Summary: We establish a correspondence between the rank of Mordell-Weil group of the complex elliptic threefold associated with a plane curve \(\mathcal{C} \subset \mathbb{P}^2(\mathbb C)\) with equation \(F=0\), certain roots of the Alexander polynomial associated with the fundamental group \(\pi_1(\mathbb{P}^2(\mathbb C)\smallsetminus\mathcal{C})\) and the polynomial solutions for the functional equation of type \[ h_1^pF_1+h_2^qF_2+h_3^rF_3=0 \] where \(F=F_1F_2F_3\). This correspondence is obtained for curves in a certain class which includes the curves having introduced here \(\delta\)-essential singularities and in particular for all curves with ADE singularities.{ } As a consequence we find a linear bound for the degree of the Alexander polynomial in terms of the degree of \(\mathcal{C}\) for curves with \(\delta\)-essential singularities and in particular arbitrary ADE singularities.