Characteristic elements in noncommutative Iwasawa theory

Abstract

In this article we construct characteristic elements for a certain class of Iwasawa modules in noncommutative Iwasawa theory. These elements live in the first K-group K_1(L_T) of the localisation L_T of the Iwasawa algebra L=L(G) of a p-adic Lie group G with respect to a certain Ore-Set T. The evaluation of the characteristic element of a module M under the Iwasawa algebra of the p-adic Lie group G is related to the (twisted) G-Euler characteristic of M. We apply these results to study the arithmetic of elliptic curves E (without CM) defined over a number field k in the tower K=k(E(p)) of fields which arises by adjoining the p-power division points to k. In particular, we relate the characteristic element of the Selmer group of E over K, i.e. the algebraic p-adic L-function of E over K, to the (classical) characteristic polynomial associated with the Selmer group over the cyclotomic Z_p-extension. Finally we discuss how the formulation of a noncommutative main conjecture could look like assuming the existence of an analytic p-adic L-function.