Relative Monodromy of Ramified Sections on Abelian Schemes

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_σ of a ramified section σ : S → A is nontrivial. Moreover, under some hypotheses on the action of the monodromy group Mon(A), we show that M^rel_σ ≅ 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.