  We provide applications to studying the behavior of Selmer groups under
specialization. We consider Selmer groups associated to four dimensional Galois
representations coming from (i) the tensor product of two cuspidal Hida
families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product
of a cusp form $f$ and the Hida family $G$, where $f$ is a classical
specialization of $F$ with weight $k \geq 2$. We prove control theorems to
relate (a) the Selmer group associated to the tensor product of Hida families
$F$ and $G$ to the Selmer group associated to its cyclotomic deformation and
(b) the Selmer group associated to the tensor product of $f$ and $G$ to the
Selmer group associated to the tensor product of $F$ and $G$. On the analytic
side of the main conjectures, Hida has constructed one variable, two variable
and three variable Rankin-Selberg $p$-adic $L$-functions. Our specialization
results enable us to verify that Hida's results relating (a) the two variable
$p$-adic $L$-function to the three variable $p$-adic $L$-function and (b) the
one variable $p$-adic $L$-function to the two variable $p$-adic $L$-function
and our control theorems for Selmer groups are completely consistent with the
main conjectures.
