Rational points on elliptic curves

By G. Everest, J. Reynolds, Shaun Stevens. 2006

359 citations recorded by the source corpus.

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an explicit fashion, and that they are effectively computable.

Subfields: Algebra and Number Theory, Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e408e09dcc1dcaf34b20e70021402be3fe7e5754

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. D. Morrison
2. Washington Taylor
3. J. Heckman
4. Tom Rudelius
5. C. Vafa
6. Yi-Nan Wang

In Computational Mechanics

1. Scott A. Vanstone
2. Alfred J. Menezes
3. Darrel Hankerson

In Geometry and Topology

1. Washington Taylor
2. D. Morrison
3. Scott A. Vanstone
4. Alfred J. Menezes
5. Darrel Hankerson
6. J. Heckman
7. Tom Rudelius
8. C. Vafa

Citations in the corpus

• Introduction to Modern Cryptography, Second Edition (2014, neutral): “The book by Silverman and Tate [160] is perhaps an exception.”
• A normal form for elliptic curves (2007, neutral): “See [7], II, 1, § 8. Also, see [10], III, 4; and [ 13 ], I, 4. 4 HAROLD M. EDWARDS”
• Guide to Elliptic Curve Cryptography (2004, background): “Intermediate-level textbooks that provide proofs of many of the basic results used in elliptic curve cryptography include Charlap and Robbins [92, 93], Enge [132], Silverman and Tate [433], and Washington [474].”
• Primitive divisors of elliptic divisibility sequences (2004, background): “Let E denote an elliptic curve defined over Q, given in generalized Weierstrass form, and suppose P = (x(P ), y(P )) denotes a non-torsion rational point on E (see [5], [8], [12] or [15] for background on elliptic curves).”
• Matter and singularities (2011, background): “This implies by the usual geometric law of addition on elliptic curves [36] that (0, 0) is a point of order 2 in the group law on each elliptic curve, so that the corresponding section defines a point of order two in the Mordell–Weil group.”