Jacobi forms and a certain space of modular forms
type: Item
instance of: scholarly article
title: Jacobi forms and a certain space of modular forms
zbMATH Open document ID: 0651.10020
DOI: 10.1007/BF01394347
author: Nils-Peter Skoruppa
author: Don Zagier
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143620
review text: Let \(M_{2k-2}(m)\) be the space of holomorphic modular forms of weight \(2k-2\) on \(\Gamma_0(m)\) and let \(J_{k,m}\) be the space of Jacobi forms of weight \(k\) and index \(m\) in the sense of Eichler-Zagier [\textit{M. Eichler} and \textit{D. Zagier}, The theory of Jacobi forms. Prog. Math. 55. Basel et al.: Birkhäuser (1985; Zbl 0554.10018)]. The main point in the proof of the Saito-Kurokawa conjecture was the isomorphism between \(J_{k,1}\) and \(M_{2k-2}(1)\) as modules over the Hecke algebra. In the impressive paper under review the authors deal with the general case for the index \(m\). There exists a canonical subspace \({\mathfrak M}^{-}_{2k-2}(m)\) of \(M^{-}_{2k-2}(m)\), which can be described by properties of the Euler factors of the \(L\)-series attached to a modular form and which contains the space of newforms. Here ``-'' means that the \(L\)-series satisfies a functional equation under \(s\mapsto 2k-2-s\) with root number \(-1\). The Main Theorem says that \(J_{k,m}\) and \({\mathfrak M}^{-}_{2k-2}(m)\) are isomorphic as modules over the Hecke algebra. In \S 1 the trace of the Hecke operator \(T(\ell)\) on \(J_{k,m}\) with \(\ell\) relatively prime to \(m\) is computed as an application of the general trace formula for Jacobi forms. Then the Eichler-Selberg trace formula is used in order to express \(\operatorname{tr}(T(\ell),J_{k,m})\) as linear combinations of \(\operatorname{tr}(T(\ell),M_{2k-2}^{\text{new},-}(m'))\), \(m'\mid m\). In \S 3 the isomorphy is proved, where the proof moreover gives a collection of explicit lifting maps. In the Appendix, the authors derive a formula for a certain class number involving Gauss sums associated to binary quadratic forms.
reviewed by: Aloys Krieg
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 4061329
zbMATH Keywords: elliptic modular form
zbMATH Keywords: Jacobi forms
zbMATH Keywords: newforms
zbMATH Keywords: Hecke operator
zbMATH Keywords: trace formula
zbMATH Keywords: isomorphy
zbMATH Keywords: lifting maps
zbMATH Keywords: class number
zbMATH Keywords: Gauss sums
zbMATH Keywords: binary quadratic forms
MaRDI profile type: Publication
cites work: The theory of Jacobi forms
cites work: Heegner points and derivatives of \(L\)-series. II
cites work: A trace formula for Jacobi forms.
cites work: On the traces of Hecke operators for a normalizer of \(\Gamma_0\)(N)
OpenAlex ID: W2003608551
description: scientific article; zbMATH DE number 4061329