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

Silverman, John Tate, Rational Points on Elliptic Curves, Springer 1992. These new spkg's are mpmath for multiprecision floating-point arithmetic, and Ratpoints for computing rational points on hyperelliptic curves. 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\}∪\{∞\}$.. Silverman, Joseph H., Tate, John, Rational Points on Elliptic Curves, 1992 63. Affine space and the Zariski topology; Regular functions; Regular maps. 106, Springer 1986; Advanced Topics in the Arithmetic of Elliptic Curves Graduate Texts in Mathl. A very good book written on the subject is "Rational points on Elliptic Curves" by Silverman and Tate. Rational Points on Elliptic Curves John Tate (Auteur), J.H. Challenge 4 is a large rational function calculating the "multiply-by-m" map of a point on an elliptic curve. Devlin, Keith, The Joy of Sets – Fundamentals of Contemporary Set Theory, 1993 64. This brings the total Construct an elliptic curve from a plane curve of genus one (Lloyd Kilford, John Cremona ) — New function EllipticCurve_from_plane_curve() in the module sage/schemes/elliptic_curves/constructor.py to allow the construction of an elliptic curve from a smooth plane cubic with a rational point. If time permits, additional topics may be covered. The first thing that we should do here is to reduce this equation to the Weierstrass normal form. Rational curves; Relation with field theory; Rational maps; Singular and nonsingular points; Projective spaces. Kinsey, L.Christine, Topology of Surfaces, 1993 65. Rational functions and rational maps; Quasiprojective varieties. I compare this book to Rational Points on Elliptic Curves (RP) by Tate and Silverman, and The Arithmetic of Ellipitic Curves (AEC) by Silverman. The Zariski topology on Additional topics. Solid intermediate introduction to elliptic curves. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves.