Effective results for division points on curves in ${𝔾}_m^2$

Bérczes, Attila

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 405-437.

Soient $A:=\mathbb{Z}[z_1,\cdots ,z_r]\supset \mathbb{Z}$ un anneau de type fini sur $\mathbb{Z}$, $K$ son corps de fractions et $K^{*}$ le groupe multiplicatif des éléments non nuls de $K$. Soit $\Gamma$ un sous-groupe de type fini de $K^{*}$ et soit $\overline{\Gamma }$ le groupe de division de $\Gamma$. Soit $F(X,Y)\in A[X,Y]$ un pôlynome. En 1974, P. Liardet a prouvé que, sous certaines conditions naturelles, l’équation

$$F(x,y)=0\text{avec}x,y\in \overline{\Gamma }$$

n’admet qu’un nombre fini de solutions. La démonstration de Liardet est ineffective. En 2009, une variante effective du théorème de Liardet a été démontrée par Bérczes, Evertse, Győry and Pontreau dans le cas $\Gamma \subset \overline{\mathbb{Q}}$. Dans cet article une variante effective du théorème de Liardet est prouvée en toute generalité.

Let $A:=\mathbb{Z}[z_1,\cdots ,z_r]\supset \mathbb{Z}$ be a finitely generated integral domain over $\mathbb{Z}$, let $K$ denote its quotient field, and $K^{*}$ the multiplicative group of non-zero elements of $K$. Let $\Gamma$ be a finitely generated subgroup of $K^{*}$, and let $\overline{\Gamma }$ denote the division group of $\Gamma$. Let $F(X,Y)\in A[X,Y]$ be a polynomial. In 1974 P. Liardet proved that under some natural conditions on $F$ the equation

$$F(x,y)=0\text{with}x,y\in \overline{\Gamma }$$

has only finitely many solutions. The proof of Liardet was ineffective. In 2009 an effective version of Liardet’s Theorem has been proved by Bérczes, Evertse, Győry and Pontreau in the case when $\Gamma \subset \overline{\mathbb{Q}}$. In the present paper an effective version of Liardet’s Theorem is proved in the general case.

Classification : 11G35, 11G50, 11D99, 14G25
Mots clés : effective results, Diophantine equations, curves, division group of finitely generated groups

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