Skip to main content
SpringerLink
Log in

Asymptotic trace formula for the Hecke operators

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Given integers m, n and k, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the mth and nth Fourier coefficients of an orthonormal basis of \(S_k\left( N\right) ^*\) (the weight k newforms with fixed square-free level N) provided that \(|4 \pi \sqrt{mn}- k|=o\left( k^{\frac{1}{3}}\right) \). Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator \(\mathcal {T}_n^*\) on \(S_k\left( N\right) ^*\) averaged over k in a short interval. By bounding the second moment of the trace of \(\mathcal {T}_{n}\) over a larger interval, we show that the trace of \(\mathcal {T}_n\) is unusually large in the range \(|4 \pi \sqrt{n}- k| = o\left( n^{\frac{1}{6}}\right) \). As an application, for any fixed prime p coprime to N, we show that there exists a sequence \(\{k_n\}\) of weights such that the error term of Weyl’s law for \(\mathcal {T}_p\) is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd et al. (J Eur Math Soc 1(1):51–85, 1999) [Theorem 1.4] with an improved exponent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We use the divisor function parameterized by t, defined by \(\sigma _t\left( n\right) =\sum _{d|n} d^t\). When \(t=0\), we drop 0, and use \(\sigma \) instead of \(\sigma _0\).

  2. The geodesic flow in this case is chaotic, but Sarnak explains that one expects to see Poisson behavior due to the high multiplicity of the geodesic length spectrum.

  3. Here and elsewhere we write \(A \ll _\tau B \) when \(|A|\le C(\tau )B\) holds with some constant \(C(\tau )\) depending only on \(\tau \).

References

  1. Atkin, A.O.L., Lehner, J.: Hecke operators on \(\Gamma _{0}(m)\). Math. Ann. 185, 134–160 (1970)

    Article  MathSciNet  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. In: For sale by the Superintendent of Documents, US Government Printing Office, Washington, D.C. (1964)

  3. Bérard, P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)

    Article  MathSciNet  Google Scholar 

  4. Berry, M.V.: Semiclassical theory of spectral rigidity. Proc. R. Soc. Lond. Ser. A 400(1819), 229–251 (1985)

    Article  MathSciNet  Google Scholar 

  5. Berry, M.V.: Fluctuations in numbers of energy levels. In: Stochastic processes in classical and quantum systems (Ascona, 1985), volume 262 of Lecture Notes in Phys., pp. 47–53. Springer, Berlin (1986)

  6. Conrey, J.B., Duke, W., Farmer, D.W.: The distribution of the eigenvalues of Hecke operators. Acta Arith. 78(4), 405–409 (1997)

    Article  MathSciNet  Google Scholar 

  7. Cohen, H.: Sums involving the values at negative integers of \(L\)-functions of quadratic characters. Math. Ann. 217(3), 271–285 (1975)

    Article  MathSciNet  Google Scholar 

  8. Deligne, P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)

    Article  MathSciNet  Google Scholar 

  9. Erdös, P., Turán, P.: On a problem in the theory of uniform distribution. I. Nederl. Akad. Wetensch. Proc. 51, 1146–1154 (1948)

    MathSciNet  MATH  Google Scholar 

  10. Gamburd, A., Jakobson, D., Sarnak, P.: Spectra of elements in the group ring of \({\rm SU}(2)\). J. Eur. Math. Soc. 1(1), 51–85 (1999)

    Article  MathSciNet  Google Scholar 

  11. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, eighth edition, 2015. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition [MR2360010] (2015)

  12. Hamer, C.: A formula for the traces of the Hecke operators on certain spaces of newforms. Arch. Math. (Basel) 70(3), 204–210 (1998)

    Article  MathSciNet  Google Scholar 

  13. Heath-Brown, D.R.: A new form of the circle method, and its application to quadratic forms. J. Reine Angew. Math. 481, 149–206 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Hejhal, D.A.: The Selberg trace formula for \({\rm PSL}(2,R)\). Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer, Berlin, New York (1976)

  15. Hardy, G.H., Landau, E.: The lattice points of a circle. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 105(731), 244–258 (1924)

    MATH  Google Scholar 

  16. Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero. Ann. Math (2) 140(1), 161–181 (1994). (With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman)

    Article  MathSciNet  Google Scholar 

  17. Hörmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)

    Article  MathSciNet  Google Scholar 

  18. Iwaniec, H., Luo, W., Sarnak, P.: Low lying zeros of families of \(L\)-functions. Inst. Hautes Études Sci. Publ. Math. 91, 55–131 (2001)

    Article  MathSciNet  Google Scholar 

  19. Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of \(L\)-functions. In: Number Special Volume, Part II, pp 705–741. GAFA 2000 (Tel Aviv, 1999) (2000)

  20. Iwaniec, H.: Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1997)

  21. Knightly, A., Li, C.: A relative trace formula proof of the Petersson trace formula. Acta Arith. 122, 297–313 (2006)

    Article  MathSciNet  Google Scholar 

  22. Kloosterman, H.D.: On the representation of numbers in the form \(ax^2+by^2+cz^2+dt^2\). Acta Math. 49(3–4), 407–464 (1927)

    MathSciNet  MATH  Google Scholar 

  23. Kohnen, W., Ono, K.: Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication. Invent. Math. 135(2), 387–398 (1999)

    Article  MathSciNet  Google Scholar 

  24. Krasikov, I.: On the bessel function \(j_\nu (x)\) in the transition region. LMS J. Comput. Math. 17(1), 273–281 (2014)

    Article  MathSciNet  Google Scholar 

  25. Luo, W., Sarnak, P.: Mass equidistribution for Hecke eigenforms. volume 56, pp. 874–891. (2003) (Dedicated to the memory of Jürgen K. Moser)

  26. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.18 of 2018-03-27. In: Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds) (2019)

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

    Article  MathSciNet  Google Scholar 

  28. Petridis, Y., Toth, J.A.: The remainder in Weyl’s law for random two-dimensional flat tori. Geom. Funct. Anal. 12(4), 756–775 (2002)

    Article  MathSciNet  Google Scholar 

  29. Ram Murty, M., Sinha, K.: Effective equidistribution of eigenvalues of Hecke operators. J. Number Theory 129(3), 681–714 (2009)

    Article  MathSciNet  Google Scholar 

  30. Rudnick, Z.: A central limit theorem for the spectrum of the modular domain. Ann. Henri Poincaré 6(5), 863–883 (2005)

    Article  MathSciNet  Google Scholar 

  31. Sarnak, P.: Some Applications of Modular Forms. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  32. Sarnak, P.: Arithmetic quantum chaos. In: The Schur lectures (1992) (Tel Aviv), volume 8 of Israel Math. Conf. Proc., pp. 183–236. Bar-Ilan Univ., Ramat Gan (1995)

  33. Sarnak, P.: Values at integers of binary quadratic forms. In: Harmonic analysis and number theory (Montreal, PQ, 1996), volume 21 of CMS Conf. Proc., pp. 181–203. Am. Math. Soc., Providence (1997)

  34. Sarnak, P.: Letter to Z. Rudnick on multiplicities of eigenvalues for the modular surface (2002). https://publications.ias.edu/sarnak/paper/500

  35. Sardari, N.T.: Optimal strong approximation for quadratic forms. Duke Math. J. 168(10), 1887–1927 (2019)

    Article  MathSciNet  Google Scholar 

  36. Serre, J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_p\). J. Am. Math. Soc. 10(1), 75–102 (1997)

    Article  Google Scholar 

  37. Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. (3) 31(1), 79–98 (1975)

    Article  MathSciNet  Google Scholar 

  38. Sarnak, P., Shin, S.W., Templier, N.: Families of \(L\)-functions and their symmetry. In: Families of automorphic forms and the trace formula, Simons Symp., pp. 531–578. Springer, [Cham] (2016)

  39. Wey, H.: Über die asymptotische Verteilung der Eigenwerte. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1911, 110–117 (1911)

    MATH  Google Scholar 

Download references

Acknowledgements

J.J. thanks S.M. and Department of Mathematics of UW-Madison for invitation and support. J.J. also thanks Sug Woo Shin, Peter Jaehyun Cho, and Matthew Young for many helpful comments. J.J. was supported by NSF grant DMS-1900993, and by Sloan Research Fellowship. S.M. was supported by NSF grant DMS-1902173. N.T.S. was supported by NSF grant DMS-2015305 and is grateful to Max Planck Institute for Mathematics in Bonn and Institute For Advanced Study for their hospitalities and financial supports. N.T.S. thanks his Ph.D. advisor Peter Sarnak for several insightful and inspiring conversations regarding the error term of the Weyl law while he was a graduate student at Princeton University. N.T.S. was supported Grant no. 1902185 and S.M. was supported Grant no. 1501230

Author information

Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Junehyuk Jung

  2. The Institute For Advanced Study, Princeton, NJ, 08540, USA

    Naser Talebizadeh Sardari

Authors
  1. Junehyuk Jung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naser Talebizadeh Sardari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Naser Talebizadeh Sardari.

Additional information

Communicated by Kannan Soundararajan.

With an appendix by Simon Marshall.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jung, J., Talebizadeh Sardari, N. Asymptotic trace formula for the Hecke operators. Math. Ann. 378, 513–557 (2020). https://doi.org/10.1007/s00208-020-02054-w

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-020-02054-w

Mathematics Subject Classification

Search

Navigation