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Non-vanishing of Rankin-Selberg convolutions for Hilbert modular forms

Abstract

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg L-function of two adelic Hilbert primitive forms \(\mathbf{f}\) and \(\mathbf{g}\), both of which have varying weight parameter k. We prove that, for sufficiently large \(k\in 2{\mathbb {N}}^n\), there are at least \(\frac{\mathrm{N}(k)}{\log ^c \mathrm{N}(k)}\) adelic Hilbert primitive forms \(\mathbf{f}\) of weight k for which \(L(\frac{1}{2}, \mathbf{f}\otimes \mathbf{g})\) are nonzero.

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Acknowledgements

The authors would like to thank Professors Amir Akbary, Ram Murty, and Nathan Ng for encouraging comments and useful discussions about the topic of this paper. The authors are also grateful to the referee for many helpful suggestions that improved the exposition of the paper.

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Correspondence to Naomi Tanabe.

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Research of Alia Hamieh was partially supported by PIMS Postdoctoral Fellowship at the University of Lethbridge.

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Hamieh, A., Tanabe, N. Non-vanishing of Rankin-Selberg convolutions for Hilbert modular forms. Math. Z. 297, 81–97 (2021). https://doi.org/10.1007/s00209-020-02502-y

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Keywords

  • Hilbert modular forms
  • Rankin-Selberg L-functions
  • Non-vanishing of central L-values

Mathematics Subject Classification

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