Relative monodromy of ramified sections on abelian schemes

Paolo Dolce; Francesco Tropeano

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arXiv · 2407.19476

Relative monodromy of ramified sections on abelian schemes

Paolo Dolce·Francesco Tropeano

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Abstract

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 $M^{\textrm{rel}}_\sigma$ of a ramified section $\sigma\colon S\to\mathcal A$ is nontrivial. Moreover, under some hypotheses on the action of the monodromy group $\textrm{Mon}(\mathcal A)$ we show that $M^{\textrm{rel}}_\sigma\cong \mathbb Z^{2g}$. 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.

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Paolo Dolce, Francesco Tropeano. 2024-07-28. Relative monodromy of ramified sections on abelian schemes. https://arxiv.org/abs/2407.19476

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