Heights and Determinants over Quaternion Algebras#

Abstract

In Liebendörfer (2004) 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) 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).