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

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 ${\mathbb{P}}_x^1$ 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.

References in corpus (4)

• Equidistribution in Families of Abelian Varieties and Uniformity
• Recent developments of the Uniform Mordell-Lang Conjecture
• Division Polynomials and Intersection of Projective Torsion Points
• The Uniform Mordell-Lang Conjecture

Cited by in corpus (1)

• Intersecting the torsion of elliptic curves