## Duke Mathematical Journal

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

Ilya Khayutin

#### Abstract

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_{n}$ 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

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

Dates
Revised: 17 February 2019
First available in Project Euclid: 24 August 2019

https://projecteuclid.org/euclid.dmj/1566612022

Digital Object Identifier
doi:10.1215/00127094-2019-0016

Mathematical Reviews number (MathSciNet)
MR3999448

Subjects
Secondary: 11F23: Relations with algebraic geometry and topology

Keywords
periodic orbit torus entropy

#### Citation

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

#### References

• [1] M. Auslander and O. Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24.
• [2] M. Bhargava, Higher composition laws, II: On cubic analogues of Gauss composition, Ann. of Math. (2) 159 (2004), no. 2, 865–886.
• [3] A. Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30.
• [4] A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221.
• [5] J. W. S. Cassels, Local Fields, London Math. Soc. Stud. Texts 3, Cambridge Univ. Press, Cambridge, 1986.
• [6] B. Conrad, Finiteness theorems for algebraic groups over function fields, Compos. Math. 148 (2012), no. 2, 555–639.
• [7] I. Dolgachev, Lectures on Invariant Theory, London Math. Soc. Lecture Note Ser. 296, Cambridge Univ. Press, Cambridge, 2003.
• [8] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90.
• [9] M. Einsiedler and S. Kadyrov, Entropy and escape of mass for $\mathrm{SL}_{3}({\mathbb{Z}})\backslash\mathrm{SL}_{3}({\mathbb{R}})$, Israel J. Math. 190 (2012), 253–288.
• [10] M. Einsiedler, S. Kadyrov, and A. Pohl, Escape of mass and entropy for diagonal flows in real rank one situations, Israel J. Math. 210 (2015), no. 1, 245–295.
• [11] M. Einsiedler, A. Katok, and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513–560.
• [12] M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups, Ann. of Math. (2) 181 (2015), no. 3, 993–1031.
• [13] M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J. 148 (2009), no. 1, 119–174.
• [14] M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, Distribution of periodic torus orbits and Duke’s theorem for cubic fields, Ann. of Math. (2) 173 (2011), no. 2, 815–885.
• [15] M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, The distribution of closed geodesics on the modular surface, and Duke’s theorem, Enseign. Math. (2) 58 (2012), no. 3–4, 249–313.
• [16] J. S. Ellenberg, P. Michel, and A. Venkatesh, “Linnik’s ergodic method and the distribution of integer points on spheres” in Automorphic Representations and $L$-Functions, Tata Inst. Fundam. Res. Stud. Math. 22, Tata Inst. Fund. Res., Mumbai, 2013, 119–185.
• [17] P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge Stud. Adv. Math. 101, Cambridge Univ. Press, Cambridge, 2006.
• [18] H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385–401.
• [19] S. Kadyrov, Entropy and escape of mass for Hilbert modular spaces, J. Lie Theory 22 (2012), no. 3, 701–722.
• [20] S. Kadyrov, Positive entropy invariant measures on the space of lattices with escape of mass, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 141–157.
• [21] S. Kadyrov, D. Kleinbock, E. Lindenstrauss, and G. A. Margulis, Singular systems of linear forms and non-escape of mass in the space of lattices, J. Anal. Math. 133 (2017), 253–277.
• [22] S. Kadyrov and A. Pohl, Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 539–563.
• [23] G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316.
• [24] I. Khayutin, Large deviations and effective equidistribution, Int. Math. Res. Not. IMRN 2017, no. 10, 3050–3106.
• [25] D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), no. 4, 453–465.
• [26] D. B. Leep and G. Myerson, Marriage, magic, and solitaire, Amer. Math. Monthly 106 (1999), no. 5, 419–429.
• [27] E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1481–1500.
• [28] Y. V. Linnik, Asymptotic-geometric and ergodic properties of sets of lattice points on a sphere, Mat. Sb. N.S. 43 (1957), no. 85, 257–276.
• [29] Y. V. Linnik, Asymptotic-geometric and ergodic properties of sets of lattice points on a sphere, Amer. Math. Soc. Transl. (2) 13 (1960), 9–27.
• [30] Y. V. Linnik, Ergodic Properties of Algebraic Fields, Ergeb. Math. Grenzgeb. 45, Springer, New York, 1968.
• [31] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
• [32] M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3 (1963/1964), 369–377.
• [33] V. L. Popov and E. B. Vinberg, “Invariant theory” in Algebraic Geometry, IV, Encyclopaedia Math. Sci. 55, Springer, Berlin, 1994, 123–278.
• [34] I. Reiner, Maximal Orders, London Math. Soc. Monogr. 5, Academic Press, New York, 1975.
• [35] J.-P. Serre, Galois Cohomology. Springer Monogr. Math., Springer, Berlin, 2002.
• [36] B. F. Skubenko, The asymptotic distribution of integers on a hyperboloid of one sheet and ergodic theorems, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 721–752.
• [37] W. A. Stein et al., Sage Mathematics Software version 6.8, The Sage Development Team, 2015. http://www.sagemath.org.
• [38] M. M. Wood, Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math. 689 (2014), 169–199.