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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2412.00688 |
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| _version_ | 1866917852934569984 |
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| author | Du, Tuoping |
| author_facet | Du, Tuoping |
| contents | This research provides a formal definition of the arithmetic theta lift for cusp forms of weight $3/2$ and establishes the arithmetic inner product formula, thereby completing the Kudla program on modular curves. This formula is demonstrated to be equivalent to the Gross-Zagier formula, for which we provide a new proof.
Additionally, the authors introduce a new arithmetic representation for the central derivatives of L-functions associated with cusp forms of higher weight. Although this representation differs from Zhang's higher weight Gross-Zagier formula, it maintains a significant connection to it. This study also proposes a conjecture indicating that the vanishing of derivatives of L-functions is determined by the algebraicity of the coefficients of harmonic weak Maass forms.
A consistent approach is employed to study both parts of this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00688 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the arithmetic inner product formula and central derivatives of L-functions Du, Tuoping Number Theory This research provides a formal definition of the arithmetic theta lift for cusp forms of weight $3/2$ and establishes the arithmetic inner product formula, thereby completing the Kudla program on modular curves. This formula is demonstrated to be equivalent to the Gross-Zagier formula, for which we provide a new proof. Additionally, the authors introduce a new arithmetic representation for the central derivatives of L-functions associated with cusp forms of higher weight. Although this representation differs from Zhang's higher weight Gross-Zagier formula, it maintains a significant connection to it. This study also proposes a conjecture indicating that the vanishing of derivatives of L-functions is determined by the algebraicity of the coefficients of harmonic weak Maass forms. A consistent approach is employed to study both parts of this work. |
| title | On the arithmetic inner product formula and central derivatives of L-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.00688 |