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

TL;DR

This paper develops a new method to prove non-vanishing results for $p$-adic limits of cusp forms, extending previous finite case results to an infinite family and explicitly calculating key quantities using Morita's $p$-adic Gamma function.

Contribution

It introduces a novel approach to establish non-vanishing in an infinite family of cases, improving upon prior finite case methods and providing explicit calculations.

Findings

Proved non-vanishing results for an infinite family of cases.

Explicitly calculated key quantities using Morita's $p$-adic Gamma function.

Extended the applicability of $p$-adic limit techniques for cusp forms.

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.