On the congruence ideal associated to $p$-adic families of Yoshida lifts

We study congruences involving $p$-adic families of Hecke eigensystems of Yoshida lifts associated with two Hida families (say $\mathcal{F},\mathcal{G}$) of elliptic cusp forms. With appropriate hypotheses, we show that if a Hida family of genus two Siegel cusp forms admits a Yoshida lift at an appr