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
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/

References
Cited by

[1] E. Bombieri; D. Masser; U. Zannier Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Notices, Volume 20 (1999), pp. 1119-1140

[2] E. Bombieri; J. Pila The number of integral points on arcs and ovals, Duke Math. J., Volume 59 (1989), pp. 337-357

[3] S. David Points de petite hauteur sur les courbes elliptiques, J. Number Theory, Volume 64 (1997), pp. 104-129

[4] D. Husemöller Elliptic Curves, Springer-Verlag, 1987

[5] D. Masser Small values of the quadratic part of the Néron–Tate height on an abelian variety, Compositio Math., Volume 53 (1984), pp. 153-170

[6] D. Masser Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc., Volume 311 (1989), pp. 413-424

[7] D. Masser Counting points of small height on elliptic curves, Bull. Soc. Math. France, Volume 117 (1989), pp. 247-265

[8] J. Pila Integer points on the dilation of a subanalytic surface, Quart. J. Math., Volume 55 (2004), pp. 207-223

[9] J. Pila; A. Wilkie The rational points of a definable set, Duke Math. J., Volume 33 (2006), pp. 591-616

[10] J. Pila, U. Zannier, Rational points in periodic analytic sets and the Manin–Mumford conjecture, Rend. Lincei Mat. Appl. (RML), in press

[11] R. Pink, A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, manuscript dated 17th April 2005 (13 pages)

[12] J.H. Silverman Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math., Volume 342 (1983), pp. 197-211

[13] B. Zilber Exponential sums equations and the Schanuel conjecture, J. London Math. Soc., Volume 65 (2002), pp. 27-44

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