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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2601.15271 |
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| _version_ | 1866917216044187648 |
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| author | Wilms, Robert |
| author_facet | Wilms, Robert |
| contents | We compute the stable Faltings height of the hyperelliptic curve $X_n\colon y^2=x^{n}-1$ for every odd integer $n\ge 3$ in terms of special values of Euler's gamma function. In particular, we prove the bounds $$-0.975n< h_{\mathrm{Fal}}(X_n)-\tfrac{n}{8}\log n<\tfrac{9}{64}n\log\log n-0.263n.$$ As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the $n$-th cyclotomic field by $\frac{n}{8}\log n+\frac{9}{64}n\log\log n-0.136n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15271 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Faltings height of the curve $y^2=x^n-1$ Wilms, Robert Number Theory Algebraic Geometry 14G40 We compute the stable Faltings height of the hyperelliptic curve $X_n\colon y^2=x^{n}-1$ for every odd integer $n\ge 3$ in terms of special values of Euler's gamma function. In particular, we prove the bounds $$-0.975n< h_{\mathrm{Fal}}(X_n)-\tfrac{n}{8}\log n<\tfrac{9}{64}n\log\log n-0.263n.$$ As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the $n$-th cyclotomic field by $\frac{n}{8}\log n+\frac{9}{64}n\log\log n-0.136n$. |
| title | On the Faltings height of the curve $y^2=x^n-1$ |
| topic | Number Theory Algebraic Geometry 14G40 |
| url | https://arxiv.org/abs/2601.15271 |