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| Main Author: | |
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| Format: | Preprint |
| Published: |
2012
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1212.1294 |
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| _version_ | 1866915551118360576 |
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| author | Mayer, Hartwig |
| author_facet | Mayer, Hartwig |
| contents | Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/ \mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N) / \mathbb{Q}$ of $X_1(N)/ \mathbb{Q}$, and, for sufficiently large N, an effective version of Bogomolov's conjecture for $X_1(N) / \mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1212_1294 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$ Mayer, Hartwig Number Theory 11G50, 11G18, 11M36 Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/ \mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N) / \mathbb{Q}$ of $X_1(N)/ \mathbb{Q}$, and, for sufficiently large N, an effective version of Bogomolov's conjecture for $X_1(N) / \mathbb{Q}$. |
| title | Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$ |
| topic | Number Theory 11G50, 11G18, 11M36 |
| url | https://arxiv.org/abs/1212.1294 |