Ternary positive definite quadratic forms are determined by their theta series
type: Item
instance of: scholarly article
title: Ternary positive definite quadratic forms are determined by their theta series
zbMATH Open document ID: 0876.11029
author: Alexander Schiemann
published in: Mathematische Annalen
publication date: 1997-12-03T00:00:00Z
review text: We consider the set \(D\) of pairs of ternary positive definite real forms which are reduced in the sense of Seeber-Eisenstein and have the same theta series. \(D\) is embedded into \(\mathbb{R}^{12}\) by the coefficients of the Gram matrices. The reduction conditions define a unique representative of every class (with respect to integral equivalence). By restricting to subsets of \(D\) with finitely many prescribed minimal vectors, the condition on the theta series yields linear equalities. We describe a series of finite coverings (one refining the other) of \(D\) with rational polyhedral cones arising from the prescribed minimal vectors and the equality of the first theta coefficients. Explicit computations (done by computer) show that the equality of the theta series of two reduced forms \(f\) and \(g\) up to a bound \(b(f)\) implies \(f=g\). We may choose \(b(f)=7/2 \cdot s_3(f)\), where \(s_3(f)\) denotes the third successive minimum of \(f\).
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11-04
zbMATH DE Number: 1057252
zbMATH Keywords: ternary positive definite quadratic forms
zbMATH Keywords: reduced forms
zbMATH Keywords: polyhedral cones
zbMATH Keywords: prescribed minimal vectors
zbMATH Keywords: representation numbers
zbMATH Keywords: theta series
MaRDI profile type: Publication
full work available at URL: s002080050086
OpenAlex ID: W2033249359
DOI: 10.1007/S002080050086
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description: scientific article; zbMATH DE number 1057252