Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

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 (2015) on the Brumer–Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l \gt 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger splitting maps in the Quillen localization sequence for $F$, extending the results of Banaszak (1992) for $K=\mathbb Q$. We also apply the Stickelberger splitting maps to construct special elements in $K_{2n}(F)_l$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$. If $n$ is odd, $l\nmid n$, 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 odd $K$-groups $K_{2n+1}(F, \mathbb{Z}/l^k)$. These are vast generalizations of Kolyvagin’s Euler system of Gauss sums (see Rubin (1991)) and of the $K$-theoretic Euler systems constructed by Banaszak and Gajda (1996) when $K=\mathbb{Q}$.