A modular framework for generalized Hurwitz class numbers II
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916478332174336 |
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| author | Beckwith, Olivia Mono, Andreas |
| author_facet | Beckwith, Olivia Mono, Andreas |
| contents | In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamoglu, and Tóth, $\mathcal{G}$ maps to Zagier's non-holomorphic Eisenstein series and a linear combination of Pei and Wang's generalized Cohen--Eisenstein series under the Bruinier--Funke operator $ξ_{\frac{1}{2}}$. In this paper, we realize $\mathcal{G}$ as the output of a regularized Siegel theta lift of $1$ whenever $N=p$ is an odd prime building on more general work by Bruinier, Funke and Imamoglu. In addition, we supply the computation of the square-indexed Fourier coefficients of $\mathcal{G}$. This yields explicit identities between the Fourier coefficients of $\mathcal{G}$ and all quadratic traces of $1$. Furthermore, we evaluate the Millson theta lift of $1$ and consider spectral deformations of $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A modular framework for generalized Hurwitz class numbers II Beckwith, Olivia Mono, Andreas Number Theory 11F37 (Primary), 11F12, 11F27, 11F30 (Secondary) In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamoglu, and Tóth, $\mathcal{G}$ maps to Zagier's non-holomorphic Eisenstein series and a linear combination of Pei and Wang's generalized Cohen--Eisenstein series under the Bruinier--Funke operator $ξ_{\frac{1}{2}}$. In this paper, we realize $\mathcal{G}$ as the output of a regularized Siegel theta lift of $1$ whenever $N=p$ is an odd prime building on more general work by Bruinier, Funke and Imamoglu. In addition, we supply the computation of the square-indexed Fourier coefficients of $\mathcal{G}$. This yields explicit identities between the Fourier coefficients of $\mathcal{G}$ and all quadratic traces of $1$. Furthermore, we evaluate the Millson theta lift of $1$ and consider spectral deformations of $1$. |
| title | A modular framework for generalized Hurwitz class numbers II |
| topic | Number Theory 11F37 (Primary), 11F12, 11F27, 11F30 (Secondary) |
| url | https://arxiv.org/abs/2411.07962 |