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

Manfred Einsiedler*, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


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

Original languageAmerican English
Pages (from-to)815-885
Number of pages71
JournalAnnals of Mathematics
Issue number2
StatePublished - Mar 2011


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