Abstract: Given an algebraic family of elliptic curves with a non-torsion section, one can locally define its elliptic logarithm, after viewing the elliptic curves of the family as complex tori. An old theorem of Bertrand and André provides, among other things, the transcendency of these elliptic logarithms, over the field of regular functions on the base. An even older, and apparently unrelated, theorem of Shioda proves that the so called 'modular elliptic surfaces' admit no section of infinite order.

In a joint work with U. Zannier we provide a unification and an improvement of these results. The outcome is a statement on the relative monodromy of elliptic logarithms.