Cusp forms without complex multiplication as $p$-adic limits

TL;DR

This paper extends the study of $p$-adic limits of cusp forms to those without complex multiplication, building on prior work that focused on forms with CM, and provides new results in this area.

Contribution

It proves $p$-adic limit results for one-dimensional cusp form spaces without complex multiplication, expanding the scope of previous studies.

Findings

Established $p$-adic limits for cusp forms without CM

Extended previous results to non-CM cusp form spaces

Strengthened understanding of $p$-adic properties of modular forms

Abstract

In 2016, Ahlgren and Samart used the theory of holomorphic modular forms to obtain lower bounds on $p$-adic valuations related to the Fourier coefficients of three cusp forms. In particular, their work strengthened a previous result of El-Guindy and Ono which expresses a cusp form as a $p$-adic limit of weakly holomorphic modular forms. Subsequently, Hanson and Jameson extended Ahlgren and Samart's result to all one-dimensional cusp form spaces of trivial character and having a normalized form that has complex multiplication. Here we prove analogous $p$-adic limits for several one-dimensional cusp form spaces of trivial character but whose normalized form does not have complex multiplication.