Rational points on elliptic curves by John Tate, Joseph H. Silverman

Rational points on elliptic curves



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Rational points on elliptic curves John Tate, Joseph H. Silverman ebook
Format: djvu
Page: 296
ISBN: 3540978259, 9783540978251
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K


The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. Smyth, Minimal polynomials of algebraic numbers with rational parameters. 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). From the formula for doubling a point we get that. Here's what this looks like: Image001. Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . Theorem 5 (on page vi) of Diem's thesis states that the discrete logarithm problem in the group of rational points of an elliptic curves E( F_{p^n} ) can be solved in an expected time of \tilde{O}( q^{2 – 2/n} ) bit operations. The only rational solution of which is x = 0. Silverman, Lehmer's Conjecture and points on elliptic curves that are congruent to torsion points. Thich corresponds to the points (0,1) and (0,-1) on the elliptic curve.