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

Abstract

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 so that f is ordinary at p and is irreducible. Under some additional hypotheses, we prove that , where S is the Pontryagin dual of the Selmer group associated to 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.