Cusp forms as p-adic limits

TL;DR

This paper extends previous work on p-adic limits of cusp forms with complex multiplication, providing a general framework for one-dimensional cusp form spaces with trivial Nebentypus, and introduces a duality between related modular forms.

Contribution

It generalizes Ahlgren and Samart's results to all one-dimensional cusp form spaces with trivial Nebentypus, using purely modular form and Hecke operator theory.

Findings

Extended p-adic limit results to broader cusp form spaces.

Established a duality between two families of modular forms.

Strengthened connections between CM forms and weakly holomorphic modular forms.

Abstract

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional and has trivial Nebentypus. Along the way, we obtain a duality result relating two families of modular forms that arise naturally in each case.