LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY 7

Also, from now on we assume that not only K but also all Kn , 0 n oo , satisfy Leopoldt's

conjecture. This has the consequence that G(M00/K00) is i?-torsion (Proposition 4.4).

Now we take any element Coo G

(R~1ZIGOG)X

such that

doo = c00((7/e - l)e + (1 - e)) belongs to AGoo ,

choose, in (4.oo), a preimage ?/oo of doo and set

# : Z/Goo - Foo , 1 ^

y o o

.

Proposition 4.5 states that ^ is injective and that coker \I/ is .R-torsion and has projective

dimension 1. It is related to the "Iwasawa module" G(M00/K00) by the short exact sequence

G{M00/K00) -+ coker^ -» coker ^

of .R-torsion Z/Goo-modules with no finite submodule. Here, ip is the map Z/Goo —» • AG?oo ,

so we can regard its cokernel to be known from the construction of doo-

The aim of "equivariant Iwasawa theory" is to describe coker \I/. More precisely, let KQR(ZIG OQ)

be the Grothendieck group of all finitely generated i?-torsion Z/Goo-modules of finite projec-

tive dimension. It is part of the localization sequence

JTifaGoo) - tfi^ZjGoo) ^ ^ofi(ZjGoo) - ^o(Z;Goo).

The map ^ gives rise to the element [coker ^J G KQR(ZIGOO). We define

Us = [coker tf] - d f / T ^ G o o , Coo] G KQR{ZiGoo).

As a matter of fact, then Us depends only on the choice of 7^. (and on S) (Proposition 4.6).

The connection to fi$n, for any £n : Z/ — G(Mn/Kn), is given in Theorem B below, in which

0"n denotes the image of Us under the deflation map KQR(ZIGOO) — KoT(ZiGn) of Lemma

4B.1.

THEOREM B. 15n-d[(Qi®AGn)e,%-i\ = £lqn-d{Qi, |$

n

|] , where | $

n

| G Z/ zs determined

by $

n

( l ) having image yk

n |

under G(Mn/Kn) —• G^^/Kn) —• G(kOQ/k).

In chapter 5 we assume that G? is abelian. Then (700 = # X T with T ~ T^. We define

JW = 0 Z

z

G o o - e

x

x

with x running through the abelian (Q/C)x-valued characters of H, modulo Galois conjuga-

tion over Q/. Above, ex = J^ ex r and ex r is the primitive central idempotent

reG(Qi(x)/Qi)

associated to

\a•

Hence e = e\H. Also, M. is the maximal ZjT-order in Quot(Z/T)[if].

Since ZjGoo • e

x

~ Z/[x]T is the Iwasawa algebra of Y with respect to the base ring Z/[x]

7

,

we can exploit the theory of finitely generated modules over such algebras. Note that each

such has finite projective dimension, since Z/[x]T is regular.

We introduce in Lemma 5.1 the map KQR^IGOQ) — KQR(M.) and, moreover, show that

finite .M-modules become 0 in KQR(M). Applying the map KQR{Z[G00) —• KQR(M) to our

7Z,[X] = ^[

X

(/i):hetf ]