2011/06/02 by Grzegorz Banaszak, Banaszak, Grzegorz, Cristian D. Popescu +1
Mathematics · #11G30 #19D10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT #msc:11G30 #msc:19D10
paper · pdf · doi:10.48550/arxiv.1106.0513
29 pages
arxiv created 2011/06/02 · openalex publication_date 2011/06/02 · arxiv updated 2011/06/06 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
For a CM abelian extension F/K of an arbitrary totally real number field K, we construct the Stickelberger splitting maps (in the sense of \citeBa1) for both the étale and the Quillen K--theory of F and we 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 of the groups of divisible elements div K2n(F)l of the even K--theory of the top field by higher Stickelberger elements, for all odd primes l. This generalizes the results of \citeBa1, which only deals with CM abelian extensions of \Bbb Q. The techniques involved in constructing our Euler systems at this level of generality are quite different from those used in \citeBG1, where an Euler system in the odd K--theory with finite coefficients of abelian CM extensions of \Bbb Q was given. We work under the assumption that the Iwasawa μ--invariant conjecture holds. This permits us to make use of the recent results of Greither-Popescu \citeGP on the étale Coates-Sinnott conjecture for arbitrary abelian extensions of totally real number fields, which are conditional upon this assumption. 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. It is known that the structure of these groups is intimately related to some of the deepest unsolved problems in algebraic number theory, e.g. the Kummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields. We make these connections explicit in the introduction.