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We develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.
This paper aims to give a survey on Fukaya and Kato's article  which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) of Burns and Flach  and the noncommutative Iwasawa Main Conjecture (MC) (with p-adic L-function) as formulated by Coates, Fukaya, Kato, Sujatha and the author . Moreover, we compare their approach with that of Huber and Kings  who formulate an Iwasawa Main Conjecture (without p-adic L-functions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over ℚ. Nevertheless we formulate the conjectures for general motives over ℚ as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [9, 24]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See  for a more down to earth introduction to the GL2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and Kato makes this statement precise as we are going to explain in these notes. For the convenience of the reader we have given some of the proofs here which had been left as an exercise in  whenever we had the feeling that the presentation of the material becomes more transparent thereby.
Binz, Neukirch and Wenzel proved a profinite version of the Kurosh subgroup theorem. In the paper a subgroup theorem for certain profinite fundamental groups of graphs of profinite groups is shown. In particular, the case of free products with amalgamation is discussed and illustrated in the context of real function fields in one variable.
In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k∞ of number fields k ‘up to pseudo-isomorphism’. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k∞ as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.
In this paper we study Iwasawa modules arising from Galois cohomology over general $p$-adic Lie extensions both in the local and global case. In particular we calculate their $\Lambda$-ranks. We then apply the results to abelian varieties.
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