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

TL;DR

This paper constructs Stickelberger splitting maps and Euler systems in the K-theory of CM abelian extensions of totally real fields, advancing understanding of algebraic number theory and related conjectures.

Contribution

It introduces new methods to construct Euler systems in K-theory for general CM abelian extensions, extending prior results beyond rational fields.

Findings

Proves annihilation of divisible K-groups by higher Stickelberger elements.

Constructs Euler systems in even Quillen K-theory for these extensions.

Relates K-theory groups to deep conjectures like Kummer-Vandiver and Iwasawa.

Abstract

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 \cite{Ba1}) for both the \'etale 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 $div K_{2n}(F)_l$ of the even $K$--theory of the top field by higher Stickelberger elements, for all odd primes $l$. This generalizes the results of \cite{Ba1}, 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 \cite{BG1}, 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 $\mu$--invariant conjecture holds. This permits us to make use of the recent results of Greither-Popescu \cite{GP} on the \'etale 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 $div K_{2n}(F)_l$, 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.