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
${\mathrm SL}_2(\mathbb Z)$
so that f is ordinary at p and
$\overline \rho _{f,\mathfrak p}$
is irreducible. Under some additional hypotheses, we prove that
${\mathrm ord}_p(L_{\mathrm alg}(k,\ f )) \leq {\mathrm ord}p(\#S)$
, where S is the Pontryagin dual of the Selmer group associated to
$\rho_{f,\mathfrak p}\ \otimes \ \varepsilon^{1-k}$
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.