Torsion sections of elliptic surfaces (Q1897586)

A semistable elliptic surface is a fibration \(p : X \to C\) of a smooth compact surface \(X\) onto a smooth curve \(C\) whose general fibre is elliptic and with semistable singular fibres (here \(p\) is assumed to have a section). The zero section \(S_0\) meets only a component \(C^0_J\) of a singular fibre \(F^J\). The authors consider a torsion section \(S\) of order \(p\) and those fibres \(F^J\) such that their component hit by \(S\) is still \(C^0_J\). They build up a number taking into account these components and, among them, those hit in a point of \(C^0_J\) having a ``fixed'' coordinate in a previously given suitable coordinate choice in \(C^0_J\). In the main result the authors prove that this number is \(2/p - 1\) and so independent of \(S\) and of the coordinate ``fixed'' among the possible ones. They give a proof based on computation on elliptic modular surfaces and another one via an explicit computation.