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We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.
Proposition 2.7.4. The generalized Beilinson-Flach 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 non-torsion 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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