Two Modularity Lifting Conjectures for Families of Siegel Modular Forms

For a prime $p$ and a positive integer $n$, using certain lifting procedures, we study some constructions of $p$-adic families of Siegel modular forms of genus $n$. Describing $L$-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from $\operatorname{GSp}_{r}\times \operatorname{GSp}_{2m}$ to $\operatorname{GSp}_{r+2m}$ for some positive integers $r$ and $m$.