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
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