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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.01688 |
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| _version_ | 1866916772870881280 |
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| author | Li, Yingkun Zhang, Mingkuan |
| author_facet | Li, Yingkun Zhang, Mingkuan |
| contents | In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and square-free, the values at Heegner points of Borcherds product on $X_0(N)^2$ with effective divisors are not integral units when the discriminants are sufficiently large. This generalizes a result of the first author to higher levels. In the process, we explicitly describe the Rankin-Selberg type L-function that appeared in the work of Bruinier-Kudla-Yang when the quadratic space has signature (2, 2), and give a new construction of fundamental invariant vectors appearing in Weil representations of finite quadratic modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01688 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert Eisenstein series as Doi-Naganuma lift Li, Yingkun Zhang, Mingkuan Number Theory In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and square-free, the values at Heegner points of Borcherds product on $X_0(N)^2$ with effective divisors are not integral units when the discriminants are sufficiently large. This generalizes a result of the first author to higher levels. In the process, we explicitly describe the Rankin-Selberg type L-function that appeared in the work of Bruinier-Kudla-Yang when the quadratic space has signature (2, 2), and give a new construction of fundamental invariant vectors appearing in Weil representations of finite quadratic modules. |
| title | Hilbert Eisenstein series as Doi-Naganuma lift |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.01688 |