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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. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, elliptic curves over finite fields, the complex numbers, local fields, and global fields. The last two chapters deal with integral and rational points, including Siegel's theorem and explicit computations for the curve Y 2 = X 3 + DX. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

Arithmetic. --- Curves, Algebraic. --- Curves, Elliptic. --- Curves, Elliptic --- Curves, Algebraic --- Courbes algébriques --- 512.74 --- Elliptic curves --- Numbers, Real --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Courbes algébriques --- Arithmetic --- Algebraic curves --- Algebraic varieties --- Mathematics --- Set theory --- Calculators --- Algebraic geometry --- Courbes elliptiques --- Arithmétique --- Arithmetical algebraic geometry --- Géométrie algébrique arithmétique --- Géométrie algébrique --- Géométrie algébrique arithmétique. --- Géométrie algébrique --- Nombres, Théorie des

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Algebraic fields --- Duality theory (Mathematics) --- Homology theory --- 511.33 --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Algebra --- Mathematical analysis --- Topology --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Algebraic fields. --- Homology theory. --- Duality theory (Mathematics). --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Geometry, Algebraic. --- Géométrie algébrique --- Galois cohomology --- Cohomologie galoisienne. --- Géométrie algébrique --- Cohomologie galoisienne --- Geometrie algebrique --- Cohomologie