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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.20520 |
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| _version_ | 1866915949126352896 |
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| author | Tajima, Ryota |
| author_facet | Tajima, Ryota |
| contents | Let $F^{+}$ be a mock modular form associated to a normalized newform $g$. K. Bringmann et. al. obtained a $p$-adic modular form starting from $F^{+}$ by adding a suitable linear combination of Eichler integrals of $g(q)$ and $g(q^{p})$. We denote the coefficients of the Eichler integrals of $g(q)$ and $g(q^{p})$ by $γ_{g}$ and $δ_{g}$. These constants are important in the $p$-adic theory of mock modular forms, but relatively little is known about them at present. For instance, K. Bringmann et. al. raised the question of whether $δ_{g}$ vanishes when $g$ has CM by an imaginary quadratic field in which $p$ is inert. In previous work, the non-vanishing of $δ_{g}$ has been proved mainly when $g$ is associated to an elliptic curve. In higher weight, only one example was known for which $δ_{g}\neq 0$. In this paper, we show that $δ_{g}\neq 0$ under mild assumptions when all the Fourier coefficients of $g \in S_{k}(Γ_{0}(N), χ)$ are real, without assuming that $g$ has CM. In particular, this provides a class of higher-weight examples for which $δ_{g}\neq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20520 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-vanishing of the $p$-adic constant for mock modular forms associated to a newform with real Fourier coefficients Tajima, Ryota Number Theory Let $F^{+}$ be a mock modular form associated to a normalized newform $g$. K. Bringmann et. al. obtained a $p$-adic modular form starting from $F^{+}$ by adding a suitable linear combination of Eichler integrals of $g(q)$ and $g(q^{p})$. We denote the coefficients of the Eichler integrals of $g(q)$ and $g(q^{p})$ by $γ_{g}$ and $δ_{g}$. These constants are important in the $p$-adic theory of mock modular forms, but relatively little is known about them at present. For instance, K. Bringmann et. al. raised the question of whether $δ_{g}$ vanishes when $g$ has CM by an imaginary quadratic field in which $p$ is inert. In previous work, the non-vanishing of $δ_{g}$ has been proved mainly when $g$ is associated to an elliptic curve. In higher weight, only one example was known for which $δ_{g}\neq 0$. In this paper, we show that $δ_{g}\neq 0$ under mild assumptions when all the Fourier coefficients of $g \in S_{k}(Γ_{0}(N), χ)$ are real, without assuming that $g$ has CM. In particular, this provides a class of higher-weight examples for which $δ_{g}\neq 0$. |
| title | Non-vanishing of the $p$-adic constant for mock modular forms associated to a newform with real Fourier coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.20520 |