On old and new Jacobi forms
type: Item
instance of: scholarly article
title: On old and new Jacobi forms
zbMATH Open document ID: 0958.11033
author: Ralf Schmidt
published in: Journal of Number Theory
publication date: 2001-04-17T00:00:00Z
review text: \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms, Prog. Math. 55 (1985; Zbl 0554.10018)] introduced certain Hecke operators \(U_d\) and \(V_d\) on classical Jacobi forms. They change the index and are the basis for a theory of Jacobi old- and newforms. In the paper under review the author introduces certain index shifting operators for local and global representations of the Jacobi groups, which turn out to be compatible with the classical one. Moreover a purely local conjecture is formulated about the dimension of spaces of spherical vectors in representations of the \(p\)-adic Jacobi group. Evidence for this conjecture also comes from the ``certain space'' of modular forms by \textit{N. Skoruppa} and \textit{D. Zagier} [Invent. Math. 94, 113-146 (1988; Zbl 0651.10020)] and its local analogs.
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1389289
zbMATH Keywords: Jacobi forms
zbMATH Keywords: Hecke operators
zbMATH Keywords: index shifting operator
zbMATH Keywords: oldforms
zbMATH Keywords: newforms
zbMATH Keywords: spherical vectors
zbMATH Keywords: local representation
zbMATH Keywords: global representation
reviewed by: Aloys Krieg
MaRDI profile type: Publication
OpenAlex ID: W2012466703
cites work: Elements of the representation theory of the Jacobi group
cites work: Q4234990
cites work: On some results of Atkin and Lehner
cites work: \(L\)-functions for Jacobi forms on \(\text{H} \times \mathbb{C}\)
cites work: The theory of Jacobi forms
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: L-functions attached to Jacobi forms of degree n. Part I. The basic identity.
cites work: Q4382649
cites work: Spherical representations of the Jacobi group
cites work: Jacobi forms and a certain space of modular forms
cites work: Correspondances de Shimura et quaternions
DOI: 10.1006/JNTH.1999.2423
description: scientific article; zbMATH DE number 1389289