Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

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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 more

Product details

  • Electronic book text
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • 1139243675
  • 9781139243674

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; Theory; 10. The isogeny; Applications; 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 more

Review quote

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

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