Manin's and Peyre's conjectures on rational points and adelic mixing
[Conjectures de Manin et de Peyre sur des points rationnels et mélange adélique]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 385-437.

Soit X la compactification merveilleuse d’un groupe semi-simple 𝐆, connexe, de type adjoint, algébrique défini sur un corps de nombre K. Nous démontrons l’asymptotique conjecturée par Manin du nombre de points K-rationnels sur X de hauteur plus petite que T, lorsque T+, et construisons de manière explicite une mesure sur X(𝔸), généralisant celle de Peyre, qui décrit la répartition asymptotique des points rationnels 𝐆(K) sur X(𝔸). Ce travail repose sur la propriété de mélange de L 2 (𝐆(K)𝐆(𝔸)), qui est démontrée avec une estimée de vitesse.

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K. We prove Manin’s conjecture on the asymptotic (as T) of the number of K-rational points of X of height less than T, and give an explicit construction of a measure on X(𝔸), generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points 𝐆(K) on X(𝔸). Our approach is based on the mixing property of L 2 (𝐆(K)𝐆(𝔸)) which we obtain with a rate of convergence.

@article{ASENS_2008_4_41_3_385_0,
     author = {Gorodnik, Alex and Maucourant, Fran\c{c}ois and Oh, Hee},
     title = {Manin's and {Peyre's} conjectures on rational points and adelic mixing},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {385--437},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {3},
     year = {2008},
     doi = {10.24033/asens.2071},
     mrnumber = {2482443},
     zbl = {1161.14015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2071/}
}
TY  - JOUR
AU  - Gorodnik, Alex
AU  - Maucourant, François
AU  - Oh, Hee
TI  - Manin's and Peyre's conjectures on rational points and adelic mixing
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2008
SP  - 385
EP  - 437
VL  - 41
IS  - 3
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2071/
DO  - 10.24033/asens.2071
LA  - en
ID  - ASENS_2008_4_41_3_385_0
ER  - 
%0 Journal Article
%A Gorodnik, Alex
%A Maucourant, François
%A Oh, Hee
%T Manin's and Peyre's conjectures on rational points and adelic mixing
%J Annales scientifiques de l'École Normale Supérieure
%D 2008
%P 385-437
%V 41
%N 3
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2071/
%R 10.24033/asens.2071
%G en
%F ASENS_2008_4_41_3_385_0
Gorodnik, Alex; Maucourant, François; Oh, Hee. Manin's and Peyre's conjectures on rational points and adelic mixing. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 385-437. doi : 10.24033/asens.2071. http://archive.numdam.org/articles/10.24033/asens.2071/

[1] V. V. Batyrev & Y. I. Manin, Sur le nombre des points rationnels de hauteur bornée des variétés algébriques, Math. Ann. 286 (1990), 27-43. | Zbl

[2] V. V. Batyrev & Y. Tschinkel, Height zeta functions of toric varieties, in Algebraic geometry, 5 (Manin's Festschrift), J. Math. Sci. 82, 1996, 3220-3239. | Zbl

[3] V. V. Batyrev & Y. Tschinkel, Manin's conjecture for toric varieties, J. Algebraic Geom. 7 (1998), 15-53. | Zbl

[4] V. V. Batyrev & Y. Tschinkel, Tamagawa numbers of polarized algebraic varieties, in Nombre et répartition de points de hauteur bornée (Paris, 1996), Astérisque 251, 1998, 299-340. | Zbl

[5] I. N. Bernstein, All reductive 𝔭-adic groups are of type I, Funkcional. Anal. i Priložen. 8 (1974), 3-6, English translation: Funct. Anal. Appl. 8 (1974), 91-93. | MR | Zbl

[6] A. Borel, Linear algebraic groups, second éd., Graduate Texts in Math. 126, Springer, 1991. | MR | Zbl

[7] A. Borel & L. Ji, Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser, 2006. | Zbl

[8] A. Borel & J. Tits, Groupes réductifs, Publ. Math. I.H.É.S. 27 (1965), 55-150. | Numdam | Zbl

[9] M. Brion & S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics 231, Birkhäuser, 2005. | Zbl

[10] D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, 1997. | MR | Zbl

[11] M. Burger & P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), 1-11. | Zbl

[12] A. Chambert-Loir & Y. Tschinkel, Fonctions zêta des hauteurs des espaces fibrés, in Rational points on algebraic varieties, Progr. Math. 199, Birkhäuser, 2001, 71-115. | Zbl

[13] A. Chambert-Loir & Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math. 148 (2002), 421-452. | Zbl

[14] L. Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. 15 (1982), 45-115. | Numdam | MR | Zbl

[15] L. Clozel, Démonstration de la conjecture τ, Invent. Math. 151 (2003), 297-328. | MR | Zbl

[16] L. Clozel, H. Oh & E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), 327-351. | Zbl

[17] L. Clozel & E. Ullmo, Équidistribution des points de Hecke, in Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, 193-254. | Zbl

[18] C. De Concini & C. Procesi, Complete symmetric varieties, in Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, Springer, 1983, 1-44. | Zbl

[19] C. De Concini & T. A. Springer, Compactification of symmetric varieties (dedicated to the memory of Claude Chevalley), Transform. Groups 4 (1999), 273-300. | Zbl

[20] J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), 991-1008. | MR | Zbl

[21] J. Dixmier, Les C * -algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars, 1964. | MR | Zbl

[22] W. Duke, Z. Rudnick & P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), 143-179. | Zbl

[23] A. Eskin & C. Mcmullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181-209. | Zbl

[24] A. Eskin & H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26 (2006), 163-167. | Zbl

[25] D. Flath, Decomposition of representations into tensor products 1979, 179-183. | MR | Zbl

[26] J. Franke, Y. I. Manin & Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421-435. | Zbl

[27] W. T. Gan & H. Oh, Equidistribution of integer points on a family of homogeneous varieties: a problem of Linnik, Compositio Math. 136 (2003), 323-352. | Zbl

[28] S. Gelbart & H. Jacquet, A relation between automorphic representations of GL (2) and GL (3), Ann. Sci. École Norm. Sup. 11 (1978), 471-542. | Numdam | Zbl

[29] A. Gorodnik, H. Oh & N. Shah, Integral points on symmetric varieties and Satake boundary, to appear in Amer. J. Math..

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

[31] M. Hindry & J. H. Silverman, Diophantine geometry: an introduction, Graduate Texts in Math. 201, Springer, 2000. | Zbl

[32] R. E. Howe & C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), 72-96. | Zbl

[33] H. Jacquet & R. P. Langlands, Automorphic forms on GL (2), Lecture Notes in Math. 114, Springer, 1970. | Zbl

[34] A. W. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series 36, Princeton University Press, 1986. | MR | Zbl

[35] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics 125, Birkhäuser, 1994. | MR | Zbl

[36] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 17, Springer, 1991. | MR | Zbl

[37] G. A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer, 2004. | MR | Zbl

[38] F. Maucourant, Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices, Duke Math. J. 136 (2007), 357-399. | MR | Zbl

[39] H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France 126 (1998), 355-380. | Numdam | MR | Zbl

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

[41] E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), 101-218. | MR | Zbl

[42] E. Peyre, Points de hauteur bornée, topologie adélique et mesures de Tamagawa, in Les XXII es Journées Arithmétiques (Lille, 2001), J. Théor. Nombres Bordeaux 15 (2003), 319-349. | Numdam | MR | Zbl

[43] V. Platonov & A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press Inc., 1994. | Zbl

[44] J. D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton University Press, 1990. | MR | Zbl

[45] S. Schanuel, On heights in number fields, Bull. Amer. Math. Soc. 70 (1964), 262-263. | MR | Zbl

[46] J. Shalika, R. Takloo-Bighash & Y. Tschinkel, Rational points and automorphic forms, in Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, 733-742. | Zbl

[47] J. Shalika, R. Takloo-Bighash & Y. Tschinkel, Rational points on compactifications of semi-simple groups, J. Amer. Math. Soc. 20 (2007), 1135-1186. | Zbl

[48] J. Shalika, R. Takloo-Bighash & Y. Tschinkel, Rational points on compactifications of semisimple groups, to appear in JAMS. | Zbl

[49] J. Shalika & Y. Tschinkel, Height zeta functions of equivariant compactifications of the Heisenberg group, in Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, 743-771. | Zbl

[50] A. J. Silberger, Introduction to harmonic analysis on reductive p-adic groups, Mathematical Notes 23, Princeton University Press, 1979. | MR | Zbl

[51] J. H. Silverman, The theory of height functions, in Arithmetic geometry (Storrs, Conn., 1984), Springer, 1986, 151-166. | MR | Zbl

[52] M. Strauch & Y. Tschinkel, Height zeta functions of toric bundles over flag varieties, Selecta Math. (N.S.) 5 (1999), 325-396. | Zbl

[53] R. Takloo-Bighash, Bounds for matrix coefficients and arithmetic applications, in Einstein Series and Applications, Progress in Mathematics 258, Birkhäuser, 2008. | MR | Zbl

[54] J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., 1966, 33-62. | MR | Zbl

[55] J. Tits, Reductive groups over local fields 1979, 29-69. | MR | Zbl

[56] Y. Tschinkel, Fujita's program and rational points, in Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud. 12, Springer, 2003, 283-310. | MR | Zbl

[57] Y. Tschinkel, Geometry over nonclosed fields, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 637-651. | MR | Zbl

[58] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer, 1972. | MR | Zbl

[59] A. Weil, Adeles and algebraic groups, Progress in Mathematics 23, Birkhäuser, 1982. | MR | Zbl

[60] D. V. Widder, The Laplace transform, Princeton, 1946. | JFM | Zbl

[61] R. J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics 81, Birkhäuser, 1984. | MR | Zbl

Cité par Sources :