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

Through Bhargava's work with Arul Shankar and Chris Skinner, he has proven that a positive proportion of elliptic curves have infinitely many rational points and a positive proportion have no rational points. Rational points on elliptic curves book download Download Rational points on elliptic curves The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. We discuss its resolved elliptic fibrations over a general base B. We prove that the presentation of a general elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. What we now know as the Hasse-Weil theorem implies that the number N(p) of rational points of an elliptic curve over the finite field Z/pZ, where p is a prime, can differ from the mean value p+1 by at most twice the square root of p. 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. 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 ilde{O}( q^{2 – 2/n} ) bit operations. 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. Some sample rational points are shown in the following graph. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. We give geometric criteria which relate these models to the minimal proper regular models of the Jacobian elliptic curves of the genus one curves above. Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field.