Rendiconti del Seminario Matematico della Università di Padova

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Volume 130, 2013, pp. 221–282
DOI: 10.4171/RSMUP/130-9

Le théorème de Schanuel pour un corps non commutatif

Gaël Rémond[1] and Christine Zehrt-Liebendörfer[2]

(1) Institut Fourier, Université Grenoble I, B.P. 74, 38402, Saint-Martin-d’Hères CEDEX, France
(2) Departement Mathematik und Informatik, Fachbereich Mathematik, Universität Basel, Spiegelgasse 1, 4051, Basel, Switzerland

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.

Keywords: Hauteur, corps non commutatif, théorème de Schanuel, ordre maximal, anti-involution

Rémond Gaël, Zehrt-Liebendörfer Christine: Le théorème de Schanuel pour un corps non commutatif. Rend. Sem. Mat. Univ. Padova 130 (2013), 221-282. doi: 10.4171/RSMUP/130-9