ELLIPTIC CURVES WITH COMMON TORSION x-COORDINATES AND HYPERELLIPTIC TORSION PACKETS

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