Iwasawa Theory, Euler Systems and Arithmetic Applications

椭圆曲线是可以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。尽管椭圆曲线的定义很简单，但它具有非常丰富的算术结构，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，尤其是涉及加密货币的交易。因此，很好地理解这些曲线的算术性质是极其重要的。在20世纪60年代，Birch和Swinnerton-Dyer提出了一个猜想，描述了给定椭圆曲线上可以有多少个点。这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7个千禧年奖问题之一，正确解决该问题的人将获得100万美元的奖金。这个问题只解决了一些特殊情况。在过去的几年里，为了解决这个猜想，数学研究取得了许多进展。针对各种特殊情况的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建一在20世纪60年代首创的一种技术。岩泽的洞察力是“一次一个质数”地研究算术对象。例如，我们不是把一个数字看成一个整体，而是看它能被一个选定的质数整除多少次。随着质数的变化，我们推断出原始数的不同信息。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，更多的“本地”信息是可用的。这种技术与Andrew Wiles在20世纪90年代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之间存在着密切的联系，模形式是非常特殊的解析函数，在各种特殊情况下为BSD问题带来了巨大的进步。另一个有希望的方法是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。在这个研究计划中，我们将：-构建特殊对象（称为欧拉系统）来研究使用iwasawa理论技术的BSD问题的新案例；-完善岩川理论中长期存在的猜想的技术和公式；-研究椭圆曲线和其他相关数学对象的算术不变量的渐近行为；-使用计算方法研究BSD问题的新案例；-采用Bhargava的技术研究岩川理论对象的统计现象；-发现岩泽理论的新算法应用。

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.