Communications in Mathematical Physics

, Volume 358, Issue 3, pp 895–917 | Cite as

Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

Article
  • 36 Downloads

Abstract

Let Q(X) be any integral primitive positive definite quadratic form in k variables, where \({k\geq4}\), and discriminant D. For any integer n, we give an upper bound on the number of integral solutions of Q(X) = n in terms of n, k, and D. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus \({\mathbb{T}^d}\) for \({d\geq 5}\). This conjecture is motivated by the work of Berry [2,3] on the semiclassical eigenfunction hypothesis.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrianov A.N.: Action of Hecke operator T(p) on theta series. Math. Ann. 247(3), 245–254 (1980)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berry, M.: Semiclassical mechanics of regular and irregular motion. In: Chaotic Behavior of Deterministic Systems (Les Houches, 1981), pp. 171–271. North-Holland, Amsterdam, (1983)Google Scholar
  3. 3.
    Berry M.V.: Regular and irregular semiclassical wave functions. J. Phys. A 10(12), 2083 (1977)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blomer V.: Uniform bounds for Fourier coefficients of theta-series with arithmetic applications. Acta Arith. 114(1), 1–21 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blomer, V.: Ternary quadratic forms, and sums of three squares with restricted variables. In: Anatomy of Integers, vol. 46 of CRM Proc. Lecture Notes, pp. 1–17. Am. Math. Soc., Providence, RI, (2008)Google Scholar
  6. 6.
    Blomer V., Michel P.: Hybrid bounds for automorphic forms on ellipsoids over number fields. J. Inst. Math. Jussieu 12(4), 727–758 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Browning T.D., Dietmann R.: On the representation of integers by quadratic forms. Proc. Lond. Math. Soc. 96(2), 389–416 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cassels, J.W.S.: Rational quadratic forms, vol. 13 of London Mathematical Society Monographs. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, (1978)Google Scholar
  9. 9.
    Colin de Verdière, Y.: Ergodicitéet fonctions propres du laplacien. In: Bony–Sjöstrand–Meyer seminar, 1984–1985, pages Exp. No. 13, 8. École Polytech., Palaiseau, (1985)Google Scholar
  10. 10.
    Hanke J.: Local densities and explicit bounds for representability by a quadratic form. Duke Math. J. 124(2), 351–388 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hezari, H., Riviere, G.: Quantitative equidistribution properties of toral eigenfunctions. Accepted for publication by J. Spectral Theory, (March 2015)Google Scholar
  12. 12.
    Iwaniec H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87(2), 385–401 (1987)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Iwaniec H., Kowalski E.: Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2004)Google Scholar
  14. 14.
    Kudla S.S., Rallis S.: On the Weil-Siegel formula. J. Reine Angew. Math. 387, 1–68 (1988)MathSciNetMATHGoogle Scholar
  15. 15.
    Kudla S.S., Rallis S.: On the Weil–Siegel formula. II. The isotropic convergent case. J. Reine Angew. Math. 391, 65–84 (1988)MathSciNetMATHGoogle Scholar
  16. 16.
    Lester, S., Rudnick, Z.: Small scale equidistribution of eigenfunctions on the torus. Commun. Math. Phys., 350(1), 279–300 (2017)Google Scholar
  17. 17.
    Marklof J., Rudnick Z.: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2(1), 107–113 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Petersson H.: über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58(1), 169–215 (1932)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schulze-Pillot R.: On explicit versions of tartakovski’s theorem. Archiv der Mathematik 77(2), 129–137 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shnirelman A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6), 181–182 (1974)MathSciNetGoogle Scholar
  21. 21.
    Siegel C.L.: über die analytische Theorie der quadratischen Formen. Ann. Math. (2) 36(3), 527–606 (1935)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Siegel C.L.: über die analytische Theorie der quadratischen Formen. II. Ann. Math. (2) 37(1), 230–263 (1936)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Siegel C.L.: über die analytische Theorie der quadratischen Formen. III. Ann. Math. (2) 38(1), 212–291 (1937)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.University of Wisconsin-MadisonMadisonUSA

Personalised recommendations