Duality of heights over quaternion algebras
type: Item
instance of: scholarly article
title: Duality of heights over quaternion algebras
zbMATH Open document ID: 1104.11035
author: Christine Liebendörfer
author: Gaël Rémond
published in: Monatshefte für Mathematik
publication date: 2005-06-14T00:00:00Z
review text: It is well-known that the height \(H(V)\) of a linear subspace \(V\) of \(K^n\) (\(K\) algebraic number field) is equal to the height of its orthogonal complement. This fact is called the duality property of the height. The simple proof of this duality property uses, that the height of \(V\) can be expressed in terms of the Grassmann coordinates of \(V\). In the present paper, the authors introduce a height for vector spaces over a quaternion algebra and show that it has a duality property analogous to the one for vector spaces over an algebraic number field. The proof of the former is however rather different, since vector spaces over a non-commutative division algebra cannot be expressed by means of Grassmann coordinates. More precisely, let \(D\) be a positive definite quaternion algebra over \({\mathbb Q}\) (i.e., a \(4\)-dimensional \({\mathbb Q}\)-algebra whose tensor product with \({\mathbb R}\) is isomorphic to the Hamiltonian skew field \({\mathbb H}={\mathbb R}\oplus i{\mathbb R}\oplus j{\mathbb R}\oplus k{\mathbb R}\)), and let \({\mathcal O}\) be an order in \(D\). The involution \(x=a+bi+cj+dk\mapsto x^*=a-bi-cj-dk\) on \({\mathbb H}\) induces an involution on \(D\). For a matrix \(A\) with entries in \(D\), define \(A^*\) to be the matrix obtained by transposing \(A\) and applying the above involution to the entries of \(A\). The authors define a height \(H^{{\mathcal O}}(V)\) for a right \(D\)-linear subspace \(V\) of \(D^n\) which depends on the choice of the order \({\mathcal O}\). This definition is as follows: if \(V\) has dimension \(n-m\), choose an \(m\times n\)-matrix \(A\) with elements in \({\mathcal O}\) of left row rank \(m\) such that \(V=\{ {\mathbf x}\in D^n:\, A{\mathbf x}=0\}\) and put \[ \mathbb H^{{\mathcal O}}(V):= \big((nr\, AA^*) \cdot[{\mathcal O}^m:A({\mathcal O}^n)]^{-1}\big)^{1/4}, \] where \(nr\) denotes the reduced norm map from \(M_m(D)\) to \({\mathbb Q}\). The authors prove the following \textbf{Duality Theorem:} if \({\…
reviewed by: Jan-Hendrik Evertse
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 2175900
zbMATH Keywords: heights
zbMATH Keywords: duality
zbMATH Keywords: quaternion algebras
MaRDI profile type: Publication
full work available at URL: s00605-004-0280-1
OpenAlex ID: W2166447393
DOI: 10.1007/S00605-004-0280-1
description: scientific article; zbMATH DE number 2175900