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

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

y2+A1xy+A3y=x3+A2x2+A4x+A6;Ai.

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=ϕm(M)ψn2(m),ωm(M)ψm3(M)=ϕ^md2ψ^m2,ω^md3ψ^m3,

where ϕ m ,ψ_m,ω m [A 1 ,,A 6 ,x,y] and ϕ ^ m , ψ ^ m , 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 (ψ ^ m ) is at least equal to three. Denote this rank by r=r(p) and let ν p (ψ ^ r )=e 0 1. We show that the sequence (ψ ^ m ) is periodic (mod p N ) for every N1. Denote this period by Π N , then there exists a rank N 1 effectively computable, 1N 1 e 0 , such that π 1 ==π N 1 and π N+1 =pπ N for NN 1 . These considerations are used to find S-integral points on elliptic curves.

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

y2+A1xy+A3y=x3+A2x2+A4x+A6;Ai.

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=ϕm(M)ψn2(m),ωm(M)ψm3(M)=ϕ^md2ψ^m2,ω^md3ψ^m3,

ϕ m ,ψ_m,ω m [A 1 ,,A 6 ,x,y] et ϕ ^ m , ψ ^ m , 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 (ψ ^ m ) est supérieur ou égal à trois. Notons ce rang par r=r(p) et soit ν p (ψ ^ r )=e 0 1. Nous montrons que la suite (ψ ^ m ) est périodique (mod p N ) pour tout N1. Notons cette période par π N , alors il existe un rang N 1 effectivement calculable, avec 1N 1 e 0 , tel que π 1 ==π N 1 et π N+1 =pπ N pour NN 1 . Ces considérations sont utilisées pour déterminer les points S-entiers sur les courbes elliptiques.

@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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {3},
     year = {1993},
     pages = {585-618},
     doi = {10.5802/aif.1349},
     zbl = {0781.11007},
     mrnumber = {94f:11009},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1993__43_3_585_0}
}
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. http://www.numdam.org/item/AIF_1993__43_3_585_0/

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