Rational points on elliptic curves ebook download

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

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). Wei Ho delivered a very Ho talked about how Bhargava and his school are approaching different conjectures on the ranks of elliptic curves. Silverman, Joseph H., Tate, John, Rational Points on Elliptic Curves, 1992 63. The book surveys some recent developments in the arithmetic of modular elliptic curves. Theorem (Uniform Boundedness Theorem).Let K be a number field of degree d . Silverman, John Tate, Rational Points on Elliptic Curves, Springer 1992. Devlin, Keith, The Joy of Sets – Fundamentals of Contemporary Set Theory, 1993 64. 106, Springer 1986; Advanced Topics in the Arithmetic of Elliptic Curves Graduate Texts in Mathl. 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. Then there is a constant B(d) depending only on d such that, if E/K is an elliptic curve with a K -rational torsion point of order N , then N < B(d) . Kinsey, L.Christine, Topology of Surfaces, 1993 65. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. 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?”. Hey, now we know that this is a question in arithmetic statistics! 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.