Exceptional integers of some ternary quadratic forms
type: Item
instance of: scholarly article
title: Exceptional integers of some ternary quadratic forms
zbMATH Open document ID: 0493.10022
DOI: 10.1016/S0001-8708(82)80007-8
author: Dennis R. Estes
author: J. S. Hsia
published in: Advances in Mathematics
publication date: 1982-01-01T00:00:00Z
review text: An integer \(c\), if everywhere locally representable by an integral ternary quadratic form, is known to be either representable by every form or by precisely half of all the inequivalent forms in the genus \(\mathrm{gen}(f)\) of \(f\). In the latter case such an integer will be called an exceptional integer for \(\mathrm{gen}(f)\). An integer \(c\) is called an exceptional integer for \(g\), \(g\in\mathrm{gen}(f)\), if \(g\) belongs to the``bad'' half-genus that fails to represent \(c\), i.e. if \(c\) is representable by some form in the genus but not by \(g\) itself. The paper gives characterizations for exceptional integers for ternary unimodular lattices. Especially the question: ``Which formally nonreal algebraic number field \(k\) has all its integers expressible as sums of three integral squares in \(k\)'' is in principle completely solved and explicitly solved for complex quadratic fields.
reviewed by: Meinhard Peters
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 3775628
zbMATH Keywords: ternary quadratic forms
zbMATH Keywords: spinor genus
zbMATH Keywords: exceptional integers
zbMATH Keywords: ternary unimodular lattices
MaRDI profile type: Publication
cites work: Spinor norms of local integral rotations. I
cites work: Representations by spinor genera
cites work: Q4175628
cites work: On the classification of unimodular quadratic forms
cites work: Q4063245
cites work: On Indefinite Ternary Quadratic Forms
cites work: Darstellungsmaße indefiniter quadratischer Formen
cites work: Q5612629
cites work: Q3293461
cites work: Sums of \(m\)th powers of algebraic integers
cites work: Darstellung durch Spinorgeschlechter ternärer quadratischer Formen
full work available at URL: s0001-8708(82)80007-8
OpenAlex ID: W2094037939
description: scientific article; zbMATH DE number 3775628