On the p-adic deformations of Saito-Kurokawa liftings

E. Urban

Abstract

This is a joint work with C. Skinner. Let f be a cuspidal eigenform of weight $2k-2\geq 2$ and level 1. Suppose p is an ordinary prime for 1 and $V_f$ is the p-adic representation of weight $2k-3$ associated to f. We show that if the zeta function of f vanishes at $s=k-1$ to odd order, then the Selmer group $H^1_f(\mathbb{Q},V_f(k-1))$ is infinite. To prove this result we construct a suitable 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.