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

Authors:

• Grzegorz Marian Banaszak,
• Cristian D. Popescu

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_K2n(F)l 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 div_K2n(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. © 2011 Elsevier Inc.