On the \(p\)-adic deformations of Saito-Kurokawa lifts (Q1852821)

Let \(f\) be a normalized cusp form of weight \(2k-2\) and level 1; let \(L(f,s)\) denote the associated \(L\)-function. Let \(p\) be an ordinary prime for \(f\). Denote by \(V_f\) the \(p\)-adic Galois representation associated to \(f\) by Deligne. Consider the Selmer groups \(H^1_f(\mathbb Q,V_f(n))\) \((n\in\mathbb Z)\) defined by Bloch and Kato. The authors prove that if \(L(f,s)\) vanishes at \(s=k-1\) to odd order, then \(H^1_f(\mathbb Q,V_f(k-1))\) is infinite. To prove the result they construct an extension of \(V_f\) using Galois representations associated to Siegel modular forms that are congruent modulo large powers of \(p\) to a suitable Saito-Kurokawa lift of \(f\).