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On Siegel eigenvarieties at Saito-Kurokawa points

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(Γ_0(N))$ ordinary at a prime $p\nmid N$ with sign $ε_f=-1$ and write $α$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $π_α$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,\mathrm{unr}}(\mathbb{Q},ρ_f(k-1))$ attached to $f$ is at most one and some mild assumptions on the automorphic representation attached to $f$, we show that $\mathcal{E}$ is smooth at the point corresponding to $π_α$, and that the irreducible component of $\mathcal{E}$ specializing to $π_α$ is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of $\mathcal{E}$ at $π_α$ yields that $\dim H^1_{f,\mathrm{unr}}(\mathbb{Q},ρ_f(k-1))\geq 2$.