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Mohamed AyadPériodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiquesAnnales de l'institut Fourier, 43 no. 3 ( 1993), p. 585-618, doi: 10.5802/aif.1349
Article PDF | Reviews MR 94f:11009 | Zbl 0781.11007
Let $E$ be an elliptic curve defined over ${\Bbb Q}$ by a generalized Weierstrass equation:
$$y^2+A_1 xy+A_3 y= x^3+A_2 x^2+ A_4 x+A_6; \qquad A_i\in {\Bbb Z}.$$
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:
$$mM= \left( {{\varphi _m(M)} \over {\psi ^ 2_n(m)}}, {{\omega _m(M)} \over {\psi ^ 3_m(M)}} \right)= \left( {{\widehat{\varphi }_m} \over {d^2\widehat{\psi }^2_ m}}, {{\widehat{\omega }_m} \over {d^3 \widehat{\psi }^3_ m}} \right),$$
where $\varphi _m, \psi \_ m, \omega _ m\in {\Bbb Z}[A_1, \dots , A_6,x,y]$ 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 $M~({\rm mod}\,p)$ is non singular and that the rank of apparition of $p$ in the sequence of integer $(\widehat{\psi }_m)$ is at least equal to three. Denote this rank by $r=r(p)$ and let $\nu _ p(\widehat{\psi }_r)=e_0\ge 1$. We show that the sequence $(\widehat{\psi }_m)$ 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 $\pi _1=\dots =\pi _{N_1}$ 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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