Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves

Abstract

Abstract We establish a correspondence between the rank of Mordell–Weil group of the complex elliptic threefold associated with a plane curve 𝒞 ⊂ ℙ2(𝔻) with equation F = 0, certain roots of the Alexander polynomial associated with the fundamental group π1(ℙ2(𝔻)∖𝒞) and the polynomial solutions for the functional equation of type h 1 p F 1 + h 2 q F 2 + h 3 r F 3 = 0 where F = F 1 F 2 F 3. This correspondence is obtained for curves in a certain class which includes the curves having introduced here δ-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 𝒞 for curves with δ-essential singularities and in particular arbitrary ADE singularities.