Complex Multiplication New Mathematical Monographs Series Number 15 1st Edition by Reinhard Schertz ISBN 0521766680 978-0521766685 by Reinhard Schertz 9780521766685, 0521766680 instant download after payment.

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Product details:

ISBN 10: 0521766680

ISBN 13: 978-0521766685 

Author: Reinhard Schertz

This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers.

Table of contents: 

1. Frontmatter
2. Contents
3. Preface
4. Elliptic functions
5. Modular functions
6. Basic facts from number theory
7. Factorisation of singular values
8. The Reciprocity Law
9. Generation of ring class fields and ray class fields
10. Integral basis in ray class fields
11. Galois module structure
12. Berwick's congruences
13. Cryptographically relevant elliptic curves
14. The class number formulae of Curt Meyer
15. Arithmetic interpretation of class number formulae
16. References
17. Index of Notation
18. Index

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Tags: Reinhard Schertz, Complex Multiplication, New Mathematical Monographs, Series Number 15