Bicyclotomic polynomials and impossible intersections

Masser, David; Zannier, Umberto

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 $(2,\sqrt{2(2-t)})$ 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 $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. 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 $t\ne 0,1$ 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 $\mathbf{C}\left(t\right)$. 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.

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