Complex Multiplication

Erscheinungsjahr 2012
ISBN: 978-0-521-76668-5
Verlag: Cambridge University Press

Buch, Englisch, Band 15, 375 Seiten, Format (B × H): 157 mm x 235 mm, Gewicht: 760 g

Reihe: New Mathematical Monographs

This is a self-contained 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.

Autoren/Hrsg.

Schertz, Reinhard

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik

Inhaltsverzeichnis

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.

Über Autor(innen)

Schertz, Reinhard
Reinhard Schertz was Professor of Mathematics at the University of Augsburg in Germany until his retirement in 2008.