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.;Algebraic Varieties -- Algebraic Curves -- The Geometry of Elliptic Curves -- The Formal Group of an Elliptic Curve -- Elliptic Curves over Finite Fields -- Elliptic Curves over C -- Elliptic Curves over Local Fields -- Elliptic Curves over Global Fields -- Integral Points on Elliptic Curves -- Computing the Mordell-Weil Group -- Appendix A: Elliptic Curves in Characteristics 2 and 3. -- Appendix B: Group Cohomology (H0 and H1) -- Appendix C.: Further Topics: An Overview -- Notes on Exercises -- Bibliography -- List of Notation -- Index.