Let L/K be a Galois extension of local fields of characteristic 0 with Galois group G. If F is a formal group over the ring of integers in K, one can associate to F and each positive integer n a G-module F n L which as a set is the n-th power of the maximal ideal of the ring of integers in L. We give explicit necessary and sufficient conditions under which F n L is a cohomologically trivial G-module. This has applications to elliptic curves over local fields and to ray class groups of number fields.
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.
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