Linear equations and heights over division algebras
In 1983, \textit{E. Bombieri} and \textit{J. Vaaler} proved a version of Siegel's Lemma for systems of linear equations to be solved in vectors whose coordinates lie in the ring of integers of a given algebraic number field [Invent. Math. 73, 11--32 (1983; Zbl 0533.10030)]. In the present paper, the author suggests to prove a similar result for vectors whose coordinates lie in a given maximal order of a given division algebra \(D\) over \(\mathbb{Q}\), and in fact she obtains such a result in case that \(D\) is a positive definite quaternion algebra over \(\mathbb{Q}\), i.e., \(D=\binom {a~b}{\mathbb{Q}}\) with \(a,b\in\mathbb{Q}_{<0}\). Given a maximal order \({\mathcal O}\) of \(D\), the author introduces norms \(\| X\| \) for vectors \(X\in{\mathcal O}^N\) and heights \(H^{{\mathcal O}}(A)\) for \(M\times N\)-matrices \(A\) with entries in \(D\) where \(M<N\) which are natural analogues of norms and heights used by Bombieri and Vaaler. Here a maximal order of \(D\) is not uniquely determined and so the height depends on \({\mathcal O}\). Then the author's version of Siegel's Lemma is as follows: Let \({\mathcal O}\) be any maximal order of \(D\), and \(A\) and \(M\times N\)-matrix with entries in \(D\), where \(N>M>0\) and \(A\) has left row rank \(M\). Then there exist right-linearly independent vectors \(X_1,\ldots ,X_{N-M}\in {\mathcal O}^N\) satisfying \(AX_i=0\) for \(i=1,\ldots ,N-M\) and \[ \prod_{i=1}^{N-M}\| X_i\| \leq \Big( {2\over\pi^2}\sqrt{\Delta_D}\Big)^{(N-M)/4}H^{{\mathcal O}}(A), \tag \(*\) \] where \(\Delta_D\) is the discriminant of \(D\). The author's proof is based on an argument by \textit{J. L. Thunder} [Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21--26, 2000. Natick, MA: A K Peters, 253--259 (2002; Zbl 1042.11044)]. Thus, unlike Bombieri and Vaaler who used geometry of numbers over the adeles, the author deduces her result directly from Minkowski's second convex body theorem. The author shows that in (*) the dependence on \(H^{{\mathcal O}}(A)\) is best possible and that the dependence on \(\Delta_D\) cannot be removed.
scientific article; zbMATH DE number 2082497
scientific article