Elliptic Curves

Elliptic Curves : Function Theory, Geometry, Arithmetic

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The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 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.show more

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'The book is a welcome extension of the existing literature about this important topic ... It is recommended to students of mathematics and physics interested in the applications of the theory and the theory itself.' European Mathematical Society 'With an easy mind the reviewer can recommend this book to those who want to become acquainted with the subject and to those who look for a book which can serve as guide for a course on the subject ... the exemplary way in which Elliptic Curves is written, made reviewing a pleasure.' Niew Archief voor Wiskundeshow more

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.show more