167 research outputs found
On the equivariant Tamagawa number conjecture in tame CM-extensions, II
We use the notion of non-commutative Fitting invariants to give a reformulation of the equivariant Iwasawa main conjecture (EIMC) attached to an extension F=K of totally real fields with Galois group G, where K is a global number field and G is a p-adic Lie group of dimension 1 for an odd prime p. We attach to each finite Galois CM-extension L=K with Galois group G a module SKu(L=K) over the center of the group ring ZG which coincides with the Sinnott-Kurihara ideal if G is abelian. We state a conjecture on the integrality of SKu(L=K) which follows from the equivariant Tamagawa number conjecture (ETNC) in
many cases, and is a theorem for abelian G. Assuming the validity of the EIMC and the vanishing of the Iwasawa Ī¼-invariant, we compute Fitting invariants of certain Iwasawa modules, and we show that this implies the minus part of the ETNC at p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that (an appropriate p-part of) the integrality conjecture holds
On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results
Let L/K be a finite Galois extension of number fields with Galois group G.
Let p be a rational prime and let r be a non-positive integer. By examining the
structure of the p-adic group ring Z_p[G], we prove many new cases of the
p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair
(h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other
conjectures concerning the vanishing of certain elements in relative algebraic
K-groups. We then prove a conjecture of Burns concerning the annihilation of
class groups as Galois modules for a wide class of interesting extensions,
including cases in which the full ETNC in not known. Similarly, we construct
annihilators of higher dimensional algebraic K-groups of the ring of integers
in L.Comment: 33 pages, error in section 3.4 corrected. To appear in Transactions
of the AM
Tamagawa Numbers for Motives with (Non-Commutative) Coefficients
Let be a motive which is defined over a number field and admits an action of a finite dimensional semisimple \bq-algebra . We formulate and study a conjecture for the leading coefficient of the Taylor expansion at of the -equivariant -function of . This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Frƶhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order \A in for which there exists a `projective \A-structure' on . The existence of such a structure is guaranteed if \A is a maximal order, and also occurs in many natural examples where \A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in by making use of the category of virtual objects introduced by Deligne
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