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 ≤ 0 one can associate Stickelberger elements in the centre of the rational group ring 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 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.