Elliptic Curves: Function Theory, Geometry, Arithmetic

by Henry McKean , Victor Moll

• ISBN13: 9780521658171
• ISBN10: 0521658179
• Format: Paperback
• Copyright: 1999-08-13
• Publisher: Cambridge University Press

Summary

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.

Table of Contents

1. First ideas: complex manifolds, Riemann surfaces, and projective curves

2. Elliptic functions and elliptic integrals

3. Theta functions

4. Modular groups and molecular functions

5. Ikosaeder and the quintic

6. Imaginary quadratic fields

7. The arithmetic of elliptic fields.