Elliptic Curves

1. Front Matter

   Pages I-XV

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2. Introduction to Rational Points on Plane Curves

   • Dale Husemöller

   Pages 1-21

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

   • Dale Husemöller

   Pages 22-42

4. Plane Algebraic Curves

   • Dale Husemöller

   Pages 43-61

5. Elliptic Curves and Their Isomorphisms

   • Dale Husemöller

   Pages 62-80

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

   • Dale Husemöller

   Pages 81-98

7. Reduction mod p and Torsion Points

   • Dale Husemöller

   Pages 99-119

8. Proof of Mordell’s Finite Generation Theorem

   • Dale Husemöller

   Pages 120-137

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

   • Dale Husemöller

   Pages 138-151

10. Descent and Galois Cohomology

    • Dale Husemöller

    Pages 152-161

11. Elliptic and Hypergeometric Functions

    • Dale Husemöller

    Pages 162-182

12. Theta Functions

    • Dale Husemöller

    Pages 183-201

13. Modular Functions

    • Dale Husemöller

    Pages 202-221

14. Endomorphisms of Elliptic Curves

    • Dale Husemöller

    Pages 222-241

15. Elliptic Curves over Finite Fields

    • Dale Husemöller

    Pages 242-261

16. Elliptic Curves over Local Fields

    • Dale Husemöller

    Pages 262-271

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

    • Dale Husemöller

    Pages 272-289

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

    • Dale Husemöller

    Pages 290-305

19. Remarks on the Birch and Swinnerton-Dyer Conjecture

    • Dale Husemöller

    Pages 306-314

20. Back Matter

    Pages 315-350

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The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A general outline ofmuch ofthe material can be found in Tate's colloquium lectures reproduced as an article in Inven­ tiones [1974]. The first part grew out of Tate's 1961 Haverford Philips Lectures as an attempt to write something for publication c10sely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This inc1udes parts of the Introduction and the first six chapters The aim ofthis part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate teturned to Haverford to give again, in revised form, the originallectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan forthe book.

• Finite
• Forth
• Galois theory
• Grad
• Hypergeometric function
• Natural
• algebraic curve
• complex analysis
• elliptic curve
• form
• presentation
• proof
• recording
• theorem
• theta function

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)

• Department of Mathematics, Haverford College, Haverford, USA

  Dale Husemöller

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