Positive definite binary Hermitian forms with finitely many exceptions
type: Item
instance of: scholarly article
title: Positive definite binary Hermitian forms with finitely many exceptions
zbMATH Open document ID: 1131.11027
author: Wai Kiu Chan
author: Anna Rokicki
published in: Journal of Number Theory
publication date: 2007-05-04T00:00:00Z
review text: Let \(E/F\) be a CM extension of algebraic number fields and let \(L\) be a positive definite Hermitian lattice over the ring of integers \(\mathfrak O\) of \(E\). An element \(a\) of \(F\) is said to be exceptional for \(L\) if \(a\) is represented by all localizations of \(L\) but not by \(L\) itself. When the exceptional set is empty (finite), the lattice is said to be regular (almost regular, respectively). If the rank of \(L\) exceeds 2, it is known that every such lattice is almost regular [\textit{J. S. Hsia} and \textit{J. P. Prieto-Cox}, J. Number Theory 47, 175--189 (1994; Zbl 0797.11040)], but this is no longer the case for lattices of rank 2. Another Hermitian \(\mathfrak O\)-lattice \(M\) is said to be similar to \(L\) if \(L\) is isometric to \(M^{\alpha}\) for some nonzero \(\alpha \in F\). Similarity is an equivalence relation on the set of \(\mathfrak O\)-lattices, and similar lattices have exceptional sets of equal size. In this paper, it is shown that if a CM extension \(E/F\) and a non-negative integer \(k\) are fixed, then there are at most finitely many similarity classes of positive definite binary Hermitian \(\mathfrak O\)-lattices for which the exceptional set has size at most \(k\). This generalizes a finiteness result for the case \(F=\mathbb Q\) proved by the reviewer and \textit{A. Khosravani} [J. Number Theory 62, 368--374 (1997; Zbl 0871.11028)]. It is also proven here that for a fixed totally real field \(F\) of odd degree over \(\mathbb Q\), there are at most finitely many CM extensions \(E/F\) for which there exists a positive definite regular normal (i.e., norm and scale ideals are equal) binary Hermitian lattice over the ring of integers of \(E\).
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11E39
zbMATH DE Number: 5149377
zbMATH Keywords: Hermitian form
zbMATH Keywords: Hermitian lattice
zbMATH Keywords: CM extension
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.07.016
OpenAlex ID: W2056552920
cites work: Discriminant Bounds for Spinor Regular Ternary Quadratic Lattices
cites work: Finiteness theorems for positive definite 𝑛-regular quadratic forms
cites work: Positive ternary quadratic forms with finitely many exceptions
cites work: Q4665003
cites work: The Representation of Binary Quadratic Forms by Positive Definite Quaternary Quadratic Forms
cites work: Q4942332
cites work: Representation of integers by positive definite binary hermitian lattices over imaginary quadratic fields
cites work: Universal positive quaternary quadratic lattices over totally real number fields
cites work: Universal binary Hermitian forms
cites work: Integral Decomposition of Hermitian Forms
cites work: Representations of positive definite Hermitian forms with approximation and primitive properties
cites work: Universal binary positive definite Hermitian lattices
cites work: Hermitian Forms Over Local Fields
cites work: Integral representations of hermitian forms over local fields.
cites work: Q4254917
cites work: Q3293461
cites work: Arithmetic of unitary groups
DOI: 10.1016/J.JNT.2006.07.016
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description: scientific article; zbMATH DE number 5149377