Elliptic curves with common torsion 𝑥-coordinates and hyperelliptic torsion packets

Abstract

We establish a connection between torsion packets on curves of genus  2 2 and pairs of elliptic curves realized as double covers of the projective line  P x 1 \mathbb {P}^1_x that have many common torsion x x -coordinates. This can be used to show that the set of common torsion x x -coordinates has size at least  22 22 infinitely often and has 34 34 elements in some cases. We also explain how we obtained the current record example of a hyperelliptic torsion packet on a genus  2 2 curve.