Bicyclotomic polynomials and impossible intersections

David Masser; Umberto Zannier

doi:10.5802/jtnb.851

David Masser ¹; Umberto Zannier ²

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

Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 635-659.

Abstract (Original language)
Résumé

In a recent paper we proved that there are at most finitely many complex numbers $t \neq 0, 1$ such that the points $(2, \sqrt{2 (2 - t)})$ and $(3, \sqrt{6 (3 - t)})$ are both torsion on the Legendre elliptic curve defined by $y^{2} = x (x - 1) (x - t)$ . In a sequel we gave a generalization to any two points with coordinates algebraic over the field $Q (t)$ and even over $C (t)$ . Here we reconsider the special case $(u, \sqrt{u (u - 1) (u - t)})$ and $(v, \sqrt{v (v - 1) (v - t)})$ with complex numbers $u$ and $v$ .

Nous avons déjà démontré qu’il n’existe qu’un nombre fini de nombres complexes $t \neq 0, 1$ tels que les points $(2, \sqrt{2 (2 - t)})$ et $(3, \sqrt{6 (3 - t)})$ soient d’ordre fini sur la courbe elliptique de Legendre définie par $y^{2} = x (x - 1) (x - t)$ . Nous avons généralisé ensuite ce résultat aux couples de points algébriques quelconques sur $C (t)$ . Nous revenons ici aux points $(u, \sqrt{u (u - 1) (u - t)})$ et $(v, \sqrt{v (v - 1) (v - t)})$ avec des nombres complexes $u$ et $v$ quelconques.

MR Zbl

DOI: 10.5802/jtnb.851

Classification: 11G05, 14H52

Author's affiliations:

David Masser ¹; Umberto Zannier ²

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

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David Masser; Umberto Zannier. Bicyclotomic polynomials and impossible intersections. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 635-659. doi : 10.5802/jtnb.851. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.851/

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