p-adic Iwasawa theory for Galois representations

Iwasawa theory is a subfield of number theory whose fundamental questions relate arithmetic problems (e.g. rational solutions to polynomial equations in several variables) to analytic information (e.g. values of certain complex-analytic functions, so-called L-functions). The merging of these two disparate ideas makes Iwasawa theory a mysterious and powerful field. Via its main conjectures, Iwasawa theory remains the only systematic method known today for studying the deep connections between the arithmetic problems and special values of L-functions, typified by the conjecture of Birch and Swinnerton-Dyer.

One can use Iwasawa theory to study the arithmetic of elliptic curves, which are a class of polynomial equations in two variables. Questions about the arithmetic of elliptic curves are linked to some of the oldest problems in number theory, such as the congruent number problem which concerns the existence of right-angled triangles with certain properties. Elliptic curves also played a central role in Wiles' proof of Fermat's last theorem.

In the proposed research I plan to investigate some aspects of the Iwasawa theory of elliptic curves using recently developed tools from an another area of mathematics called 'p-adic analysis'. This should give new insight into some important problems (such as the Birch--Swinnerton-Dyer conjecture) in algebraic number theory.

The proposed research promises to have both academic and non-academic impact in several ways.

Since it would be the first application of very recently developed tools from p-adic analysis in Iwasawa theory, it would profoundly change our understanding of the interplay between the two different branches of mathematics. This would have far-reaching consequences for further research in these areas. In particular, it would play a role in developing momentum for research in this area, and possibly in attracting researchers from abroad. It would also give rise to many small-scale projects which would be ideally suited for post-docs and PhD students. P-adic Hodge theory is not well-established in the UK, and a successful completion of the project would establish the UK as one of the leading places for research in this area.

Also, number theory has many important applications in everyday life. An example is the role played by elliptic curves in many cryptographic algorithms, which are crucial in many aspects of everyday life (for example in internet transactions, such as online shopping). Since Iwasawa theory is one of the crucial tools in studying the arithmetic of elliptic curves, advances in Iwasawa theory clearly have the potential to lead to developments in elliptic curve cryptography, and in the other areas of applied science (such as coding theory) where the arithmetic of algebraic varieties has come to play a central role. This might, for example, be interesting to IT companies wanting to improve their encryption systems.

I also very much hope that I can contribute positively to society by acting as a role model for female students. It is well-known that female academics are still substantially under-represented in the sciences, and in mathematics in particular, and that many female students are insecure about the potential to succeed in academia. By conducting this research, I would therefore hope to promote the role of women scientists in academia. To maximise the social benefits of my work, I shall engage in a programme of outreach activities, such as visits to schools.