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

Ayad, Mohamed

Périodicité (mod

q

) des suites elliptiques et points

S

-entiers sur les courbes elliptiques

Ayad, Mohamed

Annales de l'Institut Fourier, Tome 43 (1993), p. 585-618 / Harvested from Numdam

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Résumé
Abstract

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

y^{2} + A_{1} x y + A_{3} y = x^{3} + A_{2} x^{2} + A_{4} x + A_{6}; A_{i} \in 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 $m M$ sous la forme :

m M = (\frac{ϕ_{m} (M)}{ψ_{n}^{2} (m)}, \frac{ω_{m} (M)}{ψ_{m}^{3} (M)}) = (\frac{{\hat{ϕ}}_{m}}{d^{2} {\hat{ψ}}_{m}^{2}}, \frac{{\hat{ω}}_{m}}{d^{3} {\hat{ψ}}_{m}^{3}}),

où $ϕ_{m}, ψ_m, ω_{m} \in ℤ [A_{1}, \dots, A_{6}, x, y]$ et ${\hat{ϕ}}_{m}$ , ${\hat{ψ}}_{m}$ , ${\hat{ω}}_{m}$ sont déduits par multiplication par des puissances convenables de $d$ .

Soit $p$ un nombre premier impair et supposons que $M (\mod p)$ est non singulier et que le rang d’apparition de $p$ dans la suite d’entiers $({\hat{ψ}}_{m})$ est supérieur ou égal à trois. Notons ce rang par $r = r (p)$ et soit $ν_{p} ({\hat{ψ}}_{r}) = e_{0} \geq 1$ . Nous montrons que la suite $({\hat{ψ}}_{m})$ est périodique (mod $p^{N}$ ) pour tout $N \geq 1$ . Notons cette période par $π_{N}$ , alors il existe un rang $N_{1}$ effectivement calculable, avec $1 \leq N_{1} \leq e_{0}$ , tel que $π_{1} = \dots = π_{N_{1}}$ et $π_{N + 1} = p π_{N}$ pour $N \geq N_{1}$ . Ces considérations sont utilisées pour déterminer les points $S$ -entiers sur les courbes elliptiques.

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

y^{2} + A_{1} x y + A_{3} y = x^{3} + A_{2} x^{2} + A_{4} x + A_{6}; A_{i} \in 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 $m M$ in the form:

m M = (\frac{ϕ_{m} (M)}{ψ_{n}^{2} (m)}, \frac{ω_{m} (M)}{ψ_{m}^{3} (M)}) = (\frac{{\hat{ϕ}}_{m}}{d^{2} {\hat{ψ}}_{m}^{2}}, \frac{{\hat{ω}}_{m}}{d^{3} {\hat{ψ}}_{m}^{3}}),

where $ϕ_{m}, ψ_m, ω_{m} \in ℤ [A_{1}, \dots, A_{6}, x, y]$ and ${\hat{ϕ}}_{m}$ , ${\hat{ψ}}_{m}$ , ${\hat{ω}}_{m}$ are obtained from these by multiplying by appropriate powers of $d$ .

Let $p$ be a rational odd prime and suppose that $M (\mod p)$ is non singular and that the rank of apparition of $p$ in the sequence of integer $({\hat{ψ}}_{m})$ is at least equal to three. Denote this rank by $r = r (p)$ and let $ν_{p} ({\hat{ψ}}_{r}) = e_{0} \geq 1$ . We show that the sequence $({\hat{ψ}}_{m})$ is periodic (mod $p^{N}$ ) for every $N \geq 1$ . Denote this period by $Π_{N}$ , then there exists a rank $N_{1}$ effectively computable, $1 \leq N_{1} \leq e_{0}$ , such that $π_{1} = \dots = π_{N_{1}}$ and $π_{N + 1} = p π_{N}$ for $N \geq N_{1}$ . These considerations are used to find $S$ -integral points on elliptic curves.

Publié le : 1993-01-01

MR 94f:11009 | Zbl 0781.11007

DOI : https://doi.org/10.5802/aif.1349

@article{AIF_1993__43_3_585_0,
     author = {Ayad, Mohamed},
     title = {P\'eriodicit\'e (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques},
     journal = {Annales de l'Institut Fourier},
     volume = {43},
     year = {1993},
     pages = {585-618},
     doi = {10.5802/aif.1349},
     mrnumber = {94f:11009},
     zbl = {0781.11007},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_3_585_0}
}

Ayad, Mohamed. Périodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques. Annales de l'Institut Fourier, Tome 43 (1993) pp. 585-618. doi : 10.5802/aif.1349. http://gdmltest.u-ga.fr/item/AIF_1993__43_3_585_0/

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