A family of Thue equations involving powers of units of the simplest cubic fields

Levesque, Claude; Waldschmidt, Michel

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 537-563.

E. Thomas fut l’un des premiers à résoudre une famille infinie d’équations de Thue, lorsqu’il a considéré les formes $F_n(X,Y)=X^3-(n-1)X^{2}Y-(n+2)X{Y}^2-Y^3$ et la famille d’équations $F_n(X,Y)=\pm 1$, $n\in \mathbb{N}$. Cette famille est associée à la famille des corps cubiques les plus simples $\mathbb{Q}\left(\lambda \right)$ de D. Shanks, $\lambda$ étant une racine de $F_n(X,1)$. Nous introduisons dans cette famille un second paramètre en remplaçant les racines du polynôme minimal $F_n(X,1)$ de $\lambda$ par les puissances $a$-ièmes des racines et nous résolvons de façon effective la famille d’équations de Thue que nous obtenons et qui dépend maintenant des deux paramètres $n$ et $a$.

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X,Y)=X^3-(n-1)X^{2}Y-(n+2)X{Y}^2-Y^3$ and the family of equations $F_n(X,Y)=\pm 1$, $n\in \mathbb{N}$. This family is associated to the family of the simplest cubic fields $\mathbb{Q}\left(\lambda \right)$ of D. Shanks, $\lambda$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X,1)$ of $\lambda$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters $n$ and $a$.

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