Mon premier blog

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

Akhil Mathew - August 17, 2009. It had long been known that the rational points on an elliptic curve, defined over the rationals, form a group Γ under a chord and tangent construction; Mordell proved that Γ has a finite basis. There is no integral solution (x,y,z) to x^4 + y^4 = z^4 satisfying xyz \neq 0. In Chapter 1: Rational Points on Elliptic Curves, the authors state two propositions: Proposition 1.1. This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E',C,C',φ), where E and E' are elliptic curves, C and C' are cyclic 13-subgroups, and φ is an isomorphism between C and C'. 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). The first thing that we should do here is to reduce this equation to the Weierstrass normal form. A very good book written on the subject is "Rational points on Elliptic Curves" by Silverman and Tate. In the elliptic curve E: y^2+y=x^3-x , the rational points form a group of rank 1 (i.e., an infinite cyclic group), and can be generated by P =(0,0) under the group law. Who tells the story in the first half of the book narrates how a young volunteer came up to him and Rational Points on Elliptic Curves - Google Books This book stresses this interplay as it develops the basic theory,. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. If you're interested in algebraic geometry from an elementary point of view, Tate and Silverman's Rational Points on Elliptic Curves is also worth checking out. Some sample rational points are shown in the following graph.