秩的一个几何公式，并将其与一个称为L-函数的辅助对象联系起来。本项目的目标是使用称为欧拉系统的数学工具，在椭圆曲线的Birch-Swinnerton-Dyer猜想及其推广到高维对象（“阿贝尔变量”）方面取得新的进展。

项目成果

On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces

关于模阿贝尔曲面的 Birch-Swinnerton-Dyer 猜想

• 发表时间: 2021
• 作者: Loeffler, D