TY - JOUR

T1 - Distribution of periodic torus orbits and Duke's theorem for cubic fields

AU - Einsiedler, Manfred

AU - Lindenstrauss, Elon

AU - Michel, Philippe

AU - Venkatesh, Akshay

PY - 2011/3

Y1 - 2011/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79953195197&partnerID=8YFLogxK

U2 - 10.4007/annals.2011.173.2.5

DO - 10.4007/annals.2011.173.2.5

M3 - Article

AN - SCOPUS:79953195197

SN - 0003-486X

VL - 173

SP - 815

EP - 885

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -