5 J. TATE

unique to an inner automorphism of WF by an element of the connected component

W% = Ker (p in the R-cases.

To prove (1.5.1) in the leases, we note first that, since p is surjective, we are

reduced immediately to the case of the corollary: We must show that if (1, a)e

Aut(F, WF)9 then a = Inn(vv) for some w e W$. Going back to the proof of (1.3.1)

with WF = WF we find that the group of these a's is given by

proj lim (CE/CF) = proj lim (CE/CF) (1 for norm 1)

E E

= proj lim C^/proj lim CF (by compacity)

E E

= w°'i/(z n w°*)

= w°/z

(existence theorem; 0 for con-

nected component)

as claimed, where Z is the center of W.

Suppose now we are in a Z-case. Since p is injective, i.e., WF c GF, it is clear

that Aut(F, WF) consists only of the pairs (7, aa). The center of GF is 1, because

GF/GE « Gal(F/F) acts faithfully on CE c Gf for each finite Galois E/F. Hence,

since WF is dense in GFt aa is not an inner automorphism of WF unless a € WF.

However, aa does induce an inner automorphism of WE/F for finite E/F. Since WF

is dense in GF it suffices to prove this last statement for o close to 1, say o e GE.

Then aff induces an isomorphism of the group extension 0 -+ CE -• WE/F -•

Gal(F/F) -• 0 which is identity on the extremities, and hence is an inner automor-

phism by an element of CEy since Hl (Gal(F/F), CE) = 0.

(1.6) The local-global relationship. Suppose now Fis global. Let v be a place of F

and Fv the completion of Fat v. Let F(resp. Fv) be a separable algebraic closure of

F(resp. Fv) and let WF (resp. WF) be a Weil group for F/F(resp. for FJFV).

(1.6.1)

PROPOSITION.

Let iv : F -» Fv be an F-homomorphism. For each finite ex-

tension E of F in F, let Ev = i(E)F0 be the induced completion of E. There exists a

continuous homomorphism 0P: WFv -+ WF such that the following diagrams are com-

mutative

W

Fv

induced

by iv

Wt

induced

by iv

WP w$

where nv maps a e E* to the class of the idele whose v-component is a and whose other

components are 1.

If Fis a function field, then^ is unique. In the number field case, dv is unique up

to composition with an inner automorphism of WF defined by an element of the

connected component IV$ = Ker p.

The proof of this is analogous to the proof of (1.3.1) and (1.5.1), using the stand-