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

TL;DR

This paper proves that ideal classes of cubic totally real fields become uniformly distributed in a specific modular space, extending Duke's theorem to a higher-dimensional setting using advanced analytic and measure-theoretic techniques.

Contribution

It establishes a cubic analogue of Duke's theorem, demonstrating the equidistribution of certain orbits related to cubic fields in a higher-dimensional modular space.

Findings

Equivalence classes of periodic torus orbits are uniformly distributed.

Maximal compact flats of volume less than V become equidistributed as V increases.

The proof combines subconvexity estimates, measure classification, and harmonic analysis.

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.