  We determine the relative monodromy group of abelian logarithms with respect
to periods in the cases of fibered products of elliptic schemes. This gives
rise to a result stronger than a theorem due to Y. Andr\'e and implies in
particular the algebraic independence of the logarithm of any non-torsion
section and the periods. We then conjecture an analogous result for the general
case of an abelian scheme of arbitrary relative dimension. This generalizes a
theorem of Corvaja and Zannier which determines the said group in the case of a
single elliptic scheme.
