Le théorème de Schanuel pour un corps non commutatif

Abstract

We prove a version of Schanuel's theorem in the noncommutative case: we provide an asymptotic formula for the number of one-dimensional left subspaces of $D^N$ of height at most $H$, where $D$ is a finite dimensional rational division algebra, $N$ a positive integer and $H$ a real number. The height, as considered in a previous paper, is defined with the help of a maximal order in $D$ and a positive anti-involution. We give a completely explicit main term involving class number, regulator, discriminant and zeta function of $D$. We also compute an explicit error term.