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<p>In 2016, Ahlgren and Samart used the theory of holomorphic modular forms to obtain lower bounds on <i>p</i>-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 <i>p</i>-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 <i>p</i>-adic limits for several one-dimensional cusp form spaces of trivial character but whose normalized form does not have complex multiplication.</p>

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Cusp forms without complex multiplication as p-adic limits

  • 期刊论文

  • 发表日期：

    2025年2月7日

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  • Dalen Dockery

摘要

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.