Hecke characters and the $K$-theory of totally real and CM number fields

Abstract

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.