Saito–Kurokawa lifts and applications to the Bloch–Kato conjecture

Abstract

Let f be a newform of weight 2k−2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if | Lalg(k, f) then p | #Hf (Q,Wf (1− k)) where p is a suitably chosen prime and a uniformizer of a finite extension K/Qp. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift Ff of f and a cuspidal Siegel eigenform G that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in Hf (Q,Wf (1− k)).