Modular lattices in Euclidean spaces
type: Item
instance of: scholarly article
title: Modular lattices in Euclidean spaces
zbMATH Open document ID: 0874.11038
published in: Journal of Number Theory
publication date: 1997-11-11T00:00:00Z
review text: The Fricke group \(\Gamma_x(\ell)\), \(\ell\in\mathbb{N}\), is the extension of the congruence subgroup \(\Gamma_0(\ell)\) of the modular group \(SL(2,\mathbb{Z})\) by the element \(T_\ell=\left(\begin{smallmatrix} 0 & (\sqrt\ell)^{-1}\\ -\sqrt\ell & 0\end{smallmatrix}\right)\). If \(\Lambda\) is an even lattice on an Euclidean \(n\)-space the theta function of \(\Lambda\) is defined as \(\Theta_\Lambda(z)= \sum_{v\in\Lambda}q^{(v\cdot v)/2}\), \(q= e^{2\pi iz}\). If \(n=2k\) and \(\Lambda\) is an even \(\sigma\)-modular lattice for some similarity \(\sigma\) of norm \(\ell\) then \(\Theta_\Lambda(z)\) is an automorphic form of weight \(k\) for \(\Gamma_*(\ell)\). This connection is discussed in detail.
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 809117
zbMATH Keywords: modular lattices
zbMATH Keywords: modular group
zbMATH Keywords: Fricke group
zbMATH Keywords: automorphic form
author: Heinz-Georg Quebbemann
reviewed by: Gerhard Rosenberger
MaRDI profile type: Publication
full work available at URL: jnth.1995.1111
OpenAlex ID: W2035808197
DOI: 10.1006/JNTH.1995.1111
description: scientific article; zbMATH DE number 809117