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