Multiples of Points on Elliptic Curves and Continued Fractions

Abstract

-6ax*-8bx + c), (1.1)with respec tto an ar^adic valuation an,d the geometry of the ellipticcurve(1.2)where we assume that a,b,c belon to ag field k. The curve (1.2) has tworational points 0 and P at infinity. Takin 0g as the origin of the Mordell-Weil grou op f (1.2) w, e find that the multiples of P in thi s grou arpeintimately connected with the convergents and complete quotients of thecontinued fraction o y.fAlthough our arguments are purely algebraic w,e are able to recovertheorems of Abel [1] and Chebychev [3], prove byd analysing the periodsof integrals, to the effect that the continued fraction of y is periodi if ancdonly if the point P is of finite order. Ther ise a very simple relationshipbetween the period of y and the order o P.f The results we prove are validover any field and can be extende tdo curves defined ove ar ring (such asthe rational integers Z) I. n this way they can be applied to the reductionmod various primes of an elliptic curve defined ove Z orr other rings.We now describe our notation. Let C be any curve define kd ove withra non-singular ^-rationa 0. Lel point = K k(C)t be the correspondingfunction field and denote the valuation correspondin to 0 b ordgy