Noncommutative Iwasawa theory and p-adic automorphic forms.

The last fifteen years have seen two fairly disjoint developments in Iwasawa theory, and its relationship with come of the basic problems in arithmetic geometry. On the one hand, the precise formulation of the main conjectures in noncommutative Iwasawa theory reached a certain maturity in the work of Fukaya-Kato. An important case of the noncommutative main conjecture was proven by the PI (and independently by Burns-Rtter-Weiss). On the other, the theory of automorphic forms (p-adic and lambda-adic) was systematically developed and applied to prove main conjectures in commutative Iwasawa theory beyond the classical main conjectures by several authors including Hida, Tilouine, Urban and Skinner. It is therefore an appropriate time to combine these two developments to prove new results in both directions. We propose to tackle three inter-related problems in this general area. Firstly, we want to extend our methods used to prove the noncommutative main conjectures for Tate motives to prove new results on noncommutative main conjectures for motives other than Tate motives. To this end we propose to systematically study p-adic and lambda-adic automorphic forms over various totally real fields and relations between these automorphic forms as the fields vary. Secondly, implicit in the conjectures of Fukaya-Kato are certain factorisations of p-adic L-functions. These factorisations, known only in a couple of cases, have deep arithmetic implications such as towards Greenberg's L-invariant conjectures. We propose a new strategy to attack these factorisation problems using the tools developed to tackle our first question. Lastly, we propose to study main conjectures over function fields. The algebraic techniques we have developed have already proven very fruitful in Iwasawa theory over function fields in the work of Burns. There is, however, another family of main conjectures over function fields (e.g. in the work of Trihan and his collaborators). Our third project is to use our algebraic results and techniques used by Burns to attack these main conjectures.

Aside from the academic impacts mentioned previously our proposal may be beneficial in many other ways. Number theory has always been an area of strength for the UK. Progress in noncommutative Iwasawa theory has been led by UK mathematician and the UK also has a very strong research group working on p-adic automorphic forms. However, there has been very little research on applying the latter to (noncommutative) Iwasawa theory even though this has generated immense interest in the US and in Europe in recent years. Our proposed research, which compliments the existing expertise in the UK very well, therefore will ensure that the UK maintains and reinforces its leadership in noncommutative Iwasawa theory by exploiting recent developments in other parts of number theory and arithmetic geometry. This may also lead to more international recognition, for example, through increased share of ERC grants.

Number theory is an area of mathematics considered to be most accessible to general public. Therefore it is an appropriate means to introduce a wider audience to the beauty of mathematics especially to encourage enthusiastic young student to study mathematics in their pursuit of higher education. We will take opportunities provided by King's College's various outreach programs to explain number theory related to our proposed research to general public.

We believe that in the later stages of the proposed research computational number theory packages (such as PARI, FLINT) will be useful. These packages are not used just by mathematicians but also by other scientists and engineers. Numerically verifying the predictions we make may lead to a further development of these packages and can have economic benefit for the UK.