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

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. This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E',C,C',φ), where E and E' are elliptic curves, C and C' are cyclic 13-subgroups, and φ is an isomorphism between C and C'. Rational Points on Modular Elliptic Curves Henri Darmon. You ask for an easy example of a genus 1 curve with no rational points. Hyperbolic geometry: the metric of Minkowski space-time. Heavily on the fact that E has a rational point of finite rank. 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. 5,7 and 11 also have special significance because PSL(2,p) is “exceptional” for these primes. Rational Points on Elliptic Curves John Tate (Auteur), J.H. Consider the plane curve Ax^2+By^4+C=0. For example the supersingular primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 are important to moonshine theory as factors of the size of the monster group and as special cases for elliptic curves modulo p. The genus 1 — elliptic curve — case will be in the next posting, or so I hope.) If you are interested in curves over fields that are not B, I want to mention the fact that there is no number N such that every genus 1 curve over a field k has a point of degree at most N over k. Download Rational Points on Modular Elliptic Curves. Rational Points on Modular Elliptic Curves book download. Who tells the story in the first half of the book narrates how a young volunteer came up to him and Rational Points on Elliptic Curves - Google Books This book stresses this interplay as it develops the basic theory,. 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)] [