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Elliptic divisibility sequences over certain curves. (English) Zbl 1170.11010

A divisibility sequence is a sequence of integers \(C=\{C_n\}_{n\geq 1}\) such that \(C_n\mid C_m\) whenever \(n\mid m\). The paper deals with divisibility sequences arising from non-torsion rational points of an elliptic curve \(E\): namely if \(P\in E(\mathbb{Q})\) is a point of infinite order, one puts \(x(nP)=\frac{A_n}{B_n}\) (in reduced form and with \(B_n>0\)) and \(B(E,P)=\{B_n\}_{n\geq 1}\) is an elliptic divisibility sequence. A prime divisor \(p\) of \(B_n\) is called a primitive divisor if \(p\nmid B_m\) for any \(m<n\). The goal of the paper is to give informations (and upper bounds) for the Zsigmondy bound \(Z(B(E,P))=\sup\{m: B_m \) has no primitive divisor\(\}\) when \(E\) has \(j\)-invariant 0 or 1728.
Mainly by direct computations the author shows that if \(B_n\) has no primitive divisor then \(x(P)=\frac{a}{b^2}\) is associated to a solution of an explicit system of Thue equations depending on the division polynomials of the elliptic curve. Solving these equations for a fixed \(n\) provides the list of all curves \(E\) and points \(P\) such that \(B_n\) has no primitive divisor.
In the case of \(E: y^2=x^3-N^2x\) with squarefree \(N\) these computations allow the author to sharpen some of the bounds obtained by G. Everest, G. Mclaren and T. Ward [J. Number Theory 118, No. 1, 71–89 (2006; Zbl 1093.11038)].

MSC:

11G05 Elliptic curves over global fields
14G05 Rational points
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 1093.11038

Software:

PARI/GP
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Full Text: DOI

References:

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