On positive definite Hermitian forms
type: Item
instance of: scholarly article
title: On positive definite Hermitian forms
zbMATH Open document ID: 0729.11020
DOI: 10.1007/BF02568415
author: Detlev Hoffmann
published in: Manuscripta Mathematica
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155624
review text: This work has originated from a question posed by Serre, in connection with his study of the maximal number of rational points on curves of genus 3 over finite fields, to Kneser io 1983. Over which orders in imaginary quadratic fields does an indecomposable positive definite unimodular Hermitian form of rank 3 exist? The corresponding question for rank 2 had been settled earlier by \textit{T. Hayashida} and \textit{M. Nishi} [J. Math. Soc. Japan 17, 1-16 (1965; Zbl 0132.417)]. The answers to both can be found in this paper: Up to 4 (resp. 3) exceptions for rank 3 (resp. 2), such a form always exists and is given explicitly. The author applies Kneser's method of constructing neighbour lattices. After developing the background of the method for maximal orders in greater generality, he mainly deals with the classification over such orders in imaginary quadratic fields. Representatives for the classes of unimodular lattices of rank 2 and 3 are listed for discriminant D down to -20, and the case \(D=-7\) is treated in detail.
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11E39
zbMATH DE Number: 4204438
zbMATH Keywords: class numbers
zbMATH Keywords: unimodular Hermitian form of rank 3
zbMATH Keywords: Kneser's method
zbMATH Keywords: neighbour lattices
zbMATH Keywords: orders
zbMATH Keywords: imaginary quadratic fields
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4204438