1. INTRODUCTION

The intent of this paper is to exhibit a close connection between an equivariant generalization

of Iwasawa theory and multiplicative Galois structure, the beginnings of which were already

observed in [RW2] and whose significance was emphasized in [GRW1]. The main object in

Galois module theory here for us is the Lifted Root Number Conjecture as introduced in

[GRW2]. The point of the Lifted Root Number Conjecture is that it implies Chinburg's Root

Number Conjecture [Chi] and that it can be studied one prime at a time (see [GRWl or

GRW2J).

Fix a prime number /. In this paper a tripod is built which stands on

(1) the Lifted Root Number Conjecture at /,

(2) arithmetically local Z-adic objects, and

(3) an equivariant form of Iwasawa theory.

It is shown that whenever we understand one of these three items, then we already know

that the other two are equivalent. This will be described in more detail after recalling the

background in sections 1.1-1.7: see A at the bottom of page 4 .

The relation of (3) to classical Iwasawa theory and of (2) to additive Galois module theory

are then discussed at some length. A proof of the Lifted Root Number Conjecture in this way

would need strengthening of both subjects in a new well-defined sense. Making this precise

in a brief way has caused the present restrictions on generality.

We mention that the existence of relations between the Root Number Conjecture and Iwasawa

theory, in Kato's formulation ([Ka]), also motivates recent work of Burns (see e.g. [Bui,2]).

Although this is implicitly related to the material in our paper, our approach is independent

of the results and methods there. Finally there is a more general conjectural framework in

which these questions fit (see also §5 of [Ri]).

1.1 The setting

The prime number I has already been fixed. Let K/k be a finite Galois extension of number

fields with Galois group G, and let S be a finite G-stable set of primes of K containing all

archimedean primes and all primes above I. Denote the G-module of 5-units by E = Es

and let AS, respectively A/5 , be the kernel of the augmentation maps ZS —• Z, respectively

Z[S —• lii 1, which send each basis element p 6 S to 1.

*Zj is the completion of Z at /.

Received by the editor August 16, 2000

We acknowledge financial support provided by the DFG and NSERC.

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