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arXiv · 2506.07107

Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation

Pavel Guerzhoy
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Abstract

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 $\Gamma$-function.

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Pavel Guerzhoy. 2025-06-08. Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation. https://arxiv.org/abs/2506.07107

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