Pullbacks of Eisenstein Series on U(3, 3) and Non-vanishing of Shafarevich-tate Groups

In this paper we construct a pullback formula of a Siegel Eisenstein series on GU(3, 3) to GSp(4) × GL(2) and use it to study the Bloch-Kato conjecture for automorphic forms on GL(2). Let f ∈ S2k−2(SL2(Z)) be a normalized eigenform and let p > 2k− 2 be a prime so that p | Lalg(k, f). Then up to some reasonable hypotheses, we use this formula to construct a congruence between the Saito-Kurokawa lift Ff of f and a Siegel eigenform G that does not lie in the space of Maass spezialchars. We use this congruence to give evidence for a conjecture of Katsurada’s characterizing the congruence primes of Saito-Kurokawa lifts. We then use this congruence to study the 4-dimensional Galois representation attached to G. This allows us to construct a non-trivial p-torsion element in the Shafarevich-Tate group of the k-twist of the residual p-adic Galois representation associated to f . We conclude by showing how this provides evidence for the Bloch-Kato conjecture in this setting.