On Siegel eigenvarieties at Saito-Kurokawa points

We study the geometry of the Siegel eigenvariety EΔ of paramodular tame level Δ associated to a squarefree N∈N+ at certain points having a critical slope. For k≥2 let f be a cuspidal eigenform of S2k−2(Γ0(N)) ordinary at a prime p∤N with sign ϵf=−1 and write α for the unit root of the Hecke polynomial of f at p. Let SK(f)α be the semi-ordinary p-stabilization of the Saito-Kurokawa lift of the cusp form f to GSp(4) of weight (k,k) of tame level Δ. Under the assumption that the dimension of the Selmer group H1f,unr(Q,ρf(k−1)) attached to f is at most one and some mild assumptions on the mod p representation ρ¯f associated to f, we show that the rigid analytic space EΔ is smooth at the point x corresponding to SK(f)α. This means that there exists a unique irreducible component of EΔ specializing to x, and we also show that this irreducible component is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions on ρ¯f that the smoothness failure of EΔ at x yields that dimH1f,unr(Q,ρf(k−1))≥2.