Duke Mathematical Journal

Arithmetic of double torus quotients and the distribution of periodic torus orbits

Ilya Khayutin

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We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of PGLn over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus.

Using the new invariants we significantly strengthen results toward the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adèlic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates, and homogeneous dynamics.

Article information

Duke Math. J., Volume 168, Number 12 (2019), 2365-2432.

Received: 18 May 2017
Revised: 17 February 2019
First available in Project Euclid: 24 August 2019

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Mathematical Reviews number (MathSciNet)

Primary: 37A17: Homogeneous flows [See also 22Fxx]
Secondary: 11F23: Relations with algebraic geometry and topology

periodic orbit torus entropy


Khayutin, Ilya. Arithmetic of double torus quotients and the distribution of periodic torus orbits. Duke Math. J. 168 (2019), no. 12, 2365--2432. doi:10.1215/00127094-2019-0016. https://projecteuclid.org/euclid.dmj/1566612022

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