Abstract
In this paper, we consider p-adic limits of \(\beta ^{-n}g|U_{p^{2}}^{n}\) for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λ p =β+β ′ under \(T\phantom {\dot {i}\!}_{p^{2}}\) and prove that these equal classical Hecke eigenforms of the same weight. This result parallels the integral weight case, but requires a much more careful investigation due to a more complicated structure of half-integral weight weakly holomorphic Hecke eigenforms.
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Acknowledgements
The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and by the Deutsche Forschungsgemeinschaft (DFG) Grant No. BR 4082/3-1. The research of the second author is supported by Simons Foundation Collaboration Grant. The research of the third author was supported by grant project number 27300314 of the Research Grants Council.
The authors would like to thank Kevin Buzzard for helpful comments concerning Conjecture 1.2.
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Bringmann, K., Guerzhoy, P. & Kane, B. Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms. Res. number theory 1, 26 (2015). https://doi.org/10.1007/s40993-015-0027-1
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DOI: https://doi.org/10.1007/s40993-015-0027-1