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

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

These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve. [math.NT/0606003] We consider the structure of rational points on elliptic curves in Weierstrass form. 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'. Graphs of curves y2 = x3 − x and y2 = x3 − x + 1. Or: the rational points on an elliptic curve have an enormous amount of deep structure, of course, starting with the basic fact that they form a finite rank abelian group. Wei Ho delivered a very Ho talked about how Bhargava and his school are approaching different conjectures on the ranks of elliptic curves. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. Hey, now we know that this is a question in arithmetic statistics! Elliptic Curves, Modular Forms,. In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. 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. Rational Points on Modular Elliptic Curves book download Download Rational Points on Modular Elliptic Curves Request a Print Examination Copy. 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). After a nice work lunch with two of my soon-to-be collaborators, I attended Wei Ho's talk in the Current Events Bulletin on “How many rational points does a random curve have?”.