Torsion anomalous points and families of elliptic curves

David Masser; Umberto Zannier

doi:10.1016/j.crma.2008.03.024

Number Theory

Torsion anomalous points and families of elliptic curves

Presented by: Jean-Pierre Serre

David Masser ¹; Umberto Zannier ²

¹ Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
² Scuola Normale, Piazza Cavalieri 7, 56126 Pisa, Italy

Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 491-494.

Abstract (Original language)
Résumé

We prove that there are at most finitely many complex $λ \neq 0, 1$ such that two points on the Legendre elliptic curve $Y^{2} = X (X - 1) (X - λ)$ with coordinates $X = 2$ and $X = 3$ both have finite order. This is a very special case of some well-known conjectures on unlikely intersections with varying semiabelian varieties.

Comme cas très spécial de certaines conjectures générales sur l'intersection d'une variété algébrique avec la réunion des sous-schémas de dimension fixée d'un schéma semi-abélien, nous montrons qu'il n'existe qu'un nombre fini de $λ \in C ∖ {0, 1}$ tels que les quatre points de la courbe elliptique $Y^{2} = X (X - 1) (X - λ)$ avec $X = 2$ et $X = 3$ soient d'ordre fini.

Received: 2008-02-14
Accepted: 2008-03-25
Published online: 2008-04-21

DOI: 10.1016/j.crma.2008.03.024

Author's affiliations:

David Masser ¹; Umberto Zannier ²

¹ Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
² Scuola Normale, Piazza Cavalieri 7, 56126 Pisa, Italy

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David Masser; Umberto Zannier. Torsion anomalous points and families of elliptic curves. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 491-494. doi : 10.1016/j.crma.2008.03.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.024/

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