The Ramanujan Journal

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

On Doi–Naganuma and Shimura liftings

  • Balesh KumarEmail author
  • Murugesan Manickam


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.


Modular forms Doi–Naganuma lift Shimura liftings Hilbert modular forms 

Mathematics Subject Classification

11F11 11F32 11F37 11F41 



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.


  1. 1.
    Bruinier, J.-H.: Borcherds product and Chern classes on Hirzebruch–Zagier divisors. Invent. Math. 132, 491–562 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bruinier, J.-H., Yang, T.: Twisted Borcherds product on Hilbert modular surfaces and their CM values. Am. J. Math. 129(3), 807–841 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruinier, J.-H., Geer, G.-V.-D., Harder, G., Zagier, D.: The 1-2-3 of modular forms. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Doi, K., Naganuma, H.: On the functional equation of certain Dirichlet series. Invent. Math. 9, 1–14 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ehlen, S.: Twisted Borcherds product on Hilbert modular surfaces and the regularized theta lift. Int. J. Number Theory 6(7), 1473–1489 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Geer, G.-V.-D.: Hilbert Modular Surfaces. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ghate, E.: Congruences between base-change and non-base-change Hilbert modular forms. Cohomology of arithmetic groups, \(L\)-functions and automorphic forms (Mumbai, 1998/1999). Tata Inst. Fund. Res. Tata Inst. Fund. Res. Stud. Math. 15, 35–62 (2001)Google 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.
    Howe, R.: \(\theta \)-Series and invariant theory in automorphic forms, representations and \(L\)-functions. In: Proceedings of Symposia in Pure Mathematics XXXIII. American Mathematical Society, Providence, RI, pp. 275–285 (1979)Google Scholar
  10. 10.
    Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  11. 11.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. American Mathematical Society Colloquium Publications, Providence (2004)zbMATHGoogle Scholar
  12. 12.
    Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271(2), 237–268 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kohnen, W.: Modular forms of half-integral weight on \(\Gamma _{0}(4)\). Math. Ann. 248(3), 249–266 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kumar, B., Manickam, M.: On Doi–Naganuma lifting. Tsukuba J. Math. 40(2), 125–137 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kudla, S.-S.: Theta-functions and Hilbert modular forms. Nagoya Math. J. 69, 97–106 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, Y.: Restriction of coherent Hilbert Eisenstein series. Math. Ann. 368, 317–338 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Manickam, M., Meher, J., Ramakrishnan, B.: Theory of newforms of half-integral weight. Pac. J. Math. 274(1), 125–139 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Naganuma, H.: On the coincidence of two Dirichlet series associated with cusp forms of Hecke’s “Neben”-type and Hilbert modular forms over a real quadratic field. J. Math. Soc. Jpn. 25, 547–555 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Niwa, S.: Modular forms of half-integral weight and integral of certain theta functions. Nagoya Math. J. 56, 147–161 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Oda, T.: On modular forms associated with indefinite quadratic forms of signature \((2, n-2)\). Math. Ann. 231, 97–144 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saito, H.: Automorphic forms and algebraic extension of number fields II. J. Math. Kyoto Univ. 19(1), 105–123 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Siegel, C.-L.: Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Gttingen Math. Phys. Kl. II 10, 84–102 (1969)zbMATHGoogle Scholar
  23. 23.
    Yang, T.: CM number fields and modular forms. Pure Appl. Math. Q. 1(2), 305–340 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zagier, D.: Modular forms associated to real quadratic fields. Invent. Math. 30(1), 1–46 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627. Springer, Berlin, pp. 105–170 (1977)Google Scholar

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Authors and Affiliations

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

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