Cusp forms without complex multiplication as $p$-adic limits
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915140781211648 |
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| author | Dockery, Dalen |
| author_facet | Dockery, Dalen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cusp forms without complex multiplication as $p$-adic limits Dockery, Dalen Number Theory 11F11 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. |
| title | Cusp forms without complex multiplication as $p$-adic limits |
| topic | Number Theory 11F11 |
| url | https://arxiv.org/abs/2407.19374 |