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Distribution of periodic torus orbits and Duke’s theorem for cubic fields. (English) Zbl 1248.37009

In 1988, Duke showed (generalizing work of Linnik and Skubenko) equidistribution of Heegner points and closed geodesics on the modular surface as their discriminants respectively lengths go to \(\infty\). A number-theoretic interpretation of this is that the integer solutions to \(x^2 + y^2 + z^2 = m\) and to \(y^2 -xz = m\) equidistribute as \(m\to \infty\) with obvious restrictions on \(m\).
The authors show generalizations of these results to \(\mathrm{PGL}_n\) for general \(n \geq 3\), and more specific results for \(n=3\). Moreover, for \(\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})\) they obtain an analogous equidistribution result for compact equivalence classes of maximal flats. They use an adelic framework which has several advantages. E.g., it allows to handle simultaneously many different equidistribution questions, and the definition of the equivalence classes on the maximal flats comes down to the location of the shadow on the infinite place of a periodic orbit of an adelic torus.
The main results are as follows: Let \(F\) be a number field with adele ring \(\mathbb{A}\). To a maximal toral set \(Y\) on \(X = \mathrm{PGL}_n(F)\backslash \mathrm{PGL}_n(\mathbb{A})\) associate a discriminant which can be understood as a measure of the arithmetic complexity. For a sequence of toral sets with discriminant going to \(\infty\), the volume on the sequence is roughly the square root of the discriminant. Moreover, if \(n=3\) and there exists a place such that the local discriminants at this place go to \(\infty\) or the associated tori are split, then any weak* limit of the natural probability measures on the toral sets (pushforwards of the Haar measures) is a homogeneous probability measure on \(X\) and invariant under the image of \(SL_3(\mathbb{A})\).
The proofs are a combination of analytic techniques and ergodic-theoretic techniques, involving subconvexity estimates, measure classification and local harmonic analysis on Lie groups. These results have a number-theoretic interpretation concerning the distribution of orders in totally real cubic fields and of integer points on specific homogeneous varieties.

MSC:

37A17 Homogeneous flows
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
22E40 Discrete subgroups of Lie groups
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11E57 Classical groups
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References:

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