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We apply an equivariant version of the p-adic Weierstrass Preparation Theorem in the context of possible non-commutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Conjecture of Iwasawa theory we prove,… (More)

As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example , if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the l-part of the Brumer-Stark conjecture for all odd primes… (More)

We give a survey of the equivariant Tamagawa number (a.k.a. Bloch-Kato) conjecture with particular emphasis on proven cases. The only new result is a proof of the 2-primary part of this conjecture for Tate-motives over abelian fields. have tried to retain the succinctness of the talk when covering generalities but have considerably expanded the section on… (More)

Let Z −→ Z ′ be a G–Galois cover of smooth, projective curves over an arbitrary algebraically closed field κ, and let S and T be G–equivariant, disjoint, finite, non-empty sets of closed points on Z, such that S contains the ramification locus of the cover. In this context, we prove that the ℓ–adic realizations T ℓ (M S,T) of the Picard 1–motive M S,T… (More)

In this paper, we systematically construct abelian extensions of CM-fields over a totally real field whose Stickelberger elements are not in the Fitting ideals of the class groups. Our evidence indicates that Pontryagin duals of class groups behave better than the class groups themselves. We also explore the behaviour of Fitting ideals under projective… (More)

In previous work, we proved a result on the equivariant Fitting ideal of the-adic realization of the Picard 1-motive attached to an abelian covering of curves defined over a finite field. In this paper, we build upon this work to deduce results on the equivariant Fitting ideal of the Tate modules of the Jacobian of the top curve, and on the equivariant… (More)

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O L is free as an O K G-module. Let C p denote the cyclic group of prime order p. We show that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with… (More)

Barry Smith has found an error in the statement and proof of Lemma 2.5 in our paper [GRT] (Math. Comp. 73 (2004), 297-315). This Lemma concerns a cyclic Galois extension K/E of CM fields of odd prime degree p. Towards the end of the proof, it is claimed that every root of unity in E is a norm from K. Our reasoning for this has a gap (the local part of the… (More)

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q ⊂ K ⊂ L forces the tower to be split in a very strong sense.