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

The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e.,$C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$.. Elliptic curves have been a focus of intense scrutiny for decades. Henri Poincaré studied them in the early years of the 20th century. Rational.points.on.elliptic.curves.pdf. Update: also, opinions on books on elliptic curves solicited, for the four or five of you who might have some! Some sample rational points are shown in the following graph. 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. Moduli spaces of elliptic curves with level structure are fundamental for arithmetic and Diophantine problems over number fields in particular. 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. Rational points on elliptic curves. K3 surfaces, level structure, and rational points.