LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
5
The details are given in Chapter 2, §§2.1-2.3.
The tripod itself is now obtained from a G-homomorphism ZS » Yip^S Kp which? first,
restricts to a monomorphism AS E (with respect to the natural map E » YlpeS Kp )
and
second, induces a monomorphism Z G(M/K), where M is the maximal abelian 5-ramified
/-extension of K. Observe that the reciprocity map of class field theory determines a natural
homomorphism TlpeS %-p ~* G(M/K) under which the image of E in Yipes ^-p vanishes.
With C denoting the /-completion functor, this data may be combined in the commutative
diagram
A/ 5 £i Zt(g)E
I I
/ x
Q
_
r
/ 7 ^
X
\ which is diagram (D2.5*) in
( * } Z z & ~ nPes£(Kp) Chapter 2.
Zi £ G(M/K)
The lifted-0 classes mentioned above depend on these three maps (and canonically related
2-extensions).
T H E O R E M A. Suppose that K is totally real and satisfies Leopoldt's conjecture. Then Q,\p' =
The proof is in section 2.4 in Chapter 2.
This theorem is reminiscent of [Ch2] with 0 $ the lifted analogue of Chinburg's H(l). The
point of our paper is that Q$ has an interpretation in Iwasawa theory.
The next chapter, 3, is of technical nature 2. We discuss the behaviour of the basic objects
£l(p ,£l(pc and Q$ with respect to restriction, deflation, change of maps and variance with S.
The main tools for this have already been introduced in [GRW2]; the only new ingredient is
the deflation map
Exti,
G
(Z
(
® AG, A) E x 4
i [ G / H |
( Z , ® A[G/H},AH)
corresponding to a normal subgroup H of G. The explicit formulas given in chapter 3 not
only allow us to pursue the study of the three O-quantities by starting out from a specific
map (f : AS - E, they also allow us to invoke induction theorems (compare [GRW2, §8]).
We now turn to a detailed discussion of Q$ and H ^ . We begin, in chapters 4 and 5, by
looking at Q$. From now on K is totally real.
Recall that Q$ is the outcome of the lifted Q-construction attached to
G(M/K) - Y -* Z/G Zi and $ : Zx ~ G{M/K) ,
with the 2-extension resulting from the translation functor applied to G(M/K) - G(M/k)
G by splicing on the short exact sequence Zx0AG - Z\G Z\. The right end, G(M/K), is
an object classical Iwasawa theory is concerned with. In chapter 4 we formulate an equivariant
Iwasawa theory
3
and establish its relation to £1$ .
2
and may be omitted on first reading
3
When / \ [K : k] we find ourselves back in the classical situation.
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