Abstract
We show that Siegel modular forms of level \(\Gamma _{0}(p^{m})\) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre [18] closely; his proofs however do not generalize to the Siegel case or need some modifications.
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Acknowledgements
Crucial work on this paper was done during our stay at the Mathematisches Forschungsinstitut Oberwolfach under the programme “Research in Pairs”; we continued our work during research visits at Kinki University and Universität Mannheim (respectively); a final revision was done, when the first author held a guest professorship at the University of Tokyo. We thank these institutions for the support. We also thank Dr.Kikuta for pointing out some gaps in our presentation and Professor T.Ichikawa for discussions about p-adic modular forms.
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Böcherer, S., Nagaoka, S. (2014). On p-Adic Properties of Siegel Modular Forms. In: Heim, B., Al-Baali, M., Ibukiyama, T., Rupp, F. (eds) Automorphic Forms. Springer Proceedings in Mathematics & Statistics, vol 115. Springer, Cham. https://doi.org/10.1007/978-3-319-11352-4_4
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DOI: https://doi.org/10.1007/978-3-319-11352-4_4
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