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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2407.19476 |
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| _version_ | 1866912399672475648 |
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| author | Dolce, Paolo Tropeano, Francesco |
| author_facet | Dolce, Paolo Tropeano, Francesco |
| contents | Let's fix a complex abelian scheme $\mathcal A\to S$ of relative dimension $g$, without fixed part, and having maximal variation in moduli. We show that the relative monodromy group $M^{\textrm{rel}}_σ$ of a ramified section $σ\colon S\to\mathcal A$ is nontrivial. Moreover, under some hypotheses on the action of the monodromy group $\textrm{Mon}(\mathcal A)$ we show that $M^{\textrm{rel}}_σ\cong \mathbb Z^{2g}$. We discuss several examples and applications. For instance we provide a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative monodromy of ramified sections on abelian schemes Dolce, Paolo Tropeano, Francesco Number Theory Let's fix a complex abelian scheme $\mathcal A\to S$ of relative dimension $g$, without fixed part, and having maximal variation in moduli. We show that the relative monodromy group $M^{\textrm{rel}}_σ$ of a ramified section $σ\colon S\to\mathcal A$ is nontrivial. Moreover, under some hypotheses on the action of the monodromy group $\textrm{Mon}(\mathcal A)$ we show that $M^{\textrm{rel}}_σ\cong \mathbb Z^{2g}$. We discuss several examples and applications. For instance we provide a new proof of Manin's kernel theorem and of the algebraic independence of the coordinates of abelian logarithms with respect to the coordinates of periods. |
| title | Relative monodromy of ramified sections on abelian schemes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.19476 |