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

作者：Einsiedler Manfred^*; Lindenstrauss Elon; Michel Philippe; Venkatesh Akshay

来源：Annals of Mathematics, 2011, 173(2): 815-885.

DOI：10.4007/annals.2011.173.2.5

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 -> infinity.
The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

• 出版日期2011-3