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

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.

Minor revisions based on referee's suggestions; to appear in The Ramanujan Journal