On Zeta Functions and Families of Siegel Modular Forms

2007/09/11 by Alexei Panchishkin, Panchishkin, Alexei
Mathematics · #11F46 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0709.1645

openalex publication_date 2007/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime, and let Γ=\Sp_g(\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 \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 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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