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Cuspidality and the growth of Fourier coefficients of modular forms

Siegfried Böcherer and Soumya Das

Abstract

We characterize Siegel cusp forms in the space of Siegel modular forms of large weight k>2n on any Siegel congruence subgroup Γ of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well-known Hecke bound) at any one of the cusps. For this, we use a ‘local’ approach as compared to our previous results on this topic. We also touch upon the question in the context of vector-valued modular forms.

Funding statement: The second author acknowledges financial support from the UGC Centre for Advanced Studies, the Department of Science and Technology (India), and the Indian Institute of Science (Bangalore).

Acknowledgements

We thank the Indian Institute of Science in Bangalore, the Harish Chandra Research Institute in Allahabad, and the University of Tokyo, where parts of this work was done, for providing a very pleasant working atmosphere. Finally, we thank the referee for comments and suggestions on the paper which improved the presentation.

References

[1] A. N. Andrianov and V. G. Zuravlev, Modular forms and Hecke operators, Transl. Math. Monogr. 145, American Mathematical Society, Providence 1995. Search in Google Scholar

[2] S. Böcherer, Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe, Abh. Math. Sem. Univ. Hamburg 56 (1986), 35–47. Search in Google Scholar

[3] S. Böcherer and S. Das, Characterization of Siegel cusp forms by the growth of their Fourier coefficients, Math. Ann. 359 (2014), 169–188. Search in Google Scholar

[4] S. Böcherer and S. Das, Cuspidality and the growth of Fourier coefficients: Small weights, Math. Z. (2016), 10.1007/s00209-015-1609-2. 10.1007/s00209-015-1609-2Search in Google Scholar

[5] S. Böcherer and W. Kohnen, On the Fourier coefficients of Siegel modular forms, submitted. Search in Google Scholar

[6] W. Duke, R. Howe and J.-S. Li, Estimating Hecke eigenvalues of Siegel modular forms, Duke Math. J. 67 (1992), 499–517. Search in Google Scholar

[7] S. A. Evdokimov, Euler products for congruence subgroups of the Siegel group of genus 2, Math. USSR Sbornik 28 (1976), 431–458. Search in Google Scholar

[8] S. A. Evdokimov, A basis of eigenfunctions of Hecke operators in the theory of Siegel modular forms of genus n, Math. USSR Sbornik 43 (1982), 299–321. Search in Google Scholar

[9] E. Freitag, Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koeffizienten, Math. Annalen 258 (1982), 419–440. Search in Google Scholar

[10] E. Freitag, Siegelsche Modulfunktionen, Grundl. Math. Wiss. 254, Springer, Berlin 1983. Search in Google Scholar

[11] E. Freitag, Singular modular forms and theta relations, Lecture Notes in Math. 1457, Springer, Berlin 1991. Search in Google Scholar

[12] E. Freitag, Siegel Eisenstein series of arbitrary level and theta series, Abh. Math. Sem. Univ. Hamburg. 66 (1996), 229–247. Search in Google Scholar

[13] R. Godement, Généralités sur les formes modulaires, I, Séminaire Henri Cartan 10 (1957/58), no. 1, 1–18. Search in Google Scholar

[14] Y. Kitaoka, Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J. 95 (1984), 73–84. Search in Google Scholar

[15] Y. Kitaoka, Siegel modular forms and representations by quadratic forms, Lecture Notes, Tata Institute of Fundamental Research, Bombay 1986. Search in Google Scholar

[16] W. Kohnen, On certain generalized modular forms, Funct. Approx. Comment. Math. 43 (2010), 23–29. Search in Google Scholar

[17] W. Kohnen and Y. Martin, A characterization of degree two cusp forms by the growth of their Fourier coefficients, Forum Math. 26 (2014), 1323–1331. Search in Google Scholar

[18] A. Krieg, Das Vertauschungsgesetz zwischen Hecke Operatoren und dem Siegelschen ϕ-Operator, Arch. Math. 46 (1986), 323–329. Search in Google Scholar

[19] B. Linowitz, Characterizing Hilbert modular cusp forms by coefficients size, Kyushu Math. J. 68 (2014), 105–111. Search in Google Scholar

[20] Y. Mizuno, On characterisation of Siegel cusp forms by the Hecke bound, Mathematika 61 (2015), 89–100. Search in Google Scholar

[21] G. Shimura, Convergence of zeta functions on symplectic and metaplectic groups, Duke Math. J. 82 (1996), 327–347. Search in Google Scholar

[22] R. Weissauer, Vektorwertige Siegelsche Modulformen kleinen Gewichtes, J. reine angew. Math. 343 (1983), 184–202. Search in Google Scholar

[23] R. Weissauer, Stabile Modulformen und Eisensteinreihen, Lecture Notes in Math. 1219, Springer, Berlin 1986. Search in Google Scholar

Received: 2014-07-25
Revised: 2015-08-14
Published Online: 2016-01-05
Published in Print: 2018-08-01

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