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Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1327–1359 | Cite as

On central L-derivative values of automorphic forms

  • Shenhui Liu
Article

Abstract

We study the average and nonvanishing of the central L-derivative values of L(sf) and \(L(s,f_{K_{\scriptscriptstyle D}})\) for f in an orthogonal Hecke eigenbasis \(\mathcal {H}_{2k}\) of weight 2k cusp forms of level 1 for large odd k. Here \(f_{K_{\scriptscriptstyle D}}\) is the base change of f to an imaginary quadratic field \(K_{\scriptscriptstyle D}=\mathbb {Q}(\sqrt{D})\) with fundamental discriminant D. We prove asymptotic formulas for the first and second moments of \(L'(\frac{1}{2},f)\), as well as the first moment of \(L'(\frac{1}{2},f_{K_{\scriptscriptstyle D}})\), over \(\mathcal {H}_{2k}\) as odd \(k\rightarrow \infty \). Further, we employ mollifiers to establish that for sufficiently large k there are positive proportion of Hecke eigenforms f in \(\mathcal {H}_{2k}\) with \(L'(\frac{1}{2},f)\ne 0\). We also give applications of our results to Heegner cycles of high weights of the modular curve \(X_0(1)\).

Keywords

Holomorphic Hecke eigenforms Central L-derivative values Mollifiers Nonvanishing Heegner cycles 

Mathematics Subject Classification

11F11 11F67 

Notes

Acknowledgements

The author thanks Professor Wenzhi Luo for suggesting the investigation of central L-derivative values and for his valuable advice and constant support, and Professor Hui Xue for answering many questions regarding Heegner cycles of high weights. The author also thanks Yongxiao Lin and Professor Sheng-Chi Liu for their encouragement and helpful conversations. Finally, the author is grateful to the referee for valuable comments.

References

  1. 1.
    Balkanova, O., Frolenkov, D.: Moments of \(L\)-functions and the Liouville–Green method. arXiv:1610.03465
  2. 2.
    Blomer, V.: On the central value of symmetric square \(L\)-functions. Math. Z. 260(4), 755–777 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Duke, W.: The critical order of vanishing of automorphic \(L\)-functions with large level. Invent. Math. 119(1), 165–174 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erdélyi, A. (ed.): Tables of Integral Transforms, vol. I. McGraw-Hill, New York (1954)zbMATHGoogle Scholar
  6. 6.
    Estermann, T.: On the representation of a number as the sum of two products. Proc. Lond. Math. Soc. s2 31(1), 123–133 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gross, B.H., Zagier, D.B.: Heegner points and derivatives of \(L\)-series. Invent. Math. 84, 225–320 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of \(L\)-series. II. Math. Ann. 278, 497–562 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Colloquium Publications, American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  10. 10.
    Iwaniec, H., Sarnak, P.: Dirichlet \(L\)-functions at the central point. In: Győry, K., et al. (eds.) Number Theory in Progress, vol. 2, pp. 941–952. de Gruyter, Berlin (1999)Google Scholar
  11. 11.
    Iwaniec, H., Sarnak, P.: The non-vanishing of central values of automorphic \(L\)-functions and Landau–Siegel zeros. Isr. J. Math. 120, 155–177 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Khan, R.: Non-vanishing of the symmetric square \(L\)-function at the central point. Proc. Lond. Math. Soc. 100(3), 736–762 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kowalski, E., Michel, P.: The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions. Duke Math. J. 100, 503–547 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kowalski, E., Michel, P.: A lower bound for the rank of \(J_0(q)\). Acta Arith. 94(4), 303–343 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kowalski, E., Michel, P., VanderKam, J.: Mollification of the fourth moment of automorphic \(L\)-functions and arithmetic applications. Invent. Math. 142, 95–151 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kowalski, E., Michel, P., VanderKam, J.: Non-vanishing of high derivatives of automorphic \(L\)-functions at the center of the critical strip. J. Reine Angew. Math. 526, 1–34 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kowalski, E., Michel, P., VanderKam, J.: Rankin–Selberg \(L\)-functions in the level aspect. Duke Math. J. 114(1), 123–191 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lau, Y.-K., Tsang, K.-M.: A mean square formula for central values of twisted automorphic \(L\)-functions. Acta Arith. 118(3), 231–262 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lau, Y.-K., Wu, J.: A density theorem on automorphic \(L\)-functions and some applications. Trans. Am. Math. Soc. 358(1), 441–472 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Luo, W.: Nonvanishing of the central \(L\)-values with large weight. Adv. Math. 285, 220–234 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Michel, P., VanderKam, J.: Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point. J. Number Theory 81, 130–148 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Olver, F.W.J., et al. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)zbMATHGoogle Scholar
  23. 23.
    Skoruppa, N., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Soundararajan, K.: Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=\frac{1}{2}\). Ann. Math. 152, 447–488 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    VanderKam, J.: The rank of quotients of \(J_0(N)\). Duke Math. J. 97, 545–577 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, Cambridge (1996) (reprint of Cambridge Mathematical Library edition)Google Scholar
  27. 27.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1969) (reprint of the 4th edition)Google Scholar
  28. 28.
    Xue, H.: Gross–Kohnen–Zagier theorem for higher weight forms. Math. Res. Lett. 17(3), 573–586 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhang, S.: Heights of Heegner cycles and derivatives of \(L\)-series. Invent. Math. 130, 99–152 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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