Abstract
We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular $5$-fold $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3$. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3$ of volume $\leq V$ becomes equidistributed as $V \rightarrow \infty$.
The proof combines subconvexity estimates, measure classification, and local harmonic analysis.
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