On local characterizations of Hida families of Siegel modular forms

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from automorphic inductions (stable Yoshida lifts), analogous to the characterizations of Hida families of CM modular forms provided by Ghate--Vatsal. Our characterizations involve (i) density of de Rham at $p$ specializations at the singular weights $(k,2)$ and (ii) local decomposability at $p$ of the associated $\Lambda$-adic Galois representation. Our approach is similar to that of Castella--Wang-Erickson who provided an alternate strategy to reproving the main results of Ghate--Vatsal by applying Ribet's method when an anti-cyclotomic class group is assumed to be pseudo-null and cyclic as a $\Lambda$-module. Along these lines, one key input to our methods involves an assumption of pseudo-nullity of Selmer groups that are defined by imposing stricter conditions at $p$ than those imposed for the usual Greenberg Selmer groups appearing in the Asai main conjectures over real quadratic fields. Following Genestier--Tilouine and Pilloni, we also prove a minimal $R=\mathbb{T}$ theorem that is essential to establishing our results at various stages.