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Elliptic curves are of central and growing importance in computational number theory, with numerous applications in such areas as cryptography, primality testing and factorisation. This book, now in its second edition, presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves, with remarks on computer implementation. It is in three parts. First, the author describes in detail the construction of modular elliptic curves, giving an explicit algorithm for their computation using modular symbols. Secondly a collection of algorithms for the arithmetic of elliptic curves is presented; some of these have not appeared in book form before. They include: finding torsion and non-torsion points, computing heights, finding isogenies and periods, and computing the rank. Finally, an extensive set of tables is provided giving the results of the author's implementation of the algorithms. These tables extend the widely used 'Antwerp IV tables' in two ways: the range of conductors (up to 1000), and the level of detail given for each curve. In particular, the quantities relating to the Birch Swinnerton-Dyer conjecture have been computed in each case and are included. All researchers and graduate students of number theory will find this book useful, particularly those interested in the computational side of the subject. That aspect will make it appeal also to computer scientists and coding theorists.
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This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves, with remarks on computer implementation. An extensive set of tables is provided giving the results of the author's implementation of the algorithms.Review:
"Although the idea of using modular symbols for computing the modular elliptic curves defined over Q with conductor N is not new, neither the complete description of the algorithm nor the description of its implementation had been available before the writing of this book; moreover, the complete list of all the modular curves defined over Q with conductor less than 999...will prove very useful for any mathematician interested in the arithmetic of elliptic curves." Philippe Satge, Mathematical Reviews
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