Relative monodromy of ramified sections on abelian schemes

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 $\sigma \text{\hspace{0.17em}⁣}:S\to \mathcal{A}$ 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.