Manin's and Peyre's conjectures on rational points and adelic mixing

Gorodnik, Alex; Maucourant, François; Oh, Hee

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 $T\to +\infty$, et construisons de manière explicite une mesure sur $X\left(𝔸\right)$, généralisant celle de Peyre, qui décrit la répartition asymptotique des points rationnels $𝐆\left(K\right)$ sur $X\left(𝔸\right)$. Ce travail repose sur la propriété de mélange de $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$, 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 $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ which we obtain with a rate of convergence.

[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 $\tau$, 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 $\mathrm{GL}\left(2\right)$ and $\mathrm{GL}\left(3\right)$, 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 $\mathrm{GL}\left(2\right)$, 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