Siegel theta series of degree 2 of extremal lattices
type: Item
instance of: scholarly article
title: Siegel theta series of degree 2 of extremal lattices
zbMATH Open document ID: 0701.11015
DOI: 10.1016/0022-314X(90)90103-X
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: Let \(\Lambda\) be an even unimodular lattice of rank N (N must satisfy \(N\equiv 0 (mod 8))\). Furthermore let \(\Lambda\) be an extremal lattice, i.e. \(\min_{x\in \Lambda,x\neq 0}Q(x)=[N/24]+1\), where Q(x) is the quadratic form of the lattice \(\Lambda\). In J. Number Theory 28, 119-131 (1988; Zbl 0638.10031), the reviewer raised the question whether the Siegel theta-series of degree 2 distinguish the equivalence classes of even unimodular extremal lattices of rank 40. In the present paper, this question is solved negatively. The author shows that there is more than one class of even unimodular positive extremal lattices of rank 40 with the same Siegel theta-series of degree 2, using (i) Igusa's structure theorem of Siegel modular forms of degree 2, (ii) a relation between the coefficients of Jacobi theta-series attached to the lattices in question and (iii) the pigeon hole principle. He also gives a sketch that the same statement holds for the case \(N=64\).
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 4148178
zbMATH Keywords: Siegel theta-series of degree 2
zbMATH Keywords: even unimodular extremal lattices of rank 40
author: Meinhard Peters
reviewed by: Michio Ozeki
MaRDI profile type: Publication
cites work: Q4039784
cites work: The theory of Jacobi forms
cites work: On Siegel Modular Forms of Genus Two
cites work: Examples of even unimodular extremal lattices of rank 40 and their Siegel theta-series of degree 2
cites work: Definite Unimodular 48-Dimensional Quadratic Forms
cites work: Q4405255
full work available at URL: 0022-314x(90)90103-x
OpenAlex ID: W1995653910
description: scientific article; zbMATH DE number 4148178