Effective equidistribution of S-integral points on symmetric varieties

Benoist, Yves; Oh, Hee

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1889-1942.

Soit $K$ un corps global de caractéristique différente de 2. Soit $\mathbf{Z}=\mathbf{H}\setminus \mathbf{G}$ 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 $\mathbf{Z}$. Nos résultats sont effectifs quand $K$ est un corps de nombre.

Let $K$ be a global field of characteristic not 2. Let $\mathbf{Z}=\mathbf{H}\setminus \mathbf{G}$ 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 $\mathbf{Z}$. Our results are effective when $K$ is a number field.

Classification : 11G35 11S82 14G05 22E40 37A25 37P30
Keywords: Counting, equidistribution, rational points, mixing, , symmetric spaces, polar decomposition, resolution of singularities.
Mot clés : Comptage, équidistribution, points rationnels, mélange, espaces symétriques, décomposition polaire, résolution des singularités.

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

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

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

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

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

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

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

[8] Clozel, L. Démonstration de la conjecture $\tau$, Invent. Math., Volume 151 (2003), pp. 297-328 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[21] Eskin, A.; McMullen, C. Mixing, counting and equidistribution in Lie groups, Duke Math. Journ., Volume 71 (1993), pp. 181-209 | DOI | MR | Zbl

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

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

[24] Godement, R. Domaines fondamentaux des groupes arithmétiques, Seminaire Bourbaki, 257, 1963 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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