Complete solutions to a family of Thue equations of degree 12

Hoshi, Akinari

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 549-568.

We considérons une famille paramétrique non galoisienne d’équations de Thue $F_m(x,y)=\lambda$ de degré $12$ où $m$ est un paramètre entier et où $\lambda$ est un diviseur de $729(m^2+3m+9)$. En utilisant la méthode d’isomorphismes de corps développée dans [15], nous montrons que ces équations ont seulement des solutions triviales avec $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

We consider a parametric non-Galois family of Thue equations $F_m(x,y)=\lambda$ of degree $12$ where $m$ is an integral parameter and $\lambda$ is a divisor of $729(m^2+3m+9)$. Using the field isomorphism method which is developed in [15], we show that the equations have only the trivial solutions with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

Classification : 11D25, 11D41, 11R16, 11R20, 12F10
Mots clés : Thue equations, simplest cubic fields, simplest sextic fields.

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