Characteristic elements in noncommutative Iwasawa theory

Abstract

Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their Hasse-Weil L-functions are not only described by the (one-variable) main conjecture corresponding to the cyclotomic Zp-extension kcyc of k, but also by the (two-variable) main conjecture corresponding to the extension k∞ = k(Ep∞) which arises by adjoining the p-power division points Ep∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for non-CM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic p-adic L-function this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic p-adic L-function, viz as an element of the first K-group K1(ΛT ) ∼= ΛT /[Λ × T ,Λ × T ] of a localization ΛT of the usual Iwasawa algebra Λ = Λ(G) of the Galois group G = G(k∞/k). Here, for a ring R, we denote by R× its group of units. By the Weil-pairing, kcyc is contained in k∞ and we put H = G(k∞/kcyc) and Γ = G(kcyc/k). Furthermore we write m(H) for the kernel of the canonical surjective ring homomorphism