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

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.

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

References in corpus (5)

• The subconvexity problem for $\GL_{2}$
• Local-global principles for representations of quadratic forms
• Sparse equidistribution problems, period bounds, and subconvexity
• The distribution of periodic torus orbits on homogeneous spaces
• Weak subconvexity for central values of $L$-functions

Cited by in corpus (22)

• The subconvexity problem for $\GL_{2}$
• Local-global principles for representations of quadratic forms
• On the convergence of arithmetic orbifolds
• Subconvex equidistribution of cusp forms: reduction to Eisenstein observables
• Integer points and their orthogonal lattices
• An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions
• Equidistribution on the space of rank two vector bundles over the projective line
• Duke's Theorem for subcollections
• On periodicity of geodesic continued fractions
• Limiting Distributions of Translates of Divergent Diagonal Orbits
• Uniformization, Unipotent Flows and the Riemann Hypothesis
• Equidistribution of primitive lattices in ${\mathbb{R}}^n$
• Explicit subconvexity for ${\mathrm{GL}}_2$
• Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits
• Equidistribution of Heegner Points and Ternary Quadratic Forms
• The structure of Lonely Runner spectra
• Equidistribution in S-arithmetic and adelic spaces
• Equidistribution of rational subspaces and their shapes
• Non-vanishing of Class Group L-functions for Number Fields with a Small Regulator
• Asymptotic growth of translation-dilation orbits
• Translates of S-arithmetic orbits and applications
• Recent progress on rigity properties of higher rank diagonalizable actions and applications