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

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Fu, Hang ; Stoll, Michael:
Elliptic curves with common torsion x-coordinates and hyperelliptic torsion packets.
In: Proceedings of the American Mathematical Society. Bd. 150 (2022) . - S. 5137-5149.
ISSN 1088-6826
DOI: https://doi.org/10.1090/proc/16102

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Abstract

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 P1x 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.