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Rational points on elliptic curves by John Tate, Joseph H. Silverman

Rational points on elliptic curves John Tate, Joseph H. Silverman ebook
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
ISBN: 3540978259, 9783540978251
Format: djvu
Page: 296

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. 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. In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. This process never repeats itself (and so infinitely many rational points may be generated in this way). From the formula for doubling a point we get that. The only rational solution of which is x = 0. 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. In the language of elliptic curves, given a rational point P we are considering the new rational point -2P . Graphs of curves y2 = x3 − x and y2 = x3 − x + 1. 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. 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. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Here's what this looks like: Image001. Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. Thich corresponds to the points (0,1) and (0,-1) on the elliptic curve.