On the modularity of the \(\text{GL}_2\)-twisted spinor \(L\)-function
type: Item
instance of: scholarly article
title: On the modularity of the \(\text{GL}_2\)-twisted spinor \(L\)-function
zbMATH Open document ID: 1223.11062
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2010-06-21T00:00:00Z
review text: In this paper, the author studies the twisted spinor \(L\)-function \(Z_{G\otimes h}(s)\) attached to Hecke eigenforms \(G\in M_{k_1}^2\) (degree 2, weight \(k_1\)) and \(h\in M_{k_2}^1\) (degree 1, weight \(k_2\)). He proves the followings: 1) The \(L\)-function \(Z_{G\otimes h}(s)\) has a meromorphic continuation to the whole complex plane and possesses a functional equation. 2) Let \(E_k^n\) be the Siegel-Eisenstein series of degree \(n\) and weight \(k\). Then \(Z_{E_k^2\otimes E_{2k-2}^1}(s)\) is modular. Here the modularity of \(Z_{G\otimes h}(s)\) means that there exists a Siegel modular form \(F:=G\boxtimes h\) of degree 3 such that \(L(s,G\boxtimes h)=Z_{G\otimes h}(s)\). Moreover, he refers to the Miyawaki conjecture concerning with Siegel modular forms of degree 3.
reviewed by: Shoyu Nagaoka
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 5723791
zbMATH Keywords: modular forms
zbMATH Keywords: automorphic L-functions
zbMATH Keywords: Miyawaki conjecture
author: Bernhard Heim
MaRDI profile type: Publication
full work available at URL: s12188-009-0028-x
OpenAlex ID: W2126628207
cites work: EULER PRODUCTS CORRESPONDING TO SIEGEL MODULAR FORMS OF GENUS 2
cites work: Über Funktionalgleichungen automorpher L-Funktionen zur Siegelschen Modulgruppe.
cites work: \(L\)-functions on \(GSp_2\times Gl_2\) of mixed weights
cites work: The rationality of holomorphic Eisenstein series
cites work: Q4263338
cites work: Miyawaki's \(F_{12}\) spinor \(L\)-function conjecture
cites work: On the lifting of elliptic cusp forms to Siegel cusp forms of degree \(2n\)
cites work: Pullback of the lifting of elliptic cusp forms and Miyawaki's conjecture
cites work: Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two
cites work: NUMERICAL EXAMPLES OF SIEGEL CUSP FORMS OF DEGREEE 3 AND THEIR ZETA-FUNCTIONS
cites work: Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\mathrm{SL}(2)\)
cites work: Siegel modular forms of genus 2 attached to elliptic curves
cites work: On modular forms of half integral weight
cites work: Stabile Modulformen und Eisensteinreihen. (Stable modular forms and Eisenstein series)
cites work: Modular elliptic curves and Fermat's Last Theorem
cites work: Motives and Siegel modular forms
DOI: 10.1007/S12188-009-0028-X
description: scientific article; zbMATH DE number 5723791