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

Alex Gorodnik; François Maucourant; Hee Oh

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 3, page 385-437
  • ISSN: 0012-9593

Abstract

top
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.

How to cite

top

Gorodnik, Alex, Maucourant, François, and Oh, Hee. "Manin’s and Peyre’s conjectures on rational points and adelic mixing." Annales scientifiques de l'École Normale Supérieure 41.3 (2008): 385-437. <http://eudml.org/doc/272104>.

@article{Gorodnik2008,
abstract = {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\rightarrow \infty $) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X(\mathbb \{A\})$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $\mathbf \{G\}(K)$ on $X(\mathbb \{A\})$. Our approach is based on the mixing property of $\{L\}^2(\mathbf \{G\}(K)\backslash \mathbf \{G\}(\mathbb \{A\}))$ which we obtain with a rate of convergence.},
author = {Gorodnik, Alex, Maucourant, François, Oh, Hee},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Rational points; Manin conjecture; algebraic groups},
language = {eng},
number = {3},
pages = {385-437},
publisher = {Société mathématique de France},
title = {Manin’s and Peyre’s conjectures on rational points and adelic mixing},
url = {http://eudml.org/doc/272104},
volume = {41},
year = {2008},
}

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
PB - Société mathématique de France
VL - 41
IS - 3
SP - 385
EP - 437
AB - 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\rightarrow \infty $) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X(\mathbb {A})$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $\mathbf {G}(K)$ on $X(\mathbb {A})$. Our approach is based on the mixing property of ${L}^2(\mathbf {G}(K)\backslash \mathbf {G}(\mathbb {A}))$ which we obtain with a rate of convergence.
LA - eng
KW - Rational points; Manin conjecture; algebraic groups
UR - http://eudml.org/doc/272104
ER -

References

top
  1. [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. Zbl0679.14008
  2. [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. Zbl0915.14013
  3. [3] V. V. Batyrev & Y. Tschinkel, Manin’s conjecture for toric varieties, J. Algebraic Geom.7 (1998), 15–53. Zbl0946.14009
  4. [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. Zbl0926.11045
  5. [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. Zbl0298.43013MR348045
  6. [6] A. Borel, Linear algebraic groups, second éd., Graduate Texts in Math. 126, Springer, 1991. Zbl0726.20030MR1102012
  7. [7] A. Borel & L. Ji, Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser, 2006. Zbl1100.22001
  8. [8] A. Borel & J. Tits, Groupes réductifs, Publ. Math. I.H.É.S. 27 (1965), 55–150. Zbl0145.17402
  9. [9] M. Brion & S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics 231, Birkhäuser, 2005. Zbl1072.14066
  10. [10] D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, 1997. Zbl0868.11022MR1431508
  11. [11] M. Burger & P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), 1–11. Zbl0774.11021
  12. [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. Zbl1077.14524
  13. [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. Zbl1067.11036
  14. [14] L. Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup.15 (1982), 45–115. Zbl0516.22010MR672475
  15. [15] L. Clozel, Démonstration de la conjecture τ , Invent. Math.151 (2003), 297–328. Zbl1025.11012MR1953260
  16. [16] L. Clozel, H. Oh & E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math.144 (2001), 327–351. Zbl1144.11301
  17. [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. Zbl1068.11042
  18. [18] C. De Concini & C. Procesi, Complete symmetric varieties, in Invariant theory (Montecatini, 1982), Lecture Notes in Math. 996, Springer, 1983, 1–44. Zbl0581.14041
  19. [19] C. De Concini & T. A. Springer, Compactification of symmetric varieties (dedicated to the memory of Claude Chevalley), Transform. Groups4 (1999), 273–300. Zbl0966.14035
  20. [20] J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math.109 (1987), 991–1008. Zbl0659.14017MR919001
  21. [21] J. Dixmier, Les C * -algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars, 1964. Zbl0174.18601MR171173
  22. [22] W. Duke, Z. Rudnick & P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J.71 (1993), 143–179. Zbl0798.11024
  23. [23] A. Eskin & C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J.71 (1993), 181–209. Zbl0798.11025
  24. [24] A. Eskin & H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26 (2006), 163–167. Zbl1092.11023
  25. [25] D. Flath, Decomposition of representations into tensor products, in Automorphic forms, representations and L -functions, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., 1979, 179–183. Zbl0414.22019MR546596
  26. [26] J. Franke, Y. I. Manin & Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math.95 (1989), 421–435. Zbl0674.14012
  27. [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. Zbl1018.22009
  28. [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. Zbl0406.10022
  29. [29] A. Gorodnik, H. Oh & N. Shah, Integral points on symmetric varieties and Satake boundary, to appear in Amer. J. Math.. Zbl1231.14041
  30. [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. Zbl1149.11017MR2427958
  31. [31] M. Hindry & J. H. Silverman, Diophantine geometry: an introduction, Graduate Texts in Math. 201, Springer, 2000. Zbl0948.11023
  32. [32] R. E. Howe & C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal.32 (1979), 72–96. Zbl0404.22015
  33. [33] H. Jacquet & R. P. Langlands, Automorphic forms on GL ( 2 ) , Lecture Notes in Math. 114, Springer, 1970. Zbl0236.12010
  34. [34] A. W. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Zbl0604.22001MR855239
  35. [35] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics 125, Birkhäuser, 1994. Zbl0826.22012MR1308046
  36. [36] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 17, Springer, 1991. Zbl0732.22008MR1090825
  37. [37] G. A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer, 2004. Zbl1140.37010MR2035655
  38. [38] F. Maucourant, Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices, Duke Math. J.136 (2007), 357–399. Zbl1117.22006MR2286635
  39. [39] H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Soc. Math. France126 (1998), 355–380. Zbl0917.22008MR1682805
  40. [40] H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J.113 (2002), 133–192. Zbl1011.22007MR1905394
  41. [41] E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J.79 (1995), 101–218. Zbl0901.14025MR1340296
  42. [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. Zbl1057.14031MR2019019
  43. [43] V. Platonov & A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press Inc., 1994. Zbl0841.20046
  44. [44] J. D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton University Press, 1990. Zbl0724.11031MR1081540
  45. [45] S. Schanuel, On heights in number fields, Bull. Amer. Math. Soc.70 (1964), 262–263. Zbl0122.04202MR162787
  46. [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. Zbl1096.11022
  47. [47] J. Shalika, R. Takloo-Bighash & Y. Tschinkel, Rational points on compactifications of semi-simple groups, J. Amer. Math. Soc.20 (2007), 1135–1186. Zbl1122.14019
  48. [48] J. Shalika, R. Takloo-Bighash & Y. Tschinkel, Rational points on compactifications of semisimple groups, to appear in JAMS. Zbl1122.14019
  49. [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. Zbl1182.14013
  50. [50] A. J. Silberger, Introduction to harmonic analysis on reductive p -adic groups, Mathematical Notes 23, Princeton University Press, 1979. Zbl0458.22006MR544991
  51. [51] J. H. Silverman, The theory of height functions, in Arithmetic geometry (Storrs, Conn., 1984), Springer, 1986, 151–166. Zbl0604.14022MR861975
  52. [52] M. Strauch & Y. Tschinkel, Height zeta functions of toric bundles over flag varieties, Selecta Math. (N.S.) 5 (1999), 325–396. Zbl1160.14302
  53. [53] R. Takloo-Bighash, Bounds for matrix coefficients and arithmetic applications, in Einstein Series and Applications, Progress in Mathematics 258, Birkhäuser, 2008. Zbl1142.22302MR2402689
  54. [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. Zbl0238.20052MR224710
  55. [55] J. Tits, Reductive groups over local fields, in Automorphic forms, representations and L -functions, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., 1979, 29–69. Zbl0415.20035MR546588
  56. [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. Zbl1112.14021MR2011749
  57. [57] Y. Tschinkel, Geometry over nonclosed fields, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 637–651. Zbl1105.14024MR2275615
  58. [58] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer, 1972. Zbl0265.22020MR498999
  59. [59] A. Weil, Adeles and algebraic groups, Progress in Mathematics 23, Birkhäuser, 1982. Zbl0493.14028MR670072
  60. [60] D. V. Widder, The Laplace transform, Princeton, 1946. Zbl0063.08245JFM67.0384.01
  61. [61] R. J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics 81, Birkhäuser, 1984. Zbl0571.58015MR776417

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.