Monodromy of elliptic logarithms: some topological methods and effective results

We present some effective approaches in studying the relative monodromy group of elliptic logarithms with respect to periods of elliptic schemes. We provide explicit ways of constructing explicit loops which leave periods unchanged but along which logarithms have non-trivial variations. We also get some topological methods and effective results which allow to manage the ramification locus of sections. The paper was inspired by a theorem of Corvaja and Zannier which abstractly determine the relative monodromy group of non-torsion sections.