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arXiv · 0708.1113

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

Manfred Einsiedler·Elon Lindenstrauss·Philippe Michel·Akshay Venkatesh
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Abstract

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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Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh. 2010-02-03. Distribution of periodic torus orbits and Duke's theorem for cubic fields. https://doi.org/10.4007/annals.2011.173.2.5

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