Automorphisms of even unimodular lattices over number fields
type: Item
instance of: scholarly article
title: Automorphisms of even unimodular lattices over number fields
zbMATH Open document ID: 1412.11081
author: Markus Kirschmer
published in: Journal of Number Theory
publication date: 2018-12-18T00:00:00Z
review text: Let \( L \) be a lattice over a number field \( \mathbb{K} \), i.e., \( L \) is a finitely generated \( \mathcal{O}_\mathbb{K} \) module (\( \mathcal{O}_\mathbb{K} \) is the ring of algebraic integers in \( \mathbb{K} \)). Moreover, let \( L \) be equipped with a non-degenerate symmetric bilinear form \( B \). Recall that a lattice \( L \) is called even if \( B(x,x)\in 2 \mathcal{O}_\mathbb{K} \) for all \( x\in L \) and a lattice \( L \) is called unimodular, if \( L=\{ x\in \mathbb{K}L \;| \;B(x,y)\in \mathcal{O}_\mathbb{K} \;\text{ for all } y\in L \} \). In this paper the author describes the powers of irreducible polynomials occuring as characteristic polynomials of proper automorphisms of even unimodular lattices over number fields (Theorem A). The main theorem generalizes the result of \textit{E. Bayer-Fluckiger} and \textit{L. Taelman} [``Automorphisms of even unimodular lattices and equivariant Witt groups'', Preprint, \url{arxiv:1708.05540}]. The proof of the main theorem bases on the construction of a trace lattice of suitable hermitian lattice with prescribed local structure. Bayer-Fluckiger and Taelman [loc. cit.] proved the existence of such lattices over non-dyadic fields. The author extended this result for lattice over dyadic fields. The main theorem follows by applying the local-global principle for Brauer groups.
reviewed by: Alar Leibak
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 6994674
zbMATH Keywords: even unimodular lattices
zbMATH Keywords: automorphisms
zbMATH Keywords: dyadic fields
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.08.004
OpenAlex ID: W2891282528
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DOI: 10.1016/J.JNT.2018.08.004
description: scientific article; zbMATH DE number 6994674