On Zeta Functions and Families of Siegel Modular Forms

Let $p$ be a prime, and let $\mathrm{\Gamma }={\text{Sp}}_g(\mathbb{Z})$ 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 ${\text{Sp}}_g$, 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 $GS{p}_{2m}\times GS{p}_{2m}$ to $GS{p}_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda-Miyawaki constructions.