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 $\varpi \mid L_{\rm alg}(k,f)$ then $p \mid \# H_{f}(\mathbb{Q},W_{f}(1-k))$ where $p$ is a suitably chosen prime and $\varpi$ a uniformizer of a finite extension $K/\mathbb{Q}_{p}$. We demonstrate this by establishing a congruence between the Saito–Kurokawa lift $F_{f}$ 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 $H_{f}^1(\mathbb{Q},W_{f}(1-k))$.