Torsion points on elliptic curves in Weierstrass form

Habegger, Philipp

Geodesic

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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

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Résumé

We prove that there are only finitely many complex numbers $a$ and $b$ with $4 a^{3} + 27 b^{2} \neq 0$ such that the three points $(1, *), (2, *),$ and $(3, *)$ are simultaneously torsion points on the elliptic curve defined in Weierstrass form by $y^{2} = x^{3} + a x + b$ . 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.

Publié le : 2019-02-21

MR Zbl

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

@article{ASNSP_2013_5_12_3_687_0,
     author = {Habegger, Philipp},
     title = {Torsion points on elliptic curves in {Weierstrass} form},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {687--715},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {3},
     year = {2013},
     mrnumber = {3137460},
     zbl = {1281.14026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2013_5_12_3_687_0/}
}

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Habegger, Philipp. Torsion points on elliptic curves in Weierstrass form. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 687-715. http://geodesic.mathdoc.fr/item/ASNSP_2013_5_12_3_687_0/