6
J. TATE
the canonical class aE/F, as their cohomology class, 1(E) is not empty. Since
Hl(Gal(E/F),
CE) = 0, an isomorphism fe 1(E) is determined up to composing
with an inner automorphism of WE/F by an element of CE ~ W^. The center of
WE/F is CF, and CE/CF is compact. Hence 1(E), as principal homogeneous space
for CE/CF, has a natural compact topology. For Ex 3 F, the natural map I(Ei) -+
/(F) is continuous for this topology, since it is reflected in the norm map NEl/E,
once we pick an element of I(E{). Hence the projective limit proj \imE(I(E)) is not
empty. An element 6 of this limit gives an isomorphism W^ WF by (W4), and
has the required properties.
It turns out (cf. (1.5.2)) that 6 is unique up to an inner automorphism of WF by
an element w e Ker p, but we postpone the discussion of this question until after
the next section.
(1.4) Special cases. We discuss now the special features of the four cases: F local
nonarchimedean, F a global function field, F local archimedean, and F a global
number field. In the first two of these, GF is a completion of WF\ in the last two it is
a quotient of WF.
(1.4.1) F local nonarchimedean. For each F, let kE be the residue field of E and
qE = Card(^). Let k \JEkE. We can take WF to be the dense subgroup of GF
consisting of the elements a eGF which induce on k the map x -* xq*F for some
n e Z. Thus WF contains the inertia group IF (the subgroup of GF fixing k), and
WFjIF « Z. The topology in WF is that for which IF gets the profinite topology
induced from GF, and is open in WF. The map p: WF -+ GF is the inclusion, and the
maps rE: E* WEh are the reciprocity law homomorphisms. Concerning the sign
of the reciprocity law, our convention will be that rE(a) acts as x *-* xMlE on £,
where \\a\\E is the normed absolute value of an element a e F*. (If %E is a uniform-
izer in F, then \\nE\\E = #£*; thus our convention is that uniformizers correspond
to the inverse of the Frobenius automorphism, as in Deligne [D3], opposite to the
convention used in [Dl], [AT], [CF], and [SI].)
(1.4.2) F a global function field. Here the picture is as in (1.4.1). Just change "re-
sidue field" to "constant field", "inertia group JF" to "geometric Galois group
Gal(F/F&)", and define the norm ||a||£ of an idele class ae CEto be the product of
the normed absolute values of the components of an idele representing the class.
(1.4.3) Flocal archimedean. If F * C w e can take WF = F*, p the trivial map, rF
the identity.
If F « Ry we can take WF F* U jF* with the rules j
2
= 1 and
jcj~l
= c,
where c - » c is the nontrivial element of Gal(F/F). The map p takes F* to 1 and
JF* to that nontrivial element. The map rp is the identity, and rF is characterized by
rF(-l)=jWF,
rF(x) = A/~X W§9 for x e F, x 0.
(W/ is the "unit circle" of elements u e F with \\u\\ = A^/F w = 1.)
(1.4.4) Fa global number field. This is the only case in which there is, at present,
no simple description of WF, but merely the artificial construction by cocycles
described in (1.2). This construction is due to Weil in [Wl], where he emphasizes the
importance of the problem of finding a more natural construction, and proves the
following facts. The map p: WF GF is surjective. Its kernel is the connected com-
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