$p$-adic rigidity for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$

This paper establishes the $p$-adic rigidity of certain suitably refined noncuspidal automorphic Saito$\text{unicode}x2013$Kurokawa representations of ${\mathrm{G}\mathrm{S}\mathrm{p}}_4({\mathbb{A}}_{\mathbb{Q}})$, in the sense that they cannot be interpolated in a nontrivial positive dimensional $p$-adic family. The results provide a ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ analogue of Bellaïche’s rigidity theorems for $\mathrm{U}(2,1)$ and identify an obstruction to $p$-adic variation on the ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ eigenvariety.