Heights and determinants over quaternion algebras

In Liebendörfer (20047. Liebendörfer , C. ( 2004 ). Linear equations and heights over division algebras . J. Number Theory 105 : 101 – 133 . [CSA] [CROSSREF] View all references) 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 (192210. Moore , E. H. ( 1922 ). On the determinant of an Hermitian matrix of quaternionic elements . Bull. Amer. Math. Soc. 28 : 161 – 162 . [CSA] View all references) 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 (20047. Liebendörfer , C. ( 2004 ). Linear equations and heights over division algebras . J. Number Theory 105 : 101 – 133 . [CSA] [CROSSREF] View all references).