The $p$-adic constant for mock modular forms associated to CM forms II
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909431533404160 |
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| author | Tajima, Ryota |
| author_facet | Tajima, Ryota |
| contents | For a normalized newform $g \in S_{k}(Γ_{0}(N))$ with complex multiplication by an imaginary quadratic field $K$, there is a mock modular form $F^{+}$ corresponding to $g$. K. Bringmann et al. modified $F^{+}$ in order to obtain a $p$-adic modular form by a certain $p$-adic constant $α_{g}$. In addition, they showed that if $p$ is split in $\mathcal{O}_{K}$ and $p \nmid N$, then $α_{g}=0$. On the other hand, the author showed that $α_{g}$ is a $p$-adic unit for an inert prime $p$ satisfying that $p\nmid 2N$ when $\dim_{\mathbb{C}} S_{k}(Γ_{0}(N))=1$. In this paper, under mild condition, we determine the $p$-adic valuation of $α_{g}$ for an inert prime $p$ and a general CM form $g$ of weight $2$ with rational Fourier coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $p$-adic constant for mock modular forms associated to CM forms II Tajima, Ryota Number Theory 11F03, 11F11, 11F25, 11F30 For a normalized newform $g \in S_{k}(Γ_{0}(N))$ with complex multiplication by an imaginary quadratic field $K$, there is a mock modular form $F^{+}$ corresponding to $g$. K. Bringmann et al. modified $F^{+}$ in order to obtain a $p$-adic modular form by a certain $p$-adic constant $α_{g}$. In addition, they showed that if $p$ is split in $\mathcal{O}_{K}$ and $p \nmid N$, then $α_{g}=0$. On the other hand, the author showed that $α_{g}$ is a $p$-adic unit for an inert prime $p$ satisfying that $p\nmid 2N$ when $\dim_{\mathbb{C}} S_{k}(Γ_{0}(N))=1$. In this paper, under mild condition, we determine the $p$-adic valuation of $α_{g}$ for an inert prime $p$ and a general CM form $g$ of weight $2$ with rational Fourier coefficients. |
| title | The $p$-adic constant for mock modular forms associated to CM forms II |
| topic | Number Theory 11F03, 11F11, 11F25, 11F30 |
| url | https://arxiv.org/abs/2412.12811 |