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

Abstract

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 div K 2 n ( F ) l of divisible elements of K 2 n ( F ) ⊗ Z l , 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 div K 2 n ( F ) l , 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.