Minkowski’s conjecture, well-rounded lattices and topological dimension

McMullen, Curtis¹

¹ Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, Massachusetts 02138-2901

Journal of the American Mathematical Society, Tome 18 (2005) no. 3, pp. 711-734

Résumé

Let $A \subset {\operatorname {SL}}_n({\mathbb R})$ be the diagonal subgroup, and identify ${\operatorname {SL}}_n({\mathbb R})/ {\operatorname {SL}}_n({\mathbb Z})$ with the space of unimodular lattices in ${\mathbb R}^n$. In this paper we show that the closure of any bounded orbit \begin{equation*} A \cdot L \subset {\operatorname {SL}}_n({\mathbb R})/{\operatorname {SL}}_n({\mathbb Z}) \end{equation*} meets the set of well-rounded lattices. This assertion implies Minkowski’s conjecture for $n=6$ and yields bounds for the density of algebraic integers in totally real sextic fields. The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of ${\mathbb R}^n$ and $T^n$.

Détail

DOI : 10.1090/S0894-0347-05-00483-2

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