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11 Dec 2001: ij where the order. is determined by the precedence (i,j): that is, i takes precedence over j.Take this concatenated sequence together with a sequence b1,. ... Proposition 4.3 Given T-algebras A = (A,a) and B = (B,b), the T-algebra[A,B] equipped with.

22 Oct 2001: Proof. Let B = ClK1 (5). By Lemma 3.5(i) we know that H1 is the maximalunramified 5-extension of K1 which is abelian over k. ... Let B = ClK2 (5). By Lemma 3.5(ii) we know that H2 is the maximalunramified 5-extension of K2 which is abelian over k.

12 Oct 2001: a,b) 7 inv{eφ(aσ,σbτ )}σ,τNow let E be an elliptic curve over K. ... with non-vertical maps defined over K. Let b = {bσ}σ and let bσ Div0(T)with sum(bσ) = bσ.