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LMSST: 24 Lectures on Elliptic Curves

LMSST: 24 Lectures on Elliptic Curves


Part of London Mathematical Society Student Texts

  • Date Published: November 1991
  • availability: Available
  • format: Paperback
  • isbn: 9780521425308

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About the Authors
  • The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.

    • Fully class tested (based on a Cambridge course)
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    Reviews & endorsements

    '… an excellent introduction … written with humour.' Monatshefte für Mathematik

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    Product details

    • Date Published: November 1991
    • format: Paperback
    • isbn: 9780521425308
    • length: 144 pages
    • dimensions: 230 x 155 x 10 mm
    • weight: 0.22kg
    • contains: 5 b/w illus.
    • availability: Available
  • Table of Contents

    1. Curves of genus: introduction
    2. p-adic numbers
    3. The local-global principle for conics
    4. Geometry of numbers
    5. Local-global principle: conclusion of proof
    6. Cubic curves
    7. Non-singular cubics: the group law
    8. Elliptic curves: canonical form
    9. Degenerate laws
    10. Reduction
    11. The p-adic case
    12. Global torsion
    13. Finite basis theorem: strategy and comments
    14. A 2-isogeny
    15. The weak finite basis theorem
    16. Remedial mathematics: resultants
    17. Heights: finite basis theorem
    18. Local-global for genus principle
    19. Elements of Galois cohomology
    20. Construction of the jacobian
    21. Some abstract nonsense
    22. Principle homogeneous spaces and Galois cohomology
    23. The Tate-Shafarevich group
    24. The endomorphism ring
    25. Points over finite fields
    26. Factorizing using elliptic curves
    Further reading

  • Author

    J. W. S. Cassels, University of Cambridge

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