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 Mʳᵉˡ_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 Mʳᵉˡ_σ Z²ᵍ. 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.