Torsion points on elliptic curves in Weierstrass form

Habegger, Philipp

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 687-715.

We prove that there are only finitely many complex numbers $a$ and $b$ with $4a^3+27b^2\ne 0$ such that the three points $(1,*),(2,*),$ and $(3,*)$ are simultaneously torsion points 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.

Classification : 14H52, 14G40, 11G05, 11U09

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