Explicit images for the Shimura Correspondence
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arXiv
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| Format: | Preprint |
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2025
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| author | Boylan, Matthew Swati |
| author_facet | Boylan, Matthew Swati |
| contents | In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes χ_{12}$ where $G\in S^{new}_{r+2s - 1}(Γ_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$, where $\textup{Sh}_{r}$ is the $r$-th Shimura lift associated to the theta-multiplier. He proved a similar result for $(r,6) = 3$.\:His proofs rely on trace computations in integral and half-integral weights.
In this paper, we provide a constructive proof of Yang's result. We obtain explicit formulas for $\mathcal{S}_{r}(F)$, the $r$-th Shimura lift associated to the eta-multiplier defined by Ahlgren, Andersen, and Dicks, when $1\leq r\leq 23$ is odd and $N = 1$. We also obtain formulas for lifts of Hecke eigenforms multiplied by theta-function eta-quotients and lifts of Rankin-Cohen brackets of Hecke eigenforms with theta-function eta-quotients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_01018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit images for the Shimura Correspondence Boylan, Matthew Swati Number Theory 11F37, 11F11, 11F20 In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes χ_{12}$ where $G\in S^{new}_{r+2s - 1}(Γ_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$, where $\textup{Sh}_{r}$ is the $r$-th Shimura lift associated to the theta-multiplier. He proved a similar result for $(r,6) = 3$.\:His proofs rely on trace computations in integral and half-integral weights. In this paper, we provide a constructive proof of Yang's result. We obtain explicit formulas for $\mathcal{S}_{r}(F)$, the $r$-th Shimura lift associated to the eta-multiplier defined by Ahlgren, Andersen, and Dicks, when $1\leq r\leq 23$ is odd and $N = 1$. We also obtain formulas for lifts of Hecke eigenforms multiplied by theta-function eta-quotients and lifts of Rankin-Cohen brackets of Hecke eigenforms with theta-function eta-quotients. |
| title | Explicit images for the Shimura Correspondence |
| topic | Number Theory 11F37, 11F11, 11F20 |
| url | https://arxiv.org/abs/2505.01018 |