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

Fu, Hang; Stoll, Michael

doi:10.1090/proc/16102

Elliptic curves with common torsion $x$-coordinates and hyperelliptic torsion packets
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by Hang Fu and Michael Stoll

Proc. Amer. Math. Soc. 150 (2022), 5137-5149

DOI: https://doi.org/10.1090/proc/16102

Published electronically: September 9, 2022

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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 $\mathbb {P}^1_x$ 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.

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