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On zeta functions and families of Siegel modular forms

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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.

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  1. Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402, Saint-Martin d’Hères cedex, France

    Alexei Panchishkin

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  1. Alexei Panchishkin
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Correspondence to Alexei Panchishkin.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 5, pp. 139–160, 2010.

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Panchishkin, A. On zeta functions and families of Siegel modular forms. J Math Sci 180, 626–640 (2012). https://doi.org/10.1007/s10958-012-0661-2

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