Construction of Jacobi forms
type: Item
instance of: scholarly article
title: Construction of Jacobi forms
zbMATH Open document ID: 0822.11040
DOI: 10.1007/BF02572350
author: Hyun Kwang Kim
author: YoungJu Choie
author: Marvin Knopp
published in: Mathematische Zeitschrift
publication date: 1995-10-23T00:00:00Z
full work available at URL: https://eudml.org/doc/174758
review text: A Jacobi form \(\varphi\) with weight \(k\) and index \(m\) on \(SL(2, \mathbb{Z})\) is defined to be a holomorphic function \(\varphi:{\mathcal H}\times \mathbb{C}\to \mathbb{C}\) satisfying the two transformation equations, for given positive integers \(k\) and \(m\), \[ \begin{aligned} (c\tau+ d)^{-k} e^{2\pi im {{-cz^ 2} \over {c\tau+d}}} \varphi \Biggl( {{a\tau+b} \over {c\tau+d}}, {z \over {c\tau+d}} \Biggr) &= \varphi (\tau, z),\\ e^{2\pi im (\lambda^ 2\tau+ 2\lambda z)} \varphi (\tau,z+ \lambda\tau+ \mu) &= \varphi (\tau, z), \end{aligned} \] for any \((\lambda, \mu)\in \mathbb{Z}^ 2\) and \(\left( \begin{smallmatrix} a &b\\ c&d \end{smallmatrix} \right)\in SL(2, \mathbb{Z})\), and it has a Fourier expansion of the form \[ \varphi(\tau, z)= \sum^ \infty _{\substack{ n=1\\ r\in \mathbb{Z}, r^ 2\leq 4nm }} c(n,r) \xi^ r q^ n, \] where \(q= e^{2\pi i\tau}\) and \(\xi= e^{2\pi iz}\). The authors adapt the Rankin-Selberg convolution to give a construction of Jacobi forms on the modular group \(\Gamma(1)\). Given two Jacobi cusp forms on \(\Gamma (1)\), one of weight \(k\) and index \(m\) and the other of weight \(\ell\) and index 0, the construction yields a third Jacobi cusp form, of weight \(k-\ell\) and index \(m\). The method is subject to the restriction \(k>\ell +5>5\).
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 784100
zbMATH Keywords: \(L\)-series of Rankin type
zbMATH Keywords: Hecke-type estimate
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Rankin-Selberg convolution
zbMATH Keywords: modular group \(\Gamma(1)\)
zbMATH Keywords: Jacobi cusp forms
MaRDI profile type: Publication
cites work: The theory of Jacobi forms
cites work: Heegner points and derivatives of \(L\)-series. II
cites work: Über Kernfunktionen für Jacobiformen und Siegelsche Modulformen. (Kernel functions for Jacobi forms and Siegel modular forms)
cites work: Cusp forms and special values of certain Dirichlet series
cites work: Q5568135
cites work: Q3497147
OpenAlex ID: W2041008699
description: scientific article; zbMATH DE number 784100