# ANNALES DE L'INSTITUT FOURIER

Périodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques
Annales de l'Institut Fourier, Volume 43 (1993) no. 3, pp. 585-618.

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

 ${y}^{2}+{A}_{1}xy+{A}_{3}y={x}^{3}+{A}_{2}{x}^{2}+{A}_{4}x+{A}_{6};\phantom{\rule{2em}{0ex}}{A}_{i}\in ℤ.$

Let $M=\left(a/{d}^{2},b/{d}_{3}\right)$, with $\left(a,d\right)=1$, be a rational point on this curve. For every integer $m$, we express the coordinates of $mM$ in the form:

 $mM=\left(\frac{{\varphi }_{m}\left(M\right)}{{\psi }_{n}^{2}\left(m\right)},\frac{{\omega }_{m}\left(M\right)}{{\psi }_{m}^{3}\left(M\right)}\right)=\left(\frac{{\stackrel{^}{\varphi }}_{m}}{{d}^{2}{\stackrel{^}{\psi }}_{m}^{2}},\frac{{\stackrel{^}{\omega }}_{m}}{{d}^{3}{\stackrel{^}{\psi }}_{m}^{3}}\right),$

where ${\varphi }_{m},\psi _m,{\omega }_{m}\in ℤ\left[{A}_{1},\cdots ,{A}_{6},x,y\right]$ and ${\stackrel{^}{\varphi }}_{m}$, ${\stackrel{^}{\psi }}_{m}$, ${\stackrel{^}{\omega }}_{m}$ are obtained from these by multiplying by appropriate powers of $d$.

Let $p$ be a rational odd prime and suppose that $M\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}p\right)$ is non singular and that the rank of apparition of $p$ in the sequence of integer $\left({\stackrel{^}{\psi }}_{m}\right)$ is at least equal to three. Denote this rank by $r=r\left(p\right)$ and let ${\nu }_{p}\left({\stackrel{^}{\psi }}_{r}\right)={e}_{0}\ge 1$. We show that the sequence $\left({\stackrel{^}{\psi }}_{m}\right)$ 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.

Soit $E$ une courbe elliptique sur $ℚ$ par un modèle de Weierstrass généralisé :

 ${y}^{2}+{A}_{1}xy+{A}_{3}y={x}^{3}+{A}_{2}{x}^{2}+{A}_{4}x+{A}_{6};\phantom{\rule{2em}{0ex}}{A}_{i}\in ℤ.$

Soit $M=\left(a/{d}^{2},b/{d}_{3}\right)$ avec $\left(a,d\right)=1$, un point rationnel sur cette courbe. Pour tout entier $m$, on exprime les coordonnées de $mM$ sous la forme :

 $mM=\left(\frac{{\varphi }_{m}\left(M\right)}{{\psi }_{n}^{2}\left(m\right)},\frac{{\omega }_{m}\left(M\right)}{{\psi }_{m}^{3}\left(M\right)}\right)=\left(\frac{{\stackrel{^}{\varphi }}_{m}}{{d}^{2}{\stackrel{^}{\psi }}_{m}^{2}},\frac{{\stackrel{^}{\omega }}_{m}}{{d}^{3}{\stackrel{^}{\psi }}_{m}^{3}}\right),$

${\varphi }_{m},\psi _m,{\omega }_{m}\in ℤ\left[{A}_{1},\cdots ,{A}_{6},x,y\right]$ et ${\stackrel{^}{\varphi }}_{m}$, ${\stackrel{^}{\psi }}_{m}$, ${\stackrel{^}{\omega }}_{m}$ sont déduits par multiplication par des puissances convenables de $d$.

Soit $p$ un nombre premier impair et supposons que $M\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}p\right)$ est non singulier et que le rang d’apparition de $p$ dans la suite d’entiers $\left({\stackrel{^}{\psi }}_{m}\right)$ est supérieur ou égal à trois. Notons ce rang par $r=r\left(p\right)$ et soit ${\nu }_{p}\left({\stackrel{^}{\psi }}_{r}\right)={e}_{0}\ge 1$. Nous montrons que la suite $\left({\stackrel{^}{\psi }}_{m}\right)$ 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.

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title = {P\'eriodicit\'e (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques},
journal = {Annales de l'Institut Fourier},
pages = {585--618},
publisher = {Institut Fourier},
volume = {43},
number = {3},
year = {1993},
doi = {10.5802/aif.1349},
zbl = {0781.11007},
mrnumber = {94f:11009},
language = {fr},
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Ayad, Mohamed. Périodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques. Annales de l'Institut Fourier, Volume 43 (1993) no. 3, pp. 585-618. doi : 10.5802/aif.1349. https://aif.centre-mersenne.org/articles/10.5802/aif.1349/

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