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

TL;DR

This paper explores the relationship between torsion points on genus 2 curves and pairs of elliptic curves sharing many torsion $x$-coordinates, revealing large common torsion sets and providing a record example of a hyperelliptic torsion packet.

Contribution

It establishes a new connection between genus 2 torsion packets and elliptic curve pairs with shared torsion $x$-coordinates, leading to large common torsion sets and a record example.

Findings

Common torsion $x$-coordinates set size can be at least 22 infinitely often.

In some cases, the set size reaches 34 elements.

The paper explains how the current record example was obtained.

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.