On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Abstract

Let \(\kappa \ge 6\) be an even integer, \(M\) an odd square-free integer, and \(f \in S_{2\kappa -2}(\Gamma _0(M))\) a newform. We prove that under some reasonable assumptions that half of the \(\lambda \)-part of the Bloch–Kato conjecture for the near central critical value \(L(\kappa ,f)\) is true. We do this by bounding the \(\ell \)-valuation of the order of the appropriate Bloch–Kato Selmer group below by the \(\ell \)-valuation of algebraic part of \(L(\kappa ,f)\). We prove this by constructing a congruence between the Saito–Kurokawa lift of \(f\) and a cuspidal Siegel modular form.