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

Levesque, Claude¹ ; Waldschmidt, Michel²

¹ Dép. de mathématiques et de statistique Université Laval, Québec Québec CANADA G1V 0A6
² UPMC Univ Paris 06 UMR 7586-IMJ 75005 Paris France

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

Abstract (VO)

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 $a$.

Résumé

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 $a$.

Détail

DOI : 10.5802/jtnb.913

Classification : 11D61, 11D41, 11D59
Keywords: Simplest cubic fields, family of Thue equations, diophantine equations, linear forms of logarithms.

Bibliographie

[1] J.H. Chen, Efficient rational approximation of a class of algebraic numbers, (in Chinese) Chinese Science Bulletin, 41 (1996), 1643–1646. | MR

[2] G. Lettl, A. Pethő, and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc., 351 (1999), 1871–1894. | Zbl | MR

[3] C. Levesque and M. Waldschmidt, Families of cubic Thue equations with effective bounds for the solutions Springer Proceedings in Mathematics & Statistics 43 (2013) 229—243. | MR

[4] C. Levesque and M. Waldschmidt, Solving simultaneously Thue Diophantine equations: almost totally imaginary case, Proceedings of the International Meeting on Number Theory 2011, in honor of R. Balasubramanian, Lecture Notes Series in Ramanujan Mathematical Society, to appear. | MR

[5] C. Levesque and M. Waldschmidt, Solving effectively some families of Thue Diophantine equations, Moscow J. of Combinatorics and Number Theory, 3 3–4 (2013), 118–144. | MR

[6] C. Levesque and M. Waldschmidt, Families of Thue equations associated with a totally real rank $1$ subgroup of the units of a number field. Work in progress.

[7] M. Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory, 44 (1993), 172–177. | Zbl | MR

[8] M. Mignotte, A. Pethő and F. Lemmermeyer, On the family of Thue equations $x^3-(n-1)x^{2}y-(n+2)x{y}^2-y^3=k$, Acta Arith. 76 (1996), 245–269. | Zbl | MR

[9] T.N. Shorey and R. Tijdeman, Exponential Diophantine equations, vol. 87 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986. | Zbl | MR

[10] E. Thomas, Complete solutions to a family of cubic Diophantine equations, J. Number Theory, 34 (1990), pp. 235–250. | Zbl | MR

[11] I. Wakabayashi, Simple families of Thue inequalities, Ann. Sci. Math. Québec 31 (2007), no. 2, 211–232. | Zbl | MR