Bicyclotomic polynomials and impossible intersections
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 635-659.

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

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

DOI: 10.5802/jtnb.851
Classification: 11G05, 14H52
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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