Complex Multiplication
Seiten
2010
Cambridge University Press (Verlag)
978-0-521-76668-5 (ISBN)
Cambridge University Press (Verlag)
978-0-521-76668-5 (ISBN)
This self-contained 2010 account of the state of the art in classical complex multiplication provides an exhaustive treatment of the theory of elliptic functions, modular functions and quadratic number fields. The reader will find all the necessary background and tools they will need in this book.
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.
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.
Reinhard Schertz was Professor of Mathematics at the University of Augsburg in Germany until his retirement in 2008.
Preface; 1. Elliptic functions; 2. Modular functions; 3. Basic facts from number theory; 4. Factorisation of singular values; 5. The reciprocity law; 6. Generation of ring class fields and ray class fields; 7. Integral basis in ray class fields; 8. Galois module structure; 9. Berwick's congruences; 10. Cryptographically relevant elliptic curves; 11. The class number formulas of Curt Meyer; 12. Arithmetic interpretation of class number formulas; References; Index of notation; Index.
| Reihe/Serie | New Mathematical Monographs |
|---|---|
| Zusatzinfo | 4 Line drawings, unspecified |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 152 x 229 mm |
| Gewicht | 660 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 0-521-76668-0 / 0521766680 |
| ISBN-13 | 978-0-521-76668-5 / 9780521766685 |
| Zustand | Neuware |
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