Otmar Venjakob

The GL 2 Main Conjecture for Elliptic Curves without Complex Multiplication

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 Zp. 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.

On the structure of Selmer groups over -adic Lie extensions

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.

On descent theory and main conjectures in non-commutative Iwasawa theory

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 a converse to a theorem of Fukaya and Kato and also provides an important means of both deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture. 1991 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

Localizations and completions of skew power series rings

This paper is a natural continuation of the study of skew power series rings

On the ranks of Iwasawa modules over

A=R[[t;\sigma,\delta]]

p

initiated in an earlier work. We construct skew Laurent series rings

-adic Lie extensions

B

CHARACTERISTIC ELEMENTS IN NONCOMMUTATIVE IWASAWA THEORY

and show the existence of some canonical Ore sets

Local epsilon isomorphisms

S

On Kato’s local -isomorphism conjecture for rank-one Iwasawa modules

for the skew power series rings