C. Levesque - Explicit solutions of a family of Thue diophantine equations

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C. Levesque - Explicit solutions of a family of Thue diophantine equations

C. LEVESQUE, Département de mathématiques et statistique, Université de Laval, Montréal, Québec G1K 7P4, Canada

Explicit solutions of a family of Thue diophantine equations

Looking for units of certain number fields built from modular coverings X₀(m), H. Darmon obtained the family of polynomials

F_a(X,Y)= X⁵+2X⁴Y+(a+3)X³Y²+(2a+3)X²Y³+(a+1)XY⁴-Y⁵.

For a sufficiently large, O. Kihel exhibited a fundamental system of units of the field ${\bf Q}(\omega)$ where $\omega$ is a root of F_a(X,1)=0. In this lecture we will show that for a sufficiently large, the family of Thue diophantine equations of degree 5 given by

$\begin{displaymath}F_a(X,Y)=\pm1 \end{displaymath}$

has only the six trivial solutions (x,y)=(0,1),(0,-1), (1,0), (-1,0), (1,-1), (-1,1). The associated Siegel equation leads firstly to a linear form in four logarithms, but the main ingredient and the feature of the proof is to write it as a linear form in two logarithms.

Next: Kumar Murty - Zeros Up: Number Theory / Théorie Previous: Arne Ledet - Some