Torsion Points on Elliptic Curves in Weierstrass Form

Philipp Habegger

Torsion Points on Elliptic Curves in Weierstrass Form

Number Theory 2011-05-24 v1

Authors: Philipp Habegger

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.

Keywords

torsion points elliptic curves Weierstrass form Manin-Mumford conjecture Masser-Zannier problem finiteness

Cite

@article{arxiv.1105.4435,
  title  = {Torsion Points on Elliptic Curves in Weierstrass Form},
  author = {Philipp Habegger},
  journal= {arXiv preprint arXiv:1105.4435},
  year   = {2011}
}

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