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

Publication date: 20 December 2019

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

m a t h b b 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.

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