Skip to main content
SpringerLink
Log in

Non-vanishing of automorphic L-functions of prime power level

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We prove that at the minimum \(25\%\) of L-values associated to holomorphic newforms of fixed even integral weight and large prime power level do not vanish at the critical point.

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. A method of removing the harmonic weight is described in [14].

References

  1. Akbary, A.: Non-vanishing of weight \(k\) modular \(L\)-functions with large level. J. Ramanujan Math. Soc. 14(1), 37–54 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Balkanova, O., Frolenkov, D.: A uniform asymptotic formula for the second moment of primitive \(L\)-functions on the critical line. Proc. Steklov Inst. Math. 294, 13–46 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bettin, S.: The first moment of twisted Hecke \(L\)-functions with unbounded shifts. Functiones et Approximatio Commentarii Mathematici (2017). arXiv:1605.02440 [math.NT]

  4. Bykovskii, V.A.: A trace formula for the scalar product of Hecke series and its applications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226, 14–36 (1996)

    MATH  Google Scholar 

  5. Bykovskii, V.A., Frolenkov, D.A.: Asymptotic formulas for the second moments of \(L\)-series associated to holomorphic cusp forms on the critical line. arXiv:1608.00555

  6. Bykovskii, V.A., Frolenkov, D.A.: On the second moment of \(L\)-series of holomorphic cusp forms on the critical line. Doklady Math. 92(1), 1–4 (2015)

    Article  MATH  Google Scholar 

  7. Duke, W.: The critical order of vanishing of automorphic \(L\)-functions with large level. Invent. Math. 119(1), 165–174 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ellenberg, J.S.: On the error term in Duke’s estimate for the average special value of \(L\)- functions. Canad. Math. Bull. 48(4), 535–546 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, Boston (2000)

    MATH  Google Scholar 

  10. Ichihara, Y.: The first moment of \(L\) -functions of primitive forms on \(\Gamma _0(p^{\alpha })\) and a basis of old forms. J. Number Theory 131(2), 343–362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Iwaniec, H., Sarnak, P.: The non-vanishing of central values of automorphic L-functions and Landau–Siegel zeros. Israel J. Math. 120, 155–177 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jackson, J., Knightly, A.: Averages of twisted \(L\)-functions. J. Aust. Math. Soc. 99(2), 207–236 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kamiya, Y.: Certain mean values and non-vanishing of automorphic \(L\)-functions with large level. Acta Arith. 93(2), 157–176 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Kowalski, E., Michel, P.: The analytic rank of \(J_0 (q)\) and zeros of automorphic \(L\)-functions. Duke Math. J. 100(3), 503–542 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kuznetsov, N.V.: Convolution of the Fourier coefficients of Eisentein-Maass series. Automorphic functions and number theory. Part I, Zap. Nauchn. Sem. LOMI 129: Nauka, pp. 43–84. Leningrad. Otdel, Leningrad (1983)

  16. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clarke, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  17. Rouymi, D.: Formules de trace en niveau primaire et non annulation de valeurs centrales de fonctions L automorphes. Ph.D. thesis, Université Henri Poincaré (2009)

  18. Rouymi, D.: Formules de trace et non annulation de fonctions \(L\) automorphes au niveau \(p^v\). Acta Arith. 147, 1–32 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rouymi, D.: Mollification et non annulation de fonctions L automorphes en niveau primaire. J. Number Theory 132(1), 79–93 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Royer, E.: Sur les fonctions L de formes modulaires. Ph.D. thesis, Université de Paris-Sud (2001)

  21. VanderKam, J.M.: The rank of quotients of \(J_0 (N)\). Duke Math. J. 97(3), 545–577 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referee for careful reading and Sandro Bettin for extending his result [3] to the case of prime powers.

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, Khabarovsk, Russia

    Olga Balkanova

  2. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Dmitry Frolenkov

Authors
  1. Olga Balkanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Frolenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga Balkanova.

Ethics declarations

Funding

The work of Olga Balkanova (Sects. 3 and 5) is supported by the Russian Science Foundation under Grant \(14-11-00335\) and performed in the Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences. The work of Dmitry Frolenkov (Sects. 4 and 6) is supported by the Russian Science Foundation under Grant \(14-50-00005\) and performed in Steklov Mathematical Institute of Russian Academy of Sciences.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balkanova, O., Frolenkov, D. Non-vanishing of automorphic L-functions of prime power level. Monatsh Math 185, 17–41 (2018). https://doi.org/10.1007/s00605-017-1031-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-017-1031-4

Keywords

Mathematics Subject Classification

Search

Navigation