Cusp forms without complex multiplication as p-adic limits

2024/07/28 by Dalen Dockery, Dockery, Dalen
Mathematics · #11F11 #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.19374

openalex publication_date 2024/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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