On zeta functions and families of Siegel modular forms

Let $p$ be a prime, and let $\Gamma=\mathrm{Sp}_g(\mathbb Z)$ be the Siegel modular group of genus $g$. The paper is concerned with $p$-adic families of zeta functions and $L$-functions of Siegel modular forms, the latter are described in terms of motivic $L$-functions attached to $\mathrm{Sp}_g$; their analytic properties are given. Critical values for the spinor $L$-functions are discussed in relation to $p$-adic constructions. Rankin's lemma of higher genus is established. A general conjecture on a lifting of modular forms from $\mathrm{GSp}_{2m}\times\mathrm{GSp}_{2m}$ to $\mathrm{GSp}_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.