Determination of holomorphic modular forms by primitive Fourier coefficients

Abstract

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character \({\chi}\) of conductor \({\mathfrak f_\chi}\) are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of \({N/\mathfrak f_\chi}\). The cases of other major types of holomorphic modular forms are included.

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References

  1. 1

    Breulmann S., Kohnen W.: Twisted Maass–Koecher series and spinor zeta functions. Nagoya Math. J. 155, 153–160 (1999)

    MATH  MathSciNet  Google Scholar 

  2. 2

    Gan W.T., Gross B.H., Savin G.: Fourier coefficients of modular forms on G 2. Duke Math. J. 115(1), 105–169 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Heim B.: Separators of Siegel modular forms of degree two. Proc. Am. Math. Soc. 136(12), 4167–4173 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4

    Katsurada H.: On the coincidence of Hecke-eigenforms. Abh. Math. Sem. Univ. Hamburg. 70, 77–83 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5

    Miyake T.: Modular Forms. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  6. 6

    Scharlau R., Walling L.: A weak-multiplicity one theorem for Siegel modular forms. Pac. J. Math. 211, 369–374 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7

    Serre J.P., Stark H.M.: Modular forms of weight 1/2. Springer Lec. Notes Math. 627, 27–67 (1977)

    Article  MathSciNet  Google Scholar 

  8. 8

    Shimura G.: On Eisenstein series. Duke Math. J. 50, 417–476 (1983)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9

    Zagier, D.: Sur la conjecture de Saito-Kurokawa (d‘prés H. Maass). Sém Deligne-Pisot-Poitou Paris 1979/1980, (éd. M.-J. Bertin), Progr. Math. 12, Birkhauser, Boston, pp. 371–394 (1981)

  10. 10

    Zhuravrev V.G.: Hecke rings for a covering of the symplectic group. Math. Sbornik 121(163), 381–402 (1983)

    Google Scholar 

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Correspondence to Shunsuke Yamana.

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The author is supported by the Grant-in-Aid for JSPS fellows.

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Yamana, S. Determination of holomorphic modular forms by primitive Fourier coefficients. Math. Ann. 344, 853–862 (2009). https://doi.org/10.1007/s00208-008-0330-4

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Mathematics Subject Classification (2000)

  • 11F30