Abstract
In the case of Siegel modular forms of degree \(n\), we prove that, for almost all prime ideals \(\mathfrak {p}\) in any ring of algebraic integers, mod \(\mathfrak {p}^m\) cusp forms are congruent to true cusp forms of the same weight. As an application we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan’s congruence. We will conclude by giving numerical examples.
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Acknowledgments
The authors would like to thank Professor S. Nagaoka and Professor S. Böcherer for the valuable discussions about the proofs, and Professor H. Katsurada for informing them about the value of studying congruences on Fourier coefficients between Klingen-Eisenstein series and cusp forms. The authors would also like to thank the referee, whose advice was helpful in improving the presentation of this paper. The first named author is supported by JSPS Grant-in-Aid for Young Scientists (B) 26800026.
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Communicated by Jens Funke.
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Kikuta, T., Takemori, S. Ramanujan type congruences for the Klingen-Eisenstein series. Abh. Math. Semin. Univ. Hambg. 84, 257–266 (2014). https://doi.org/10.1007/s12188-014-0098-2
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DOI: https://doi.org/10.1007/s12188-014-0098-2