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The Ramanujan Journal

, Volume 48, Issue 2, pp 279–303 | Cite as

On Doi–Naganuma and Shimura liftings

  • Balesh KumarEmail author
  • Murugesan Manickam
Article
First Online:
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Abstract

In this paper, we extend the Doi–Naganuma lifting to higher levels by following the methods of Zagier and Kohnen. We prove that there is a Hecke-equivariant linear map from the space of elliptic cusp forms of integer weight k, level \(N, ((N,D)=1)\) to Hilbert cusp forms of weight k, level N associated to a real quadratic field of discriminant D (\(D\equiv 1\pmod {4}\)) with class number one. The above lifting is obtained by computing the explicit image of Poincaré series of weight k, level N for the cusp at \(\infty \). Finally, we see that the above lifting is closely related to the Dth Shimura lift on the Kohnen plus space.

Keywords

Modular forms Doi–Naganuma lift Shimura liftings Hilbert modular forms 

Mathematics Subject Classification

11F11 11F32 11F37 11F41 
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Notes

Acknowledgements

The first author would like to thank Prof. E. Ghate for the helpful discussions. Theorem 1.1 arose from a question raised by Prof. B. Ramakrishnan during a discussion meeting at KSOM in February, 2016 while the first author was giving a talk on Doi–Naganuma lifting. The first author gratefully acknowledges him for this. The first author also thanks KSOM and The Institute of Mathematical Sciences for providing nice working conditions. Finally, the authors are thankful to the referee for going through the manuscript carefully and for valuable suggestions to improve the presentation.

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Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesHBNIChennaiIndia
  2. 2.Kerala School of MathematicsKozhikodeIndia

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