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。尽管椭圆曲线的定义很简单，但它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。它们现在被广泛用于金融交易，特别是那些涉及加密货币的交易。因此，对这些曲线的算术性质有一个很好的了解是非常重要的。1960年，S、伯奇和斯温纳顿-戴尔提出了一个猜想，即一条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，它被克莱数学研究所选为七个千禧年奖问题之一，该研究所将奖励100万美元，以奖励正确解决该问题。这个问题只有一些特例得到了解决。在过去的几年里，解决这一猜想的尝试导致了数学研究的许多进步。岩泽理论是由日本数学家岩泽贤一在1960年的《S》中首创的一种方法。岩泽健一的洞见是研究算术对象“一次一个素数”。例如，我们不是把一个数字作为一个整体来看，而是看它能被一个选定的素数整除多少次。随着素数的变化，我们推导出关于原始数的不同信息。然后，我们可以将所有这些信息“拼凑”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，可以获得更多的“局部”信息。这一技巧与安德鲁·威尔斯在1990年《S》中关于费马大定理的开创性工作相结合，告诉我们椭圆曲线和模形式这两个非常特殊的解析函数之间存在着密切的联系，在各种特殊情况下给BSD问题带来了巨大的进展。另一种有希望的方法是Bhargava(2014年菲尔兹奖获得者)最近使用统计方法所做的工作，该方法被用于研究BSD问题在一大类椭圆曲线上的平均有效性。在这个研究计划中，我们将：-构造特殊的对象(称为欧拉系统)，以使用岩泽理论技术研究BSD问题的新情况；-改进岩川理论中长期存在的猜想的技术和公式；-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问题的新情况；-采用Bhargava的技术来研究岩川理论对象的统计现象；-发现岩川理论的新的算术应用。