Périodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques

Ayad, Mohamed

Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 585-618.

Soit $E$ une courbe elliptique sur $\mathbb{Q}$ par un modèle de Weierstrass généralisé :

$$y^2+A_{1}xy+A_{3}y=x^3+A_2{x}^2+A_{4}x+A_6;A_i\in \mathbb{Z}.$$

Soit $M=(a/d^2,b/d_3)$ avec $(a,d)=1$, un point rationnel sur cette courbe. Pour tout entier $m$, on exprime les coordonnées de $mM$ sous la forme :

$$mM=\left(\frac{{\varphi }_m\left(M\right)}{{\psi }_n^2\left(m\right)},\frac{{\omega }_m\left(M\right)}{{\psi }_m^3\left(M\right)}\right)=\left(\frac{{\widehat{\varphi }}_m}{d^2{\widehat{\psi }}_m^2},\frac{{\widehat{\omega }}_m}{d^3{\widehat{\psi }}_m^3}\right),$$

où ${\varphi }_m,\psi \_m,{\omega }_m\in \mathbb{Z}[A_1,\cdots ,A_6,x,y]$ et ${\widehat{\varphi }}_m$, ${\widehat{\psi }}_m$, ${\widehat{\omega }}_m$ sont déduits par multiplication par des puissances convenables de $d$.

Soit $p$ un nombre premier impair et supposons que $M\left(\mathrm{mod}p\right)$ est non singulier et que le rang d’apparition de $p$ dans la suite d’entiers $\left({\widehat{\psi }}_m\right)$ est supérieur ou égal à trois. Notons ce rang par $r=r\left(p\right)$ et soit ${\nu }_p({\widehat{\psi }}_r)=e_0\ge 1$. Nous montrons que la suite $\left({\widehat{\psi }}_m\right)$ est périodique (mod $p^N$) pour tout $N\ge 1$. Notons cette période par ${\pi }_N$, alors il existe un rang $N_1$ effectivement calculable, avec $1\le N_1\le e_0$, tel que ${\pi }_1=\cdots ={\pi }_{N_1}$ et ${\pi }_{N+1}=p{\pi }_N$ pour $N\ge N_1$. Ces considérations sont utilisées pour déterminer les points $S$-entiers sur les courbes elliptiques.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by a generalized Weierstrass equation:

$$y^2+A_{1}xy+A_{3}y=x^3+A_2{x}^2+A_{4}x+A_6;A_i\in \mathbb{Z}.$$

Let $M=(a/d^2,b/d_3)$, with $(a,d)=1$, be a rational point on this curve. For every integer $m$, we express the coordinates of $mM$ in the form:

$$mM=\left(\frac{{\varphi }_m\left(M\right)}{{\psi }_n^2\left(m\right)},\frac{{\omega }_m\left(M\right)}{{\psi }_m^3\left(M\right)}\right)=\left(\frac{{\widehat{\varphi }}_m}{d^2{\widehat{\psi }}_m^2},\frac{{\widehat{\omega }}_m}{d^3{\widehat{\psi }}_m^3}\right),$$

where ${\varphi }_m,\psi \_m,{\omega }_m\in \mathbb{Z}[A_1,\cdots ,A_6,x,y]$ and ${\widehat{\varphi }}_m$, ${\widehat{\psi }}_m$, ${\widehat{\omega }}_m$ are obtained from these by multiplying by appropriate powers of $d$.

Let $p$ be a rational odd prime and suppose that $M\left(\mathrm{mod}p\right)$ is non singular and that the rank of apparition of $p$ in the sequence of integer $\left({\widehat{\psi }}_m\right)$ is at least equal to three. Denote this rank by $r=r\left(p\right)$ and let ${\nu }_p({\widehat{\psi }}_r)=e_0\ge 1$. We show that the sequence $\left({\widehat{\psi }}_m\right)$ is periodic (mod $p^N$) for every $N\ge 1$. Denote this period by ${\Pi }_N$, then there exists a rank $N_1$ effectively computable, $1\le N_1\le e_0$, such that ${\pi }_1=\cdots ={\pi }_{N_1}$ and ${\pi }_{N+1}=p{\pi }_N$ for $N\ge N_1$. These considerations are used to find $S$-integral points on elliptic curves.

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