\(L\)-functions of \(S_{3}(\Gamma (4,8))\)
type: Item
instance of: scholarly article
title: \(L\)-functions of \(S_{3}(\Gamma (4,8))\)
zbMATH Open document ID: 1118.11026
author: Takeo Okazaki
published in: Journal of Number Theory
publication date: 2007-06-06T00:00:00Z
review text: Let \(S_3(\Gamma(4, 8))\) denote the space of Siegel modular cusp forms of weight 3 on the congruence subgroup \(\Gamma(4,8)\subset\text{Sp}_2(\mathbb{Z})\) containing all matrices \(\left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)\) such that \(\text{diag\,}B\equiv \text{diag\,}C\equiv 0\bmod 8\). In previous work, \textit{B. van Geemen} and \textit{D. van Straten} [Math. Comput. 61, No. 204, 849--872 (1993; Zbl 0787.11018)] and \textit{B. van Geemen} and \textit{N. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)] determined the Hecke eigenforms belonging to \(S_3(\Gamma(4, 8))\), and computing some Hecke operators for these forms they stated a list of seven conjectures for the Andrianov \(L\)-functions of these forms (outside of 2). The main theorem of the paper under review says that six of these conjectures are true. The corresponding formulae express the Andrianov \(L\)-functions in terms of \(L\)-functions of certain elliptic cusp forms. The Yoshida lift is a crucial tool in the author's arguments. -- Concerning the last of the aforementioned conjectures the author expresses his belief that this one will also be true though he expects that a proof will require more preparations.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 5161938
zbMATH Keywords: Siegel modular form
zbMATH Keywords: Siegel modular threefold
zbMATH Keywords: Andrianov \(L\)-function
zbMATH Keywords: Yoshida lift
zbMATH Keywords: Jacquet-Langlands theory
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.11.005
OpenAlex ID: W2020906717
cites work: Siegel modular forms and theta series attached to quaternion algebras
cites work: EULER PRODUCTS FOR CONGRUENCE SUBGROUPS OF THE SIEGEL GROUP OF GENUS 2
cites work: On the geometry and arithmetic of some Siegel modular threefolds
cites work: The Cusp Forms of Weight 3 on Γ 2 (2, 4, 8)
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
cites work: Automorphic forms on GL (2)
cites work: On modular forms associated with indefinite quadratic forms of signature \((2,n-2)\)
cites work: Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces
cites work: Cusp Forms of Weight 2 for the Group Γ 2 (4, 8)
cites work: Q5625265
cites work: Siegel's modular forms and the arithmetic of quadratic forms
DOI: 10.1016/J.JNT.2006.11.005
description: scientific article; zbMATH DE number 5161938