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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