Relative monodromy of ramified sections on abelian schemes

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

arXiv comment25 pages. The Main Proofs have been clarified. The section "Ramification and torsion" has been added