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Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori

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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.

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References

  1. Andrianov A.N.: Action of Hecke operator T(p) on theta series. Math. Ann. 247(3), 245–254 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  3. Berry M.V.: Regular and irregular semiclassical wave functions. J. Phys. A 10(12), 2083 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Blomer V.: Uniform bounds for Fourier coefficients of theta-series with arithmetic applications. Acta Arith. 114(1), 1–21 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

  6. Blomer V., Michel P.: Hybrid bounds for automorphic forms on ellipsoids over number fields. J. Inst. Math. Jussieu 12(4), 727–758 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Browning T.D., Dietmann R.: On the representation of integers by quadratic forms. Proc. Lond. Math. Soc. 96(2), 389–416 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  10. Hanke J.: Local densities and explicit bounds for representability by a quadratic form. Duke Math. J. 124(2), 351–388 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hezari, H., Riviere, G.: Quantitative equidistribution properties of toral eigenfunctions. Accepted for publication by J. Spectral Theory, (March 2015)

  12. Iwaniec H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87(2), 385–401 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Kudla S.S., Rallis S.: On the Weil-Siegel formula. J. Reine Angew. Math. 387, 1–68 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Kudla S.S., Rallis S.: On the Weil–Siegel formula. II. The isotropic convergent case. J. Reine Angew. Math. 391, 65–84 (1988)

    MathSciNet  MATH  Google Scholar 

  16. Lester, S., Rudnick, Z.: Small scale equidistribution of eigenfunctions on the torus. Commun. Math. Phys., 350(1), 279–300 (2017)

  17. Marklof J., Rudnick Z.: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2(1), 107–113 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Petersson H.: über die Entwicklungskoeffizienten der automorphen Formen. Acta Math. 58(1), 169–215 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schulze-Pillot R.: On explicit versions of tartakovski’s theorem. Archiv der Mathematik 77(2), 129–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shnirelman A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6), 181–182 (1974)

    MathSciNet  Google Scholar 

  21. Siegel C.L.: über die analytische Theorie der quadratischen Formen. Ann. Math. (2) 36(3), 527–606 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  22. Siegel C.L.: über die analytische Theorie der quadratischen Formen. II. Ann. Math. (2) 37(1), 230–263 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  23. Siegel C.L.: über die analytische Theorie der quadratischen Formen. III. Ann. Math. (2) 38(1), 212–291 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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  1. University of Wisconsin-Madison, Madison, WI, 53706, USA

    Naser T. Sardari

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  1. Naser T. Sardari
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Correspondence to Naser T. Sardari.

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Communicated by J. Marklof

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T. Sardari, N. Quadratic Forms and Semiclassical Eigenfunction Hypothesis for Flat Tori. Commun. Math. Phys. 358, 895–917 (2018). https://doi.org/10.1007/s00220-017-3044-1

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