Iwasawa theory for p-adic representations

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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, elliptic curves possess very rich arithmetic structure, which enable us to define a cryptosystem for encrypting messages. They are used extensively in online communication and financial transactions. It is therefore important to have a good understanding of the arithmetic properties of these curves. In 1960’s, Birch and Swinnerton-Dyer formulated a conjecture that describes how many points there can be on any elliptic curves. It is one of the most important problems in 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. Today, it is still an open problem and only some special cases have been solved. Many tools have been developed to tackle this conjecture. One of the more fruitful approaches is Iwasawa Theory, which studies the behaviour of an elliptic curve at one prime number at a time. More specifically, let E be an elliptic curve and p a fixed prime number. We study how the number of points on E can vary when we allow the coordinates of these points to have different algebraic structures defined using p. For example, let Q be the set of rational numbers. The natural points on E to study are the ones with coordinates in Q. But we could also ask how many points there are if we allow the coordinates to be expressions of numbers in Q and a square root. What if we relax this condition further and allow fourth roots? Eight roots? What is the asymptotic behaviour if we keep on doing this forever? Surprisingly, we are able to describe this behaviour by very explicit formulae, thanks to the algebraic tools mathematicians in Iwasawa Theory have developed over the years. In this project, we will study some of these tools and apply them to different mathematical objects. For example, instead of just studying elliptic curves, we will study abelian varieties, which are higher-dimensional avatars of elliptic curves. These abstract geometric objects have similar arithmetic structures as elliptic curves. But they are more complex and more difficult to understand because its dimension can be arbitrarily large. As a result, these objects could potentially have important applications in cryptography in the future.