4
J. TATE
induced
by a
r E
r
conjugation
by w
~*VV pa
E*
(W
3
)For£' c E the diagram
CFr w$
induced by
inclusion
E'czE
r E
transfer
-
Whh
is commutative.
(W4) The natural map
WP
proj lim{WE/F}
E
is an isomorphism of topological groups, where
(1.1.1) WE/F denotes WF/W§ {not WFjWE\
and the projective limit is taken over all E, ordered by inclusion, as E -+ F.
This concludes our definition of Weil group. It is clear from the definition that if
WF is a Weil group for F/F, then, for each finite extension E of F in F, WE (fur-
nished with the restriction of p and the isomorphisms rE, for E' = E) is a Weil
group for F/E.
If WF is a Weil group, then for each F c Er c £ the diagram
0 #
(1.2.2)
norm,
induced by
inclusion W £ c W £ '
C~-
-w&
is commutative.
This follows from the fact that, when H is a normal subgroup of finite index in G,
the composition
/fab.
induced by
inclusion
Gab
transfer
*H*h
is the map which takes an element he H into the product of its conjugates by re-
presentatives of elements of GjH. (In the notation of the first paragraph above,
hgtX = s(x) gs(x)~\ i f g e H c G . )
(1.2) Cohomology; construction of Weil groups. Let WF be a Weil group for FjF.
Then for each Galois E/F the group WE/F = WFjWE is an extension of WFjWE =
Gal(£/F) by WE/W£ * Q . Let a
E / F
e 7/2(Gal(£/F), Q ) denote the class of this
group extension. For each n e Z, let
(1.2.3) an(EIF): H»{G2A{EjT), Z) ff+2(Ga\(E/F), CE)
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