Iwasawa Theory, Euler Systems and Arithmetic Applications

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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.