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 SL3(ℤ)∖SL3(ℝ)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(ℤ)∖SL3(ℝ)/SO3 of volume ≤V becomes equidistributed as V→∞.
The proof combines subconvexity estimates, measure classification, and local harmonic analysis.