  Let $p$ be a prime, and let $\Gamma=\Sp_g(\Z)$ be the Siegel modular group of
genus $g$. We study $p$-adic families of zeta functions and Siegel modular
forms. $L$-functions of Siegel modular forms are described in terms of motivic
$L$-functions attached to $\Sp_g$, and their analytic properties are given.
Critical values for the spinor $L$-functions and $p$-adic constructions are
discussed. Rankin's lemma of higher genus is established. A general conjecture
on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$)
is formulated. Constructions of $p$-adic families of Siegel modular forms are
given using Ikeda-Miyawaki constructions.
