Introduction to Rational Points on Plane Curves * Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve * Plane Algebraic Curves * Factorial Rings and Elimination Theory * Elliptic Curves and Their Isomorphism * Families of Elliptic Curves and Geometric Properties of Torsion Points * Reduction mod p and Torsion Points * Proof of Mordell's Finite Generation Theorem * Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields * Descent and Galois Cohomology * Elliptic and Hypergeometric Functions * Theta Functions * Modular Functions * Endomorphisms of Elliptic Curves * Elliptic Curves over Finite Fields * Elliptic Curves over Local Fields * Elliptic Curves over Global Fields and l-adic Representations * L-Functions of an Elliptic Curve and Its Analytic Continuation * Remarks on the Birch and Swinnerton-Dyer Conjecture * Remarks on the Modular Curves Conjecture and Fermat's Last Theorem * Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties * Families of Elliptic Curves * Appendix I: Calabi-Yau Manifolds and String Theory * Appendix II: Elliptic Curves in Algorithmic Number Theory * Appendix III: Guide to the Exercises * Bibliography * Index

There are three new appendices, one by Stefan Theisen on the role of Calabi¿ Yau manifolds in string theory and one by Otto Forster on the use of elliptic curves in computing theory and coding theory. In the third appendix we discuss the role of elliptic curves in homotopy theory. In these three introductions the reader can get a clue to the far-reaching implications of the theory of elliptic curves in mathematical sciences. During the ?nal production of this edition, the ICM 2002 manuscript of Mike Hopkins became available. This report outlines the role of elliptic curves in ho- topy theory. Elliptic curves appear in the form of the Weierstasse equation and its related changes of variable. The equations and the changes of variable are coded in an algebraic structure called a Hopf algebroid, and this Hopf algebroid is related to a cohomology theory called topological modular forms. Hopkins and his coworkers have used this theory in several directions, one being the explanation of elements in stable homotopy up to degree 60. In the third appendix we explain how what we described in Chapter 3 leads to the Weierstrass Hopf algebroid making a link with Hopkins¿ paper.