Distribution of periodic torus orbits and Duke’s theorem for cubic fields

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.