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

Let k > 9 be an even integer and pa 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 (rho) over barf,p is irreducible. Under some additional hypotheses, we prove that ord(p)(L-alg(k, f)) <= ord(p)(# S), where S is the Pontryagin dual of the Selmer group associated to rho f,p circle times epsilon(1-k) with epsilon 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.