  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 SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger
form of) the folklore conjecture that the collection of maximal compact flats
in SL_3(Z)\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V
goes to infinity. The proof combines subconvexity estimates, measure
classification, and local harmonic analysis.
