The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.

Autorentext

Dr. Susanne Schmitt works at the Max Planck Institute for Computer Sciences at Saarbrücken, Germany. Horst-Günter Zimmer is Professor at the Mathematics Department of the University of the Saarland at Saarbrücken, Germany.

Klappentext

The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel. The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. Editor-in-Chief Guozhen Lu, University of Connecticut, USA Editorial Board Carstensen Carsten, Humboldt-Universitat zu Berlin, Germany Gavril Farkas, Humboldt-Universitat zu Berlin, Germany Nicola Fusco, Università di Napoli "Federico II", Italy Fritz Gesztesy, Baylor University, USA Zenghu Li, Beijing Normal University, China Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany René L. Schilling, Technische Universität Dresden, Germany Volkmar Welker, Philipps-Universität Marburg, Germany Please submit book proposals to Guozhen Lu

Zusammenfassung

"This is a most welcome addition to the existing literature on elliptic curves."
Ch. Baxa in: Monatshefte für Mathematik 4/2004

"[...] [the authors'] book will function well both as a textbook for students and as a reference book for researchers. [...] it nicely complements the more theoretical books [...], while providing a path for both students and researchers into the computational theory of elliptic curves." Mathematical Reviews