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Proposition 2.7.4. The generalized BeilinsonFlach element cΞm,N,j,α is the pushforward of the element α0 α0 , α0 α0 Y (M, N ), Proof. It suffices to show that the hypotheses of Theorem 7.1.4 are satisfied. By Proposition 7.2.18, the element τ required by Hypothesis Hyp(Q, T ) exists for T = TOp(f, g)∗; and the Euler system of Definition 7.3.5 satisfies Hypothesis Hyp(S{p}, V )(ii). Since T is nontrivial and irreducible, T GK = 0; and the element γ in Hypothesis Hyp(S{p}, V )(iii) clearly exists.
By Theorem 5.6.4, if Dp(f, g, 1/N )(1) 6= 0, the image of regp Ξ1,N,1 in the (f, g)isotypical quotient of Hd2R(X1(N )/Qp)/ Fil2 is nonzero. Hence, by the diagram of §5.5, the localization of the Galois cohomology class zf,g,N at p is nonzero, so in particular czf1,g,N is nontorsion as an element of H1(Q, TLp(f, g)∗)
for any c > 1. Thus we may apply Theorem 7.1.4 to the Euler system (czˆfm,g,N )m∈A of Definition 7.3.5 to obtain the finiteness of the strict Selmer group.
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