Torsion Points on Elliptic Curves in Weierstrass Form

TL;DR

This paper proves that only finitely many elliptic curves in Weierstrass form have three specified points simultaneously torsion, confirming a special case of the relative Manin-Mumford Conjecture.

Contribution

It establishes finiteness of such elliptic curves with three specified torsion points, answering a question by Masser and Zannier.

Findings

Finiteness of elliptic curves with three specified torsion points.

Confirmation of a special case of the relative Manin-Mumford Conjecture.

Provides a new result in the context of torsion points on elliptic curves.

Abstract

We prove that there are only finitely many complex numbers $a$ and $b$ with $4a^3+27b^2\not=0$ such that the three points $(1,*),(2,*),$ and $(3,*)$ are simultaneously torsion on the elliptic curve defined in Weierstrass form by $y^2=x^3+ax+b$. This gives an affirmative answer to a question raised by Masser and Zannier. We thus confirm a special case in two dimensions of the relative Manin-Mumford Conjecture formulated by Pink and Masser-Zannier.