Iwasawa Theory, Euler Systems and Arithmetic Applications

岩泽理论、欧拉系统和算术应用

• 批准号: RGPIN-2020-04259
• 负责人: Lei, Antonio
• 金额: $ 2.7万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739770
• 关键词: Iwasawa; Theory; Euler; Systems; Arithmetic

Elliptic curves are curves that can be defined using cubic equations. The study of these curves can be traced back to the ancient Greeks. Despite its simple definition, an elliptic curve possesses a very rich arithmetic structure, which enables us to define cryptosystems for encrypting electronic communications. They are now extensively used in financial transactions, especially those involving cryptocurrencies. It is therefore extremely important to have a good understanding of the arithmetic properties of these curves. In 1960's, Birch and Swinnerton-Dyer have formulated a conjecture that describes how many points there can be on a given elliptic curve. It is one of the most important open problems in modern-day Number Theory. In 2000, it has been chosen as one of the seven Millennium Prize Problems by the Clay Mathematics Institute, who will award one million US dollars for a correct solution to the problem. Only some special cases of this problem have been solved. Attempts to tackle this conjecture have led to many advancements in the research of Mathematics in the last few years. Partial solutions to the BSD problem for various special cases have come from Iwasawa Theory, a technique pioneered by the Japanese mathematician Kenkichi Iwasawa in 1960's. The insight of Iwasawa was to study arithmetic objects "one prime number at a time". For example, instead of looking at a number as a whole, we look at how many times it is divisible by one chosen prime number. As the prime number varies, we deduce different informations about the original number. We can then "patch" all these informations together to study the original number. The advantage of this approach is that when we focus on one prime number, more "local" information is available. This technique, combined with the seminal work of Andrew Wiles on Fermat's Last Theorem in 1990's, which tells us that there is an intimate link between elliptic curves and modular forms, which are very special analytic functions, have brought tremendous progress towards the BSD problem in various specials cases. Another promising approach is the recent work of Bhargava (Fields medallist in 2014) using statistical method, which is utilised to study the validity of the BSD problem on average over a large family of elliptic curves. In this research program, we shall: - construct special objects (called Euler systems) to study new cases of the BSD problem using Iwasawa-theoretic techniques; - refine techniques and formulations of longstanding conjectures in Iwasawa Theory; - study asymptotic behaviours of arithmetic invariants defined for elliptic curves and other related mathematical objects; - use computational methods to study new cases of the BSD problem; - adopt Bhargava's techniques to study statistical phenomenons of Iwasawa-theoretic objects; - find new arithmetic applications of Iwasawa Theory.

椭圆曲线是可以使用三次方程定义的曲线。对这些曲线的研究可以追溯到古希腊人。尽管定义很简单，椭圆曲线却拥有非常丰富的算术结构，这使我们能够定义用于加密电子通信的密码系统。它们现在广泛用于金融交易，特别是涉及加密货币的交易。因此，充分理解这些曲线的算术特性非常重要。 1960 年代，Birch 和 Swinnerton-Dyer 提出了一个猜想，描述给定的椭圆曲线上可以有多少个点。它是现代数论中最重要的开放问题之一。 2000年，该问题被克莱数学研究所选为七大千年奖问题之一，正确解决该问题将奖励100万美元。仅解决了该问题的一些特殊情况。过去几年，解决这一猜想的尝试导致数学研究取得了许多进展。各种特殊情况下 BSD 问题的部分解决方案来自岩泽理论，这是日本数学家岩泽健吉在 1960 年代首创的技术。岩泽的见解是“一次研究一个素数”的算术对象。例如，我们不是将一个数字视为一个整体，而是查看它可以被一个选定的素数整除多少次。随着素数的变化，我们推断出有关原始数的不同信息。然后我们可以将所有这些信息“修补”在一起以研究原始数字。这种方法的优点是，当我们关注一个素数时，可以获得更多“本地”信息。这项技术与 Andrew Wiles 在 1990 年代关于费马大定理的开创性工作相结合，它告诉我们椭圆曲线和模形式（这是非常特殊的解析函数）之间存在密切的联系，为解决 BSD 问题带来了巨大的进展。各种特殊情况。另一个有前途的方法是 Bhargava（2014 年菲尔兹奖得主）最近使用统计方法进行的工作，该方法用于研究 BSD 问题在一大群椭圆曲线上的平均有效性。在本研究计划中，我们将： - 使用岩泽理论技术构建特殊对象（称为欧拉系统）来研究 BSD 问题的新案例； - 完善岩泽理论中长期存在的猜想的技术和表述； - 研究为椭圆曲线和其他相关数学对象定义的算术不变量的渐近行为； - 使用计算方法来研究 BSD 问题的新案例； - 采用 Bhargava 的技术来研究 Iwasawa 理论物体的统计现象； - 寻找岩泽理论的新算术应用。