Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918109087006720 |
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| author | Guerzhoy, Pavel |
| author_facet | Guerzhoy, Pavel |
| contents | Several authors have recently proved results which express a cusp form as a $p$-adic limit of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart \cite{AS} found a method to prove that no exceptions happen in the specific case considered by El-Guindy and Ono, Hanson and Jameson, and (independently) Dicks. generalized this method to finitely many other cases.
In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's $p$-adic $Γ$-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07107 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation Guerzhoy, Pavel Number Theory 11F11, 11F33, 14H52 Several authors have recently proved results which express a cusp form as a $p$-adic limit of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart \cite{AS} found a method to prove that no exceptions happen in the specific case considered by El-Guindy and Ono, Hanson and Jameson, and (independently) Dicks. generalized this method to finitely many other cases. In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's $p$-adic $Γ$-function. |
| title | Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation |
| topic | Number Theory 11F11, 11F33, 14H52 |
| url | https://arxiv.org/abs/2506.07107 |