Unimodular lattices over real quadratic fields
type: Item
instance of: scholarly article
title: Unimodular lattices over real quadratic fields
zbMATH Open document ID: 0807.11021
DOI: 10.1007/BF02572333
author: Rudolf Scharlau
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174662
review text: We investigate integral even unimodular lattices \(L\) in a vector space with a totally positive definite quadratic form, defined over a real quadratic field \(F\). We give explicit constructions of a number of such lattices in dimension 4, for indeterminate field discriminant \(d\) (only depending on \(d\bmod 24\), i.e. on the ramification of 2 and 3 in \(F\)). The lattices we construct here have large automorphism groups. In most cases, the full orthogonal group \(O(L)\) is known (and essentially independent of the field). This is true in particular for the so-called reflective lattices which have a root system of maximal rank. As an application we obtain the full classification (essentially independent of the use of computers) of all even unimodular lattices in dimension 4 over the first 11 real quadratic fields with discriminants \(d=5,8,12,13,17,21,24,28,29,33,37\).
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E41
zbMATH DE Number: 704929
zbMATH Keywords: integral lattice
zbMATH Keywords: Weyl group
zbMATH Keywords: class number
zbMATH Keywords: integral even unimodular lattices
zbMATH Keywords: real quadratic field
zbMATH Keywords: reflective lattices
zbMATH Keywords: root system
MaRDI profile type: Publication
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OpenAlex ID: W2059661867
description: scientific article; zbMATH DE number 704929