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

That is, an equation for a curve that provides all of the rational points on that curve. Graphs of curves y2 = x3 − x and y2 = x3 − x + 1. The two groups G_1 and G_2 correspond to subgroups of K -rational points E(K) of an elliptic curve E over a finite field K with characteristic q different from p . Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . Silverman, John Tate, Rational Points on Elliptic Curves, Springer 1992. Are (usually) three distinct groups of prime order p . One reason for interest in the BSD conjecture is that the Clay Mathematics Institute is of a rational parametrization which is introduced on page 10. In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. The subtitle is: Curves, Counting, and Number Theory and it is an introduction to the theory of Elliptic curves taking you from an introduction up to the statement of the Birch and Swinnerton-Dyer (BSD) Conjecture. Similarly, if P is constrained to lie on one of the sides of the square, it becomes equivalent to showing that there are no non-trivial rational points on the elliptic curve y^2 = x^3 - 7x - 6 . Hmmm… The “parametrize by slopes of lines through the origin” is a standard trick to get rational or integral points on an elliptic curve. In 1922 Louis Mordell proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis. 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. An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring ZN , N=pq for two distinct odd primes p and q. 106, Springer 1986; Advanced Topics in the Arithmetic of Elliptic Curves Graduate Texts in Mathl. Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics) By Joseph H. Rational points on elliptic curves. Rational.points.on.elliptic.curves.pdf.