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

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. P_t=(2,p_t),\quad Q_t=(3,q_t These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Benedict Gross, Harvard University. Moduli spaces of elliptic curves with level structure are fundamental for arithmetic and Diophantine problems over number fields in particular. It had long been known that the rational points on an elliptic curve, defined over the rationals, form a group Γ under a chord and tangent construction; Mordell proved that Γ has a finite basis. 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. We give some examples, and list new algorithms that are due to Cremona and Delaunay. Rational points on elliptic curves. We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. Rational.points.on.elliptic.curves.pdf. Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . K3 surfaces, level structure, and rational points. The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. The Arithmetic of Elliptic Curves. 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. 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'. In 1922 Louis Mordell proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.