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.