Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

A unique insight into the topic of curves of genus 2, by two of the world's leading practitioners.

J. W. S. Cassels (Author), E. V. Flynn (Author)

9780521483704, Cambridge University Press

Paperback, published 18 April 1996

236 pages
22.8 x 15.2 x 1.6 cm, 0.342 kg

'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

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.

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 FTP.

Subject Areas: Number theory [PBH]