Abstract
The p-adic L-function for modular forms of integral weight is well-known. For certain weights the p-adic L-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin–Selberg method, and proving explicit p-adic congruences in the resultant Rankin–Selberg expression.
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Acknowledgments
I would like to acknowledge EPSRC for funding the research of this paper. Special acknowledgements go to both my Ph.D. supervisor Thanasis Bouganis for general guidance and particularly for staying patient throughout this paper’s seemingly never ending development of problems, and to the anonymous referee for their careful reading and suggestions. Funding was provided by Engineering and Physical Sciences Research Council (Grant No. 000118421).
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Mercuri, S. The p-adic L-function for half-integral weight modular forms. manuscripta math. 161, 61–91 (2020). https://doi.org/10.1007/s00229-018-1085-1
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DOI: https://doi.org/10.1007/s00229-018-1085-1