A note on cusp forms as $p$-adic limits

TL;DR

This paper explores expressing cusp forms as $p$-adic limits of weakly holomorphic modular forms, improving previous results by employing techniques from holomorphic modular form theory.

Contribution

It provides strengthened results on cusp forms as $p$-adic limits, using holomorphic modular forms instead of harmonic Maass forms techniques.

Findings

Enhanced $p$-adic limit results for cusp forms.

Utilization of holomorphic modular forms techniques.

Improved understanding of $p$-adic coupling in modular forms.

Abstract

Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.