Advanced Topics In The Arithmetic Of Elliptic Curves
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Advanced Topics in the Arithmetic of Elliptic Curves
Author: Joseph H. Silverman
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-12-01
In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
The Arithmetic of Elliptic Curves
Author: Joseph H. Silverman
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-03-09
The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic, the arithmetic of elliptic curves, there is little need for such an apologia. Considering the vast amount of research currently being done in this area, the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang (especially [La 3] and [La 5]); and there are books of Koblitz ([Kob]) and Robert ([Rob], now out of print) which concentrate mostly on the analytic and modular theory. In addition, survey articles have been written by Cassels ([Ca 7], really a short book) and Tate ([Ta 5J, which is beautifully written, but includes no proofs). Thus the author hopes that this volume will fill a real need, both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts. Our approach is more algebraic than that taken in, say, [La 3] or [La 5], where many of the basic theorems are derived using complex analytic methods and the Lefschetz principle. For this reason, we have had to rely somewhat more on techniques from algebraic geometry. However, the geom etry of (smooth) curves, which is essentially all that we use, does not require a great deal of machinery.
Elliptic Curves in Cryptography
Author: Ian F. Blake
language: en
Publisher: Cambridge University Press
Release Date: 1999-07-08
This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems.