On zeta functions and families of Siegel modular forms

Abstract

Let p be a prime, and let \( \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) \) be the Siegel modular group of genus g. This 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 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 GSp2m × GSp2m to GSp4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.