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-