Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z p . We prove the existence of a canonical Ore set S * of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S * , we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q , without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q .

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, of a p-adic analytic group G. For G without any p-torsion element we prove that is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null -module. This is classical when G = Z k p for some integer k ≥ 1, but was previously unknown in the non-commutative case. Then the category of -modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the Z p -torsion part of a finitely generated -module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.

Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k ∞ /k) is isomorphic to the semidirect product of two copies of the p-adic integers Z p . After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group X nr of the p-Hilbert class field over k ∞ . The main tool we use is a version of the Weierstrass preparation theorem, which we prove for certain skew power series with coefficients in a not necessarily commutative local ring. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)-modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra F p [[G]] with coefficients in the finite field F p is a unique factorization domain in the sense of Chatters [8].

Abstract. We prove a 'Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of noncommutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense 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.

The goal of this paper is to prove that the Pontryagin dual of the Selmer group over the trivializing extension of an elliptic curve without complex multiplication does not have any nonzero pseudo-null submodule. The main point is to extend the definition of pseudo-null to modules over the completed group ring Zp[[G]] of an arbitrary p-adic Lie group G without p-torsion. For this purpose we prove that Zp [[G]] is an Auslander regular ring. For the proof we also extend some results of Jannsen's homotopy theory of modules and study intensively higher Iwasawa adjoints.