We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in [6] and studied extensively from a Galois module structure point of view in our previous work [13] and [12]. We prove that the new Iwasawa modules are of projective dimension 1 over the appropriate profinite group rings. In the abelian case, we prove an Equivariant Main Conjecture, identifying the first Fitting ideal of the Iwasawa module in question over the appropriate profinite group ring with the principal ideal generated by a certain equivariant p-adic L-function. This is an integral, equivariant refinement of the classical Main Conjecture over totally real number fields proved by Wiles in [34]. Finally, we use these results and Iwasawa co-descent to prove refinements of the (imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away from their 2-primary components, in the most general number field setting. All of the above is achieved under the assumption that the relevant prime p is odd and that the appropriate classical Iwasawa µ-invariants vanish (as conjectured by Iwasawa.) The reader can easily check that G (∞) S = (ι • t 1 )(G S ), with notations as in §5.2. 2 The reader can easily check that D, with notations as in §5.2. this paper and its function field companion [13] in the following directions: I. An explicit construction of ℓ-adic models for Tate sequences in the general case of global fields; II. A proof of the Equivariant Tamagawa Number Conjecture for Dirichlet motives, in full generality for function fields, and on the "minus side" and away from its 2-primary part for CM-extensions of totally real number fields. A proof of the Gross-Rubin-Stark Conjecture (a vast refined generalization of the Brumer-Stark Conjecture, see Conjecture 5.3.4 in [24]) will ensue in the cases listed above; III. Finally, we will consider non-abelian versions of the Equivariant Main Conjectures proved in this paper and [13].Acknowledgement. The authors would like to thank their home universities for making mutual visits possible. These visits were funded by the DFG, the NSF, and U. C. San Diego.2. Algebraic and number theoretic preliminaries 2.1. Abstract 1-motives. Assume that J is an arbitrary abelian group, m is a strictly positive integer and p is a prime. We denote by J[m] the maximal mtorsion subgroup of J and let J[p ∞ ] := ∪ m J[p m ]. As usual, T p (J) will denote the p-adic Tate module of J. By definition, T p (J) is the Z p -module given bywhere the projective limit is taken with respect to the multiplication-by-p maps. There is an obvious canonical isomorphism of Z p -modulesNow, let us assume that J is an abelian, divisible group. J is said to be of finite local corank if there exists a positive integer r p (J) and a Z p -module isomorphismfor any prime p. The integer r p (J) is called the p-corank of J. Obviously, in this case, we have T p (J) ≃ Z rp(J) p , for all primes p. Further, one can use the "Hom"description of T p (J) gi...
We assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the dual of cl − K as a Galois module, under mild extra hypotheses on K/k. This builds on concepts and results of Tate, Burns, Ritter and Weiss. If k is the field of rational numbers, our results are unconditional.
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