Elliptic Curves

1. Front Matter

   Pages i-xxi

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2. Introduction to Rational Points on Plane Curves

   Pages 1-22

3. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

   Pages 23-43

4. Plane Algebraic Curves

   Pages 45-63

5. Elliptic Curves and Their Isomorphisms

   Pages 65-84

6. Families of Elliptic Curves and Geometric Properties of Torsion Points

   Pages 85-102

7. Reduction mod p and Torsion Points

   Pages 103-123

8. Proof of Mordell’s Finite Generation Theorem

   Pages 125-142

9. Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields

   Pages 143-156

10. Descent and Galois Cohomology

    Pages 157-166

11. Elliptic and Hypergeometric Functions

    Pages 167-187

12. Theta Functions

    Pages 189-207

13. Modular Functions

    Pages 209-231

14. Endomorphisms of Elliptic Curves

    Pages 233-252

15. Elliptic Curves over Finite Fields

    Pages 253-273

16. Elliptic Curves over Local Fields

    Pages 275-289

17. Elliptic Curves over Global Fields and ℓ-Adic Representations

    Pages 291-308

18. L-Function of an Elliptic Curve and Its Analytic Continuation

    Pages 309-324

19. Remarks on the Birch and Swinnerton-Dyer Conjecture

    Pages 325-332

20. Remarks on the Modular Elliptic Curves Conjecture and Fermat’s Last Theorem

    Pages 333-343

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.

• Dimension
• Grad
• algebraic curve

From the reviews of the second edition:

"Husemöller’s text was and is the great first introduction to the world of elliptic curves … and a good guide to the current research literature as well. … this second edition builds on the original in several ways. … it has certainly gained a good deal of topicality, appeal, power of inspiration, and educational value for a wider public. No doubt, this text will maintain its role as both a useful primer and a passionate invitation to the evergreen theory of elliptic curves and their applications" (Werner Kleinert, Zentralblatt MATH, Vol. 1040, 2004)

• Max-Planck-Institut für Mathematik, Bonn, Germany

  Dale Husemöller

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