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

Claude Levesque; Michel Waldschmidt

doi:10.5802/jtnb.913

Claude Levesque ¹ ; Michel Waldschmidt ²

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.

Résumé
Abstract

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 $F_{n} (X, Y) = \pm 1$ , $n \in ℕ$ . Cette famille est associée à la famille des corps cubiques les plus simples $ℚ (λ)$ de D. Shanks, $λ$ 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 $λ$ 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 $F_{n} (X, Y) = \pm 1$ , $n \in ℕ$ . This family is associated to the family of the simplest cubic fields $ℚ (λ)$ of D. Shanks, $λ$ 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 $λ$ 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 $a$ .

MR

DOI : 10.5802/jtnb.913

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

Affiliations des auteurs :

Claude Levesque ¹ ; Michel Waldschmidt ²

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

@article{JTNB_2015__27_2_537_0,
     author = {Claude Levesque and Michel Waldschmidt},
     title = {A family of {Thue} equations involving powers of units of the simplest cubic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {537--563},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     doi = {10.5802/jtnb.913},
     mrnumber = {3393166},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/}
}

TY  - JOUR
AU  - Claude Levesque
AU  - Michel Waldschmidt
TI  - A family of Thue equations involving powers of units of the simplest cubic fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2015
SP  - 537
EP  - 563
VL  - 27
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/
DO  - 10.5802/jtnb.913
LA  - en
ID  - JTNB_2015__27_2_537_0
ER  -

%0 Journal Article
%A Claude Levesque
%A Michel Waldschmidt
%T A family of Thue equations involving powers of units of the simplest cubic fields
%J Journal de théorie des nombres de Bordeaux
%D 2015
%P 537-563
%V 27
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/
%R 10.5802/jtnb.913
%G en
%F JTNB_2015__27_2_537_0

Claude Levesque; Michel Waldschmidt. A family of Thue equations involving powers of units of the simplest cubic fields. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 537-563. doi : 10.5802/jtnb.913. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/

Bibliographie
Cité par

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

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

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

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

Cité par Sources :