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Inventiones mathematicae

, Volume 171, Issue 2, pp 257–279 | Cite as

Local-global principles for representations of quadratic forms

Article

Abstract

We prove a local-global principle for the problem of representations of quadratic forms by quadratic forms over ℤ, in codimension ≥5. The proof uses the ergodic theory of p-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.

Keywords

Quadratic Form Spin Group Quadratic Space Open Compact Subgroup Hasse Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Artin, M.: Geometric Algebra. Interscience Publishers, Inc., New York London (1957)MATHGoogle Scholar
  2. 2.
    Baez, J.: The octonions. Bull. Am. Math. Soc. New. Ser. 39(2), 145–205 (2002) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burger, M., Sarnak, P.: Ramanujan duals. II. Invent. Math. 106(1), 1–11 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cassels, J.W.S.: Rational Quadratic Forms. Academic Press, London, New York (1978)MATHGoogle Scholar
  5. 5.
    Clozel, L.: Démonstration de la conjecture τ. Invent. Math. 151, 297–328 (2003)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Clozel, L., Ullmo, E.: Equidistribution des points de Hecke. In: Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 193–254. Johns Hopkins Univ. Press, Baltimore, MD (2004)Google Scholar
  7. 7.
    Dani, S., Margulis, G.: Limit distributions of orbits of unipotent flows and values of quadratic forms. Adv. Sov. Math. 16, 91–137 (1993)MathSciNetGoogle Scholar
  8. 8.
    Duke, W., Schulze-Pillot, R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eichler, M.: Quadratische Formen und orthogonale Gruppen. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. LXIII. Springer, Berlin Göttingen Heidelberg (1952)Google Scholar
  10. 10.
    Einsiedler, M.: Ratner’s theorem on SL(2,ℝ)-invariant measures. Jahresber. Deutsch. Math.-Verein. 108(3), 143–164 (2006)MATHMathSciNetGoogle Scholar
  11. 11.
    Einsiedler, M., Lindenstrauss, E.: Diagonalizable flows on locally homogeneous spaces and number theory. In: Proceedings of the ICM, vol. 11, pp. 1731–1759. Eur. Math. Society, Zurich (2006)Google Scholar
  12. 12.
    Einsiedler, M., Lindenstrauss, E., Michel, P., Venkatesh, A.: The distribution of periodic torus orbits on homogeneous spaces. math.DS/0607815Google Scholar
  13. 13.
    Einsiedler, M., Lindenstrauss, E., Michel, P., Venkatesh, A.: Distribution of periodic torus orbits and Duke’s theorem for cubic fields. arxiv:0708.1113Google Scholar
  14. 14.
    Einsiedler, M., Margulis, G., Venkatesh, A.: Effective equidistribution of closed orbits of semisimple groups on homogeneous spaces. arxiv:0708.4040Google Scholar
  15. 15.
    Ellenberg, J., Venkatesh, A.: Local-global principles for representations of quadratic forms. Arxived version, math.NT/0604232Google Scholar
  16. 16.
    Eskin, A., Mozes, S., Shah, N.: Unipotent flows and counting lattice points on homogeneous varieties. Ann. Math. (2) 143, 253–299 (1996)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Eskin, A., Oh, H.: Representations of integers by an invariant polynomial and unipotent flows. Duke Math. J. 135(3), 481–506 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gan, W., Gross, B.: Commutative subgroups of certain non-associative rings. Math. Ann. 314, 265–283 (1999)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. Éc. Norm. Supér., IV. Sér. 11(4), 471–542 (1978)MATHMathSciNetGoogle Scholar
  20. 20.
    Gille, P., Moret-Bailly, L.: Actions algèbriques de groupes arithmètiques. PreprintGoogle Scholar
  21. 21.
    Glasner, Y., Weiss, B.: Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7, 917–935 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Guralnick, R., Saxl, J.: Generation of finite almost simple groups by conjugates. J. Algebra 268(2), 519–571 (2003)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hsia, J.: Representations by spinor genera. Pac. J. Math. 63, 147–152 (1976)MATHMathSciNetGoogle Scholar
  24. 24.
    Hsia, J., Kitaoka, Y., Kneser, M.: Representations of positive definite quadratic forms. J. Reine Angew. Math. 301, 132–141 (1978)MATHMathSciNetGoogle Scholar
  25. 25.
    Jacobson, N.: Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo, II. Ser. 7, 55–80 (1958)MATHGoogle Scholar
  26. 26.
    Jacquet, H., Langlands, R.P.: Automorphic Forms on GL(2). Lect. Notes Math., vol. 114. Springer, Berlin New York (1970)MATHGoogle Scholar
  27. 27.
    Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge, UK (1999)MATHGoogle Scholar
  28. 28.
    Kneser, M.: Composition of binary quadratic forms. J. Number Theory 15, 406–413 (1982)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kneser, M.: Darstellungsmasse indefiniter quadratischer Formen. Math. Z. 77, 188–194 (1961)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Linnik, Y.: Ergodic properties of algebraic number fields. Ergeb. Math. Grenzgeb., vol. 45. Springer, New York (1968)Google Scholar
  31. 31.
    Lubotzky, A.: Discrete groups, expanding graphs and invariant measures. Prog. Math., vol. 125. Birkhäuser, Basel (1994)MATHGoogle Scholar
  32. 32.
    Margulis, G., Tomanov, G.: Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116, 347–392 (1994)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Mozes, S., Shah, N.: On the space of ergodic invariant measures of unipotent flows. Ergodic Theory Dyn. Syst. 15 149–159 (1995)Google Scholar
  34. 34.
    Oh, H.: Hardy–Littlewood system and representations of integers by an invariant polynomial. Geom. Funct. Anal. 14, 791–809 (2004)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    O’Meara, T.: Introduction to quadratic forms. In: Classics in Mathematics. Springer, Berlin (2000)Google Scholar
  36. 36.
    Platonov, V.P.: The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups. Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1211–1219 (1969) (Russian)MathSciNetGoogle Scholar
  37. 37.
    Prasad, G.: Elementary proof of a theorem of Bruhat–Tits–Rousseau and of a theorem of Tits. Bull. Soc. Math. Fr. 110, 197–202 (1982)MATHGoogle Scholar
  38. 38.
    Ratner, M.: Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups. Duke. Math. J. 77, 275–382 (1995)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Shimura, G.: Quadratic diophantine equations, the class number, and the mass formula. Bull. Am. Math. Soc., New Ser. 43, 285–304 (2006)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Shimura, G.: Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups. Am. Math. Soc., Providence, RI (2004)MATHGoogle Scholar
  41. 41.
    Schulze-Pillot, R.: Representation by integral quadratic forms – a survey. In: Algebraic and Arithmetic Theory of Quadratic Forms. Contemp. Math., vol. 344, pp. 303–321. Am. Math. Soc., Providence, RI (2004)Google Scholar
  42. 42.
    Tits, J.: Algebraic and abstract simple groups. Ann. Math. (2) 80, 313–329 (1964)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Weil, A.: Sur la théorie des formes quadratiques. In: Colloq. Théorie des Groupes Algébriques (Bruxelles 1962), pp. 9–22. Librairie Universitaire, Louvain, Gauthier-Villars, ParisGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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