Effective equidistribution of S-integral points on symmetric varieties
Annales de l'Institut Fourier, Volume 62 (2012) no. 5, p. 1889-1942

Let K be a global field of characteristic not 2. Let Z=HG be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.

Soit K un corps global de caractéristique différente de 2. Soit Z=HG une variété symétrique définie sur K et S un ensemble fini de places de K. Nous obtenons des résultats de comptage et d’équidistribution pour les points S-entiers de Z. Nos résultats sont effectifs quand K est un corps de nombre.

DOI : https://doi.org/10.5802/aif.2738
Classification:  11G35 11S82 14G05 22E40 37A25 37P30
Keywords: Counting, equidistribution, rational points, mixing,, symmetric spaces, polar decomposition, resolution of singularities.
@article{AIF_2012__62_5_1889_0,
     author = {Benoist, Yves and Oh, Hee},
     title = {Effective equidistribution of S-integral points on symmetric varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {5},
     year = {2012},
     pages = {1889-1942},
     doi = {10.5802/aif.2738},
     mrnumber = {3025156},
     zbl = {pre06130496},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_5_1889_0}
}
Benoist, Yves; Oh, Hee. Effective equidistribution of S-integral points on symmetric varieties. Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1889-1942. doi : 10.5802/aif.2738. http://www.numdam.org/item/AIF_2012__62_5_1889_0/

[1] Atiyah, M. Resolution of singularities and division of distributions, Comm. Pure Appl. Math., Tome 23 (1970), pp. 145-150 | Article | MR 256156 | Zbl 0188.19405

[2] Aubin, F. Nonlinear analysis on manifolds. Monge-Ampère equations, Springer, GM, Tome 252 (1982) | MR 681859 | Zbl 0512.53044

[3] Benoist, Y. Five lectures on lattices, Séminaires et Congrès, Tome 18 (2010)

[4] Benoist, Y.; Oh, H. Polar decomposition for p-adic symmetric spaces, Int. Math. Res. Not., Tome 24 (2007) (article IC 121) | MR 2377008 | Zbl 1137.22008

[5] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local I, Publ. IHES, Tome 41 (1972), pp. 5-252 | Numdam | MR 327923 | Zbl 0254.14017

[6] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local II, Publ. IHES, Tome 60 (1984), pp. 5-184 | Numdam | MR 327923 | Zbl 0254.14017

[7] Chambert-Loir, A.; Tschinkel, Yu. On the distribution of points of bounded height on equivariant compactification of vector groups, Invent. Math., Tome 48 (2002), pp. 421-452 | Article | MR 1906155 | Zbl 1067.11036

[8] Clozel, L. Démonstration de la conjecture τ, Invent. Math., Tome 151 (2003), pp. 297-328 | Article | MR 1953260 | Zbl 1025.11012

[9] Clozel, L.; Oh, H.; Ullmo, E. Hecke operators and equidistribution of Hecke points, Inv. Math., Tome 144 (2003), pp. 327-351 | Article | MR 1827734 | Zbl 1144.11301

[10] Cluckers, R. Classification of semialgebraic p-adic sets up to semi-algebraic bijection, Jour. Reine Angw. Math., Tome 540 (2001), pp. 105-114 | MR 1868600 | Zbl 0984.14018

[11] Dani, S.; Margulis, G. Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces, Proc. Indian. Acad. Sci., Tome 101 (1991), pp. 1-17 | Article | MR 1101994 | Zbl 0731.22008

[12] Dani, S.; Margulis, G. Limit distribution of orbits of unipotent flows and values of quadratic forms, Advances in Soviet Math., Tome 16 (1993), pp. 91-137 | MR 1237827 | Zbl 0814.22003

[13] Delorme, P.; Sécherre, V. An analogue of the Cartan decomposition for p-adic reductive symmetric spaces, Pacific J. Math., Tome 251 (2011), pp. 1-21 | Article | MR 2794612 | Zbl 1220.22003

[14] Denef, J. On the evaluation of certain p-adic integral, Progress in Math., Tome 59 (1985), pp. 25-47 | MR 902824 | Zbl 0597.12021

[15] Denef, J. p-adic semialgebraic sets and cell decomposition, Jour. Reine Angw. Math., Tome 369 (1986), pp. 154-166 | MR 850632 | Zbl 0584.12015

[16] Duke, W. Hyperbolic distribution problems and half integral weight Maass forms, Inven. Math., Tome 92 (1988), pp. 73-90 | Article | MR 931205 | Zbl 0628.10029

[17] Duke, W.; Rudnick, Z.; Sarnak, P. Density of integer points on affine homogeneous varieties, Duke Math. Journ., Tome 71 (1993), pp. 143-179 | Article | MR 1230289 | Zbl 0798.11024

[18] Einsiedler, M.; Lindenstrauss, E. Diagonalizable flows on locally homogeneous spaces and number theory, Int. Cong. Math. (2006), pp. 1731-1759 | MR 2275667 | Zbl 1121.37028

[19] Einsiedler, M.; Margulis, G.; Venkatesh, A. Effective equidistribution of closed orbits of semisimple groups on homogeneous spaces, Invent. Math., Tome 177 (2009), pp. 137-212 | Article | MR 2507639 | Zbl 1176.37003

[20] Einsiedler, M.; Venkatesh, A. Local-Global principles for representations of quadratic forms, Inv. Math., Tome 171 (2008), pp. 257-279 | Article | MR 2367020 | Zbl 1247.11048 | Zbl pre05248087

[21] Eskin, A.; Mcmullen, C. Mixing, counting and equidistribution in Lie groups, Duke Math. Journ., Tome 71 (1993), pp. 181-209 | Article | MR 1230290 | Zbl 0798.11025

[22] Eskin, A.; Mozes, S.; Shah, N. Unipotent flows and counting lattice points on homogeneous varieties, Annals of Math., Tome 143 (1996), pp. 149-159 | Article | MR 1381987 | Zbl 0852.11054

[23] Eskin, A.; Oh, H. Representations of integers by an invariant polynomial and unipotent flows, Duke Math. Journ., Tome 135 (2006), pp. 481-506 | Article | MR 2272974 | Zbl 1138.11011

[24] Godement, R. Domaines fondamentaux des groupes arithmétiques, Seminaire Bourbaki, Tome 257 (1963) | Numdam | MR 191899 | Zbl 0136.30101

[25] Gorodnik, A.; Maucourant, F.; Oh, H. Manin’s and Peyre’s conjectures on rational points and adelic mixing, Ann. Sci. Ecole Norm. Sup., Tome 41 (2008), pp. 47-97 | Numdam | MR 2482443 | Zbl 1161.14015

[26] Gorodnik, A.; Nevo, A. The ergodic theory of lattice subgroups, Annals Math. Studies, Tome 172 (2009) | MR 2573139 | Zbl 1186.37004

[27] Gorodnik, A.; Oh, H. Rational points on homogeneous varieties and Equidistribution for Adelic periods, GAFA, Tome 21 (2011), pp. 319-392 | Article | MR 2795511 | Zbl 1317.11069 | Zbl pre05902967

[28] Gorodnik, A.; Oh, H.; Shah, N. Integral points on symmetric varieties and Satake compactifications, Amer. J. Math., Tome 131 (2009), pp. 1-57 | Article | MR 2488484 | Zbl 1231.14041

[29] Gorodnik, A.; Weiss, B. Distribution of lattice orbits on homogeneous varieties, GAFA, Tome 17 (2007), pp. 58-115 | Article | MR 2306653 | Zbl 1112.37001

[30] Guilloux, A. Existence et équidistribution des matrices de dénominateur n dans les groupes unitaires et orthogonaux, Ann. Inst. Fourier, Tome 58 (2008), pp. 1185-1212 | Article | Numdam | MR 2427958 | Zbl 1149.11017

[31] Helminck, A.; Wang, S. On rationality properties of involutions of reductive groups, Adv. in Math., Tome 99 (1993), pp. 26-97 | Article | MR 1215304 | Zbl 0788.22022

[32] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., Tome 79 (1964), pp. 109-326 | Article | MR 199184 | Zbl 0122.38603

[33] Iwaniec, H. Fourier coefficients of modular forms of half integral weight, Inv. Math., Tome 87 (1987), pp. 385-401 | Article | MR 870736 | Zbl 0606.10017

[34] Jeanquartier, P. Integration sur les fibres d’une fonction analytique (Travaux en Cours) Tome 34 (1999), pp. 1-39

[35] Ledrappier, F. Distribution des orbites des réseaux sur le plan réel, CRAS, Tome 329 (1999), p. 61-54 | MR 1703338 | Zbl 0928.22012

[36] Linnik, (Y. V.) Additive problems and eigenvalues of the modular operators (Proc. Int. Cong. Math. Stockholm) (1962), pp. 270-284 | Zbl 0116.03604

[37] Macintyre, A. On definable subsets of p-adic fields, J. Symb. Logic, Tome 41 (1976), pp. 605-610 | Article | MR 485335 | Zbl 0362.02046

[38] Margulis, G. Discrete subgroups of semisimple Lie groups, Springer Ergebnisse (1991) | MR 1090825 | Zbl 0732.22008

[39] Margulis, G. On some aspects of the theory of Anosov systems, Springer (2004) | MR 2035655

[40] Maucourant, F. Homogeneous asymptotic limits of Haar measures of semisimple linear groups, Duke Math. Jour., Tome 136 (2007), pp. 357-399 | Article | MR 2286635 | Zbl 1117.22006

[41] Michel, P.; Venkatesh, A. Equidistribution, L -functions and ergodic theory: on some problems of Yu. Linnik, Proc. Int. Cong. Math. (2006) | MR 2275604 | Zbl 1157.11019

[42] Mozes, S.; Shah, N. On the space of ergodic invariant measures of unipotent flows, ETDS, Tome 15 (1995), pp. 149-159 | MR 1314973 | Zbl 0818.58028

[43] Oh, H. Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., Tome 113 (2002), pp. 133-192 | Article | MR 1905394 | Zbl 1011.22007

[44] Oh, H. Hardy-Littlewood system and representations of integers by invariant polynomials, GAFA, Tome 14 (2004), pp. 791-809 | MR 2084980 | Zbl 1196.11057

[45] Platonov, V.; Rapinchuk, A. Algebraic groups and number theory, Ac. Press (1994) | MR 1278263 | Zbl 0841.20046

[46] Prasad, G. Strong approximation for semisimple groups over function fields, Annals of Math., Tome 105 (1977), pp. 553-572 | Article | MR 444571 | Zbl 0348.22006

[47] Ratner, M. On Raghunathan’s measure conjecture, Annals of Math., Tome 134 (1991), pp. 545-607 | Article | MR 1135878 | Zbl 0763.28012

[48] Sarnak, P. Diophantine problems and linear groups, Proc. Int. Cong. Math. (1990), pp. 459-471 | MR 1159234 | Zbl 0743.11018

[49] Shah, N. Limit distribution of expanding translates of certain orbits on homogeneous spaces on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., Tome 106 (1996), pp. 105-125 | Article | MR 1403756 | Zbl 0864.22004