NUMBER THEORETIC BACKGROUND

7

ponent of identity in WF, isomorphic to the inverse limit, under the norm maps

NE/E of the connected components DE of 1 in CE. These norm maps DE - DE, are

surjective, and rE{DE) = (Ker p) W§/ WEis the image of Ker p in WE/F. If Ehas rx real

and r2 complex places then DE is isomorphic to the product of R with rx + r2 — 1

solenoids and r2 circles.

(1.4.5) Notice that in each of the four cases just discussed the subgroups of WF

which are of the form WE for some finite extension E of F are just the open sub-

groups of finite index. Their intersection, Ker p, is a divisible connected abelian

group, trivial in the first two cases, isomorphic to C* in the third, and enormous in

the last case.

(1.4.6) In each case there is a homomorphism w »-*

||H||

of WF into the multipli-

cative group of strictly positive real numbers which reflects the norm or normed

absolute value on CF under the isomorphism rF : CF «

WFh.

By (1.1.2) and the

rule \\NE/Fa\\F = \\a\\E, the restriction of this "norm" function ||w|| from WF to a

subgroup WE is the norm function for WE, so we can write simply ||w|| instead of

||w||F without creating confusion. In each case the kernel WF o f w ^

||H||

is com-

pact. In the first two cases, the image of w »-» • ||w|| consists of the powers of qF, and

WF is a semidirect product Z tx WF. In the last two cases

W H ||H||

is surjective,

and in fact, WF is a direct product R x W$.

Let us refer to the first two cases as the "Z-cases" and the last two as the "In-

cases". In the Z-cases, tp is injective, but not surjective; in the i?-cases, (p is surjec-

tive, but not injective.

(1.5) Automorphisms of Weil groups. Let WF be a Weil group for F/F. Let

Aut(F, WF) denote the set of pairs {a, a), where o e GF is an automorphism of F/F,

and a is an automorphism of the group WF such that the following diagrams are

commutative, the second for all E:

WF

Wr

-&F

Inn (a)

GF

Wf

Here Inn(a) denotes the inner automorphism defined by a.

We shall call an automorphism of WF essentially inner if it induces an inner au-

tomorphism on WE/F for each finite Galois E/F.

(1.5.1)

PROPOSITION.

In the R-cases Aut(F, WF) consists of the pairs (p(w),

Inn(w)), for we WF.

In the Z-cases, Aut(F, WF) consists of the pairs (?, aa), for a e GF, where aa

denotes the restriction of Inn(er) to WF, viewed as a subgroup of GF via p. This

automorphism aa of WF is not an inner automorphism if o $ WF, but it is essentially

inner in the sense of the definition above.

(1.5.2) COROLLARY. The isomorphism 0 in (1.3.1) is unique in the Z-cases, and is