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Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2


Part of London Mathematical Society Lecture Note Series

  • Date Published: April 1996
  • availability: Available
  • format: Paperback
  • isbn: 9780521483704

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About the Authors
  • The number theoretic properties of curves of genus 2 are attracting increasing attention. This book provides new insights into this subject; much of the material here is entirely new, and none has appeared in book form before. Included is an explicit treatment of the Jacobian, which throws new light onto the geometry of the Kummer surface. The Mordell–Weil group can then be determined for many curves, and in many non-trivial cases all rational points can be found. The results exemplify the power of computer algebra in diophantine contexts, but computer expertise is not assumed in the main text. Number theorists, algebraic geometers and workers in related areas will find that this book offers unique insights into the arithmetic of curves of genus 2.

    • Cassels is THE top person on this subject. His previous books have done very well
    • This book is unique
    • The subject matter is getting hotter all the time
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    Reviews & endorsements

    '… this textbook provides an excellent treatment of the subject of the basic algebraic geometry (and arithmetic) of genus 2 curves and would serve as a useful introductory text for graduate students.' A Maciocia, Proceedings of the Edinburgh Mathematical Society

    '… an explicit treatment.' L'Enseignement Mathématique

    'It is clear that this often subtle and witty book is required reading for those working in the field.' J. Schoissengeier, Monatshefte für Mathematik

    '… the authors adopt a down-to-earth approach, assuming a fair background in algebraic geometry, but explaining carefully the tools used.' Mathematika

    'The book is accessible without too many prerequisutes … As a readable introduction to an exciting area of research in which there are may interesting questions still to be answered, the book is to be strongly recommended.' Bulletin of the London Mathematics Society

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    Product details

    • Date Published: April 1996
    • format: Paperback
    • isbn: 9780521483704
    • length: 236 pages
    • dimensions: 228 x 152 x 16 mm
    • weight: 0.342kg
    • availability: Available
  • Table of Contents

    Background and conventions
    1. Curves of genus 2
    2. Construction of the jacobian
    3. The Kummer surface
    4. The dual of the Kummer
    5. Weddle's surface
    6. G/2G
    7. The jacobian over local fields
    Formal groups
    8. Torsion
    9. The isogeny
    10. The isogeny
    11. Computing the Mordell-Weil group
    12. Heights
    13. Rational points Chabauty's theorem
    14. Reducible jacobians
    15. The endomorphism ring
    16. The desingularized Kummer
    17. A neoclassical approach
    18. Zukunftsmusik
    Appendix 1. MAPLE programs
    Appendix 2. Files available by FTP.

  • Authors

    J. W. S. Cassels, University of Cambridge

    E. V. Flynn, University of Liverpool

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