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

Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. Heavily on the fact that E has a rational point of finite rank. Rational Points on Elliptic Curves - Silverman, Tate.pdf. This week the lecture series is given by Shou-wu Zhang from Columbia, and revolves around the topic of rational points on curves, a key subject of interest in arithmetic geometry and number theory. That is, an equation for a curve that provides all of the rational points on that curve. 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. It also has It has no dependencies (instead of PARI), because Mark didn't want to have to license sympow under the GPL. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Ratpoints (C library): Michael Stoll's highly optimized C program for searching for certain rational points on hyperelliptic curves (i.e. The first of three While these counterexamples are completely explicit, they were found by geometric means; for instance, Elkies' example was found by first locating Heegner points of an elliptic curve on the Euler surface, which turns out to be a K3 surface. [math.NT/0606003] We consider the structure of rational points on elliptic curves in Weierstrass form. 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. This library is very, very good and fast for doing computations of many functions relevant to number theory, of "class groups of number fields", and for certain computations with elliptic curves. 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. Points on elliptic curves over Q which are not [0:1:0] have their last coordinate =1 but sometimes this is an int (not even an Integer) which breaks some code: sage: E=EllipticCurve('37a1') sage: [type(c) for c in E(0)] [