On the Cuspidality of Pullbacks of Siegel Eisenstein Series and Applications to the Bloch–Kato Conjecture

Let k > 9 be an even integer and p a prime with p > 2k− 2. Let f be a newform of weight 2k− 2 and level SL2(Z) so that f is ordinary at p and ρ f,p is irreducible. Under some additional hypotheses, we prove that ordp(Lalg(k, f)) ≤ ordp(#S), where S is the Pontryagin dual of the Selmer group associated to ρ f,p ⊗ ε1−k with ε the p-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito–Kurokawa lift of f and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.