Noncommutative Iwasawa theory and p-adic automorphic forms.

点击翻译按钮获取中文摘要

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.