Skip to main content
SpringerLink
Log in

The twelfth moment of Hecke L-functions in the weight aspect

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We prove an upper bound for the twelfth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of weight k in a dyadic interval \(T \le k \le 2T\) as T tends to infinity. This bound recovers the Weyl-strength subconvex bound \(L(1/2,f) \ll _{\varepsilon } k^{1/3 + \varepsilon }\) and shows that for any \(\delta > 0\), the sub-Weyl subconvex bound \(L(1/2,f) \ll k^{1/3 - \delta }\) holds for all but \(O_{\varepsilon }(T^{12\delta + \varepsilon })\) Hecke cusp forms f of weight at most T. Our result parallels a related result of Jutila for the twelfth moment of Hecke L-functions associated to Hecke–Maaß cusp forms. The proof uses in a crucial way a spectral reciprocity formula of Kuznetsov that relates the fourth moment of L(1/2, f) weighted by a test function to a dual fourth moment weighted by a different test function.

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

Data Availibility

Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

Notes

  1. As stated, [8, Corollary 1.4] claims the stronger bound \(L(1/2,f) \ll k_f^{1/3} (\log k_f)^{13/6}\) in place of (1.8). However, this bound relies upon the erroneously claimed bound \(L(1,{\text {ad}}\, f) \ll (\log k_f)^2\) stated in [22, (2.9)], whereas only the weaker bound \(L(1,{\text {ad}}\, f) \ll (\log k_f)^3\) is known (cf. (1.2)).

References

  1. Jutila, M.: The Twelfth Moment of Central Values of Hecke Series. J. Num. Theo. 108(1), 157–168 (2004). https://doi.org/10.1016/j.jnt.2004.05.012

    Article  MathSciNet  MATH  Google Scholar 

  2. Lau, Y.-K., Jie, W.: A Density Theorem on Automorphic \(L\)-Functions and Some Applications. Trans. Am. Math. Soc. 358(1), 441–472 (2006). https://doi.org/10.1090/S0002-9947-05-03774-8

    Article  MathSciNet  MATH  Google Scholar 

  3. Hoffstein, J., Lockhart, P.: Coefficients of Maass Forms and the Siegel Zero, with an appendix by Dorian Goldfeld. Jeffrey Hoffstein, and Daniel Lieman. Annal. Math. 140(1), 161–181 (1994). https://doi.org/10.2307/2118544

    Article  MATH  Google Scholar 

  4. Peng, Z.: Zeros and Central Values of Automorphic\(L\)-Functions, Ph.D. Thesis, Princeton University, (2001). https://search.proquest.com/docview/276446097/

  5. Jutila, M., Motohashi, Y.: Uniform Bound for Hecke \(L\)-Functions. Acta Math. 195(1), 61–115 (2005). https://doi.org/10.1007/bf02588051

    Article  MathSciNet  MATH  Google Scholar 

  6. Soundararajan, K.: “The Distribution of Values of Zeta and \(L\)-Functions”, preprint, p. 49, (2021). arXiv:2112.03389v1

  7. Balkanova, O., Frolenkov, D.: Moments of \(L\)-Functions and the Liouville-Green Method. J. Eur. Math. Soc. 23(4), 1333–1380 (2021). https://doi.org/10.4171/JEMS/1035

    Article  MathSciNet  MATH  Google Scholar 

  8. Frolenkov, D.: The Cubic Moment of Automorphic \(L\)-Functions in the Weight Aspect. J. Num. Theo. 207, 247–281 (2020). https://doi.org/10.1016/j.jnt.2019.07.009

    Article  MathSciNet  MATH  Google Scholar 

  9. Qi, Z.: “Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-Form \(L\)-Functions”, to appear in International Mathematics Research Notices p. 32, (2023). https://doi.org/10.1093/imrn/rnad090

  10. Ivić, A.: On the Moments of Hecke Series at Central Points. Funct. Approx. Comment. Math. 30, 49–82 (2002). https://doi.org/10.7169/facm/1538186661

    Article  MathSciNet  MATH  Google Scholar 

  11. Khan, R.: The Fifth Moment of Hecke \(L\)-Functions in the Weight Aspect. Math. Proc. Cambridge Philos. Soc. 168(3), 543–566 (2020). https://doi.org/10.1017/S0305004118000944

    Article  MathSciNet  MATH  Google Scholar 

  12. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral Moments of \(L\)-Functions. Proc. London Math. Soc. 91(1), 33–104 (2005). https://doi.org/10.1112/S0024611504015175

    Article  MathSciNet  MATH  Google Scholar 

  13. Heath-Brown, D.R.: The Twelfth Power Moment of the Riemann-Function. Q. J. Math. 29(4), 443–462 (1978). https://doi.org/10.1093/qmath/29.4.443

    Article  MathSciNet  MATH  Google Scholar 

  14. Iwaniec, H.: Fourier Coefficients of Cusp Forms and the Riemann Zeta-Function. Seminaire de Théorie des Nombres de Bordeaux 18, 1–36 (1980). https://www.jstor.org/stable/44165358

  15. Jutila, M.: Lectures on a Method in the Theory of Exponential Sums, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 80, Springer–Verlag, Berlin, (1987). http://www.math.tifr.res.in/~publ/ln/tifr80.pdf

  16. Ivić, A.: The Riemann Zeta-Function. Dover Publications Inc, Mineola, New York (2003)

    MATH  Google Scholar 

  17. Milićević, D., White, D.: Twelfth Moment of Dirichlet \(L\)-Functions to Prime Power Moduli. Ann. Scuola Norm. Sci. 22(4), 1879–1898 (2021). https://doi.org/10.2422/2036-2145.201909_008

    Article  MathSciNet  MATH  Google Scholar 

  18. Nunes, Ramon M.: The Twelfth Moment of Dirichlet \(L\)-Functions with Smooth Moduli. Int. Math. Res. Notices 2021(12), 9180–9202 (2021). https://doi.org/10.1093/imrn/rnz089

    Article  MathSciNet  MATH  Google Scholar 

  19. Conrey, J.B., Snaith, N.C.: Applications of the \(L\)-Functions Ratios Conjectures. Proc. London Math. Soc. 94(3), 594–646 (2007). https://doi.org/10.1112/plms/pdl021

    Article  MathSciNet  MATH  Google Scholar 

  20. Kohnen, W., Zagier, D.: Values of \(L\)-Series of Modular Forms at the Center of the Critical Strip. Invent. Math. 64(2), 175–198 (1981). https://doi.org/10.1007/BF01389166

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim, Henry H.: “Functoriality for the Exterior Square of \({{\rm GL}}_4\) and the Symmetric Fourth of \({{\rm GL}}_2\)’’, with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Henry H. Kim and Peter Sarnak. J. Am. Math. Soc. 16(1), 139–183 (2003). https://doi.org/10.1090/S0894-0347-02-00410-1

    Article  MATH  Google Scholar 

  22. Iwaniec, H., Sarnak, P.: The Non-vanishing of Central Values of Automorphic \(L\)-Functions and Landau-Siegel Zeros. Israel J. Math. 120(1), 155–177 (2000). https://doi.org/10.1007/s11856-000-1275-9

    Article  MathSciNet  MATH  Google Scholar 

  23. Humphries, P., Khan, R.: “\(L^p\)-Norm Bounds for Automorphic Forms via Spectral Reciprocity”, preprint (2022), 57 pages. arXiv:2208.05613v1

  24. Kuznetsov, N.V.: Sums of Kloosterman Sums and the Eighth Power Moment of the Riemann Zeta-Function. In: Number Theory and Related Topics, Tata Institute of Fundamental Research: Studies in Mathematics 12, Oxford University Press, (1989), 57–117

  25. Motohashi, Y.: A Functional Equation for the Spectral Fourth Moment of Modular Hecke \(L\)-Functions. In: Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner mathematische Schriften 360, 19 p. Universität Bonn, Bonn (2003). arXiv:math/0310105v1

  26. Motohashi, Y.: Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Mathematics 127, Cambridge University Press, Cambridge, (1997). https://doi.org/10.1017/CBO9780511983399

  27. Kuznetsov, N.V.: “The Hecke Series at the Center of the Critical Strip”, preprint (1999), 27 pages

  28. Humphries, P., Khan, R.: On the Random Wave Conjecture for Dihedral Maaß Forms. Geom. Funct. Anal. 30(1), 34–125 (2020). https://doi.org/10.1007/s00039-020-00526-4

    Article  MathSciNet  MATH  Google Scholar 

  29. Blomer, V., Li, X., Miller, S.D.: A Spectral Reciprocity Formula and Non-Vanishing for \(L\)-Functions on \({{\rm GL}}(4) \times {{\rm GL}}(2)\). J. Num. Theo. Prime 205, 1–43 (2019). https://doi.org/10.1016/j.jnt.2019.05.011

    Article  MathSciNet  MATH  Google Scholar 

  30. Iwaniec, H., Luo, W., Sarnak, P.: Low Lying Zeros of Families of \(L\)-Functions. Pub. Math. Inst. Hautes Etudes Sci. 91(1), 55–131 (2000). https://doi.org/10.1007/BF02698741

    Article  MathSciNet  MATH  Google Scholar 

  31. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. In: Daniel Zwillinger and Victor Moll (editors). 8 Edition, Academic Press, Burlington (2015) https://doi.org/10.1016/C2010-0-64839-5

  32. Iwaniec, H.: Spectral Methods of Automorphic Forms, Second Edition, Graduate Studies in Mathematics 53, American Mathematical Society, Providence, (2002). https://doi.org/10.1090/gsm/053

  33. Blomer, V., Khan, R., Young, M.: Distribution of Mass of Holomorphic Cusp Forms. Duke Math. J. 162(14), 2609–2644 (2013). https://doi.org/10.1215/00127094-2380967

    Article  MathSciNet  MATH  Google Scholar 

  34. Jutila, M.: The Fourth Moment of Central Values of Hecke Series. In: M. Jutila, T. Metsänkylä, W. de Gruyter (eds). Number Theory: Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri. Berlin, p. 167–177 (2001). https://doi.org/10.1515/9783110870923.167

Download references

Acknowledgements

The authors would like to thank Yoichi Motohashi for useful comments and the anonymous referee for their detailed reading of this paper.

Funding

The first author was supported by the Simons Foundation (award 965056). The second author was supported by the National Science Foundation grants DMS-2001183/DMS-2344044 and DMS-2140604/DMS-2341239 and the Simons Foundation (award 630985).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Peter Humphries

  2. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080, USA

    Rizwanur Khan

Authors
  1. Peter Humphries
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rizwanur Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Humphries.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Humphries, P., Khan, R. The twelfth moment of Hecke L-functions in the weight aspect. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02747-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00208-023-02747-y

Mathematics Subject Classification

Search

Navigation