The Stickelberger splitting map and Euler systems in the K-theory of number fields

For a CM abelian extension F/K of a totally real number field K, we construct the Stickelberger splitting maps (in the sense of Banaszak, 1992 [1], ) for the étale and the Quillen K-theory of F and use these maps to construct Euler systems in the even Quillen K-theory of F. The Stickelberger splitting maps give an immediate proof of the annihilation by higher Stickelberger elements of the subgroups of divisible elements of K2n(F)⊗Zl, for all n>0 and all odd primes l. This generalizes the results of Banaszak (1992) [1], which only deals with CM abelian extensions of Q. Throughout, we work under the assumption that the Iwasawa μ-invariant conjecture holds. In upcoming work, we will use the Euler systems constructed in this paper to obtain information on the groups of divisible elements , for all n>0 and odd l. The structure of these groups is intimately related to some long standing open problems in number theory, e.g. the Kummer–Vandiver and Iwasawa conjectures.