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 SL3(Z)\SL3(R)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(Z)\SL3(R)/SO3 of volume ≤ V becomes equidistributed as V → ∞.The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

Equidistribution of Heegner points and ternary quadratic forms

“…Similar results have been established by Michel [25], Harcos-Michel [19] (see also [11,Thm. 4.6]) using subconvexity results for Rankin-Selberg L-functions of automorphic forms twisted by ring class characters.…”

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

“…54,(83)]), which depends on k since χ un ξ does. As remarked in the second case following Hypothesis A.2 at the bottom of [ELMV,p. 55], that the bound (7.1) holds for the (varying quadratic) CM extensions E/F follows (by quadratic base change) from the works [BHM], [DFI], and [MiV2].…”

“…Here C ∞ (χ un ξ, 1/2) is the Archimedean part of the analytic conductor of χ un ξ (see e.g. [ELMV,p. 54,(83)]), which depends on k since χ un ξ does.…”

On the convergence of arithmetic orbifolds

“…It simplifies to D F for r = 2; for r 4 it diverges because of the cusp at infinity (C = O F ), and for r = 3 we can estimate it using an elementary argument about the distribution of norms of ideal classes (note that a much more precise statement holds: by a difficult result of M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh [20] the ideal classes become equidistributed in the space of unimodular three-dimensional Euclidean lattices as the discriminant goes to infinity).…”