Abstract Let’s fix a complex abelian scheme A S of relative dimension g, without fixed part, and having maximal variation in moduli. We show that the relative monodromy group M^rel_ of a ramified section S A is nontrivial. Moreover, under some hypotheses on the action of the monodromy group Mon (A), we show that M^rel_ 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.
Dolce et al. (Tue,) studied this question.