Heights and Determinants over Quaternion Algebras#
Christine Liebendörfer · Communications in Algebra · 2005
In Liebendörfer (2004 Liebendörfer , C. ( 2004 ). Linear equations and heights over division algebras . J. Number Theory 105 : 101 – 133 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) we defined a height function for matrices over a positive definite rational quaternion algebra. In this article, we prove that this height can be expressed, like the well-known height over number fields, as a product of local factors involving the maximal minors of the matrix. Part of our proof uses a quaternionic determinant invented by Moore (1922 Moore , E. H. ( 1922 ). On the determinant of an Hermitian matrix of quaternionic elements . Bull. Amer. Math. Soc. 28 : 161 – 162 . [CSA] [Google Scholar]) and yields a non-commutative analogue of the Cauchy–Binet formula. The new formula of the height enables us to improve a result of Liebendörfer (2004 Liebendörfer , C. ( 2004 ). Linear equations and heights over division algebras . J. Number Theory 105 : 101 – 133 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]).