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

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

2007-08-08

arxiv.org/abs/0708.1113v2

Mathematics

Number Theory

Annals of Maths. (to appear) typos corrected; references updated

Scientific paper

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.

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