Non-vanishing of derivatives of L-functions of Hilbert modular forms in the critical strip

Abstract

In this paper, we show that, on average, the derivatives of L-functions of cuspidal Hilbert modular forms with sufficiently large parallel weight k do not vanish on the line segments \(\mathfrak {I}(s)=t_{0}\), \(\mathfrak {R}(s)\in (\frac{k-1}{2},\frac{k}{2}-\epsilon )\cup (\frac{k}{2}+\epsilon ,\frac{k+1}{2})\). This is analogous to the case of classical modular forms.

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References

  1. 1.

    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications Inc, New York (1964)

    MATH  Google Scholar 

  2. 2.

    Hamieh, A., Raji, W.: Non-vanishing of L-functions of Hilbert modular forms inside the critical strip. Acta Arith. 185, 333–346 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kohnen, W.: Nonvanishing of Hecke \(L\)-functions associated to cusp forms inside the critical strip. J. Number Theory 67, 182–189 (1997)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kohnen, W., Sengupta, J., Weigel, M.: Nonvanishing of derivatives of Hecke L-functions associated to cusp forms inside the critical strip. Ramanujan J. 51, 319–327 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Luo, W.: Poincaré series and Hilbert modular forms. Ramanujan J. 7, 129–140 (2003). Rankin memorial issues

    MathSciNet  Article  Google Scholar 

  6. 6.

    Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45, 637–679 (1978)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Trotabas, D.: Non annulation des fonctions \(L\) des formes modulaires de Hilbert au point central. Ann. Inst. Fourier (Grenoble) 61, 187–259 (2011)

    MathSciNet  Article  Google Scholar 

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The authors are grateful to the referee for a number of suggestions that improved the exposition of this manuscript.

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Correspondence to Wissam Raji.

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Research of the first author is partially supported by an NSERC Discovery Grant.

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Hamieh, A., Raji, W. Non-vanishing of derivatives of L-functions of Hilbert modular forms in the critical strip. Res. number theory 7, 20 (2021). https://doi.org/10.1007/s40993-021-00248-y

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Keywords

  • Hilbert modular forms
  • Derivatives of L-functions
  • Non-vanishing of L-functions

Mathematics Subject Classification

  • Primary 11F41
  • 11F67
  • Secondary 11F30
  • 11F11
  • 11F12
  • 11N75