The Ramanujan Journal

, Volume 45, Issue 3, pp 615–637 | Cite as

Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect

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Abstract

The purpose of this paper is to prove that a primitive Hilbert cusp form \(\mathbf{g}\) is uniquely determined by the central values of the Rankin–Selberg L-functions \(L(\mathbf{f}\otimes \mathbf{g}, \frac{1}{2})\), where \(\mathbf{f}\) runs through all primitive Hilbert cusp forms of weight \(k\) for infinitely many weight vectors \(k\). This result is a generalization of the work of Ganguly et al. (Math Ann 345:843–857, 2009) to the setting of totally real number fields, and it is a weight aspect analogue of our previous work (Hamieh and Tanabe in Trans Am Math Soc, arXiv:1609.07209, 2016).

Keywords

Hilbert modular forms Rankin–Selberg convolutions Special values of L-functions 

Mathematics Subject Classification

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

Acknowledgements

The authors are grateful to an anonymous referee for a number of valuable suggestions that improved the exposition of this manuscript and enhanced the main result. The authors would also like to thank Amir Akbary and Jeff Hoffstein for useful remarks and discussions about the topic of this paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA

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