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Rational points on elliptic curves John Tate, Joseph H. Silverman
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. By introducting a special point O (point is a rational function. Rational Points on Elliptic Curves John Tate (Auteur), J.H. Graphs of curves y2 = x3 − x and y2 = x3 − x + 1. E is just a set of points fulfilling an equation that is quadratic in terms of y and cubic in x . P_t=(2,p_t),\quad Q_t=(3,q_t These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Order of a pole is similar: b is a pole of order n if n is the largest integer, such that r(x)=\frac{s(x)}{(x-b . The most general definition of an elliptic curve, is. The problem is therefore reduced to proving some curve has no rational points. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. Whose rational points are precisely isomorphism classes of elliptic curves over {{\mathbb Q}} together with a rational point of order 13.