Duality of heights over quaternion algebras
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 \({\mathcal O}\) is a maximal order of \(D\), then for any non-zero right linear subspace \(V\) of \(D^N\) we have \(\mathbb H^{{\mathcal O}}(V^{\bot})=H^{{\mathcal O}}(V)\), where \(V^{\bot}:=\{ {\mathbf x}\in D^N:\, {\mathbf x}^*{\mathbf y}=0\;\forall {\mathbf y}\in V\}\). Further they give examples showing that this Duality Theorem does not hold true in general for non-maximal orders \({\mathcal O}\), but that on the other hand there are also non-maximal orders for which it does remain valid. The proofs use local arguments, as well as properties of certain non-commutative finite rings.
scientific article; zbMATH DE number 2175900