Integrality of Stickelberger elements and annihilation of natural Galois modules

Abstract

To each Galois extension $L/K$ of number fields with Galois group $G$ and each integer $r \leq 0$ one can associate Stickelberger elements in the centre of the rational group ring $\mathbb{Q}[G]$ in terms of values of Artin $L$-series at $r$. We show that the denominators of their coefficients are bounded by the cardinality of the commutator subgroup $G'$ of $G$ whenever $G$ is nilpotent. Moreover, we show that, after multiplication by $|G'|$ and away from $2$-primary parts, they annihilate the class group of $L$ if $r=0$ and higher Quillen $K$-groups of the ring of integers in $L$ if $r<0$. This generalizes recent progress on conjectures of Brumer and of Coates and Sinnott from abelian to nilpotent extensions. For arbitrary $G$ we show that the denominators remain bounded along the cyclotomic $\mathbb{Z}_p$-tower of $L$ for every odd prime $p$. This allows us to give an affirmative answer to a question of Greenberg and of Gross on the behaviour of $p$-adic Artin $L$-series at zero.