Main subject categories: • Ellipic curves • Diophantine geometry • Algebraic geometry • Number theoryMathematics Subject Classification (2000): • 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory • 11GXX Arithmetic algebraic geometry (Diophantine geometry) • 14GXX Arithmetic problems in algebraic geometry; Diophantine geometry • 11G05 Elliptic curves over global fieldsThe 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 approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.