  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.
