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By Nagnibida N. I.

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Extra resources for A basis for the space of functions analytic in a disk

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Be a cohomologically trivial submodule of N*v (resp. of N'* Let Hv (resp. (N*) F x). x ) be a subgroup of Norm T(V) * H', T(V) * r G v (resp. of N O M , (N** J ) . Define A = N*/T H and A' . , = G / v T (v) v v7 v v T(V) T (V) G N' * // T ' . N H ' . , . ;> • Let T * : K* -> G„ and 3 -rr I » T \ / v T(V) v w v T(w) , , , y dT(v) f / , : K(w) * x - > G( V/ ), b e the local Artin maps. Suppose v *v vtt: Av -• A' , . is T(V) T(V) T T(V) W V T an isomorphism with the following properties. 1 hold.

From [31; chap. XI , §3, Prop. 2) where — a where \j Xdj, (a)) = inv(H,v) (a U ax) V,H H H is the class of an element a € (N ) * in (N )*/Norirt (N*) , and v v ' H v denotes cup product. 5) shows res| H (p v )^(3 u ax) = res| H (p^ ( v ) )^( u (p) u a*) 2 in H (H,A ). 5) imply that r e s l ( p ) and res! (p' ) are injec'H v * «H T (v) * 2 tive and have equal images on the canonical classes of H (H,N*) and H (H,N'* v ) , respectively. W e conclude that T v ; inv(H,v)(p U aX) = inv(H,T(v))( u (3) u d X ) .

3. For i € SP, one has the set of ramified and infinite places S[m£] in the twist N[m i\ of N Q by m i. THE ANALYTIC THEORY OF MULTIPLICATIVE GALOIS STRUCTURE Fix an isomorphism of G = Gal(Nfm^j/Q) with HQ. For i € SP, we fix an isomorphism of Gal(N[m ] {Ji) /Q) with G X Gal (CD (Jl)/®) by identifying with Gal(N[mQ] Ul,)/Jl) ) and by identifying Gal (Q (Jz)/Q) Gal(N[mQ] (Vj&)/N[m ]). Let H 43 G with be the subgroup of G x G(Q(Vx)/$) generated by the element of order two which is in neither G x {1} nor {1} x Gal(Q(*/jj)/CD) .

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