FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 5, PAGES 139-160

On zeta functions and families of Siegel modular forms

A. A. Panchishkin

Abstract

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Let $p$ be a prime, and let $$G = Sp_g(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 $Sp$_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 $GSp$_2m ´ 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.

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Last modified: May 30, 2011