Aq6 Z

The Arithmetic of Elliptic Curves

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.

Dam Breach Modeling Technology

Dams are constructed for economic development, and their construction involves large investments of money, and natural and human resources. Of the various types of dams constructed around the globe, earth dams are the most common type and constitute the vast majority of dams. When adam fails, it culminates in the sudden release of artificially stored water which, in turn, becomes a potential menace to virtually everything downstream. The dam failure may result in loss of life and property. In recent years, instances of dam failure in the world have been too many, and the resulting loss too high. As a result, dam safety pro grams have been developed in most countries of the world since the beginning of the nineteenth eighties. · Earth dams are more susceptible to failure than other types. The cause of failure is often either overtopping or piping. The modeling of dam breaching due to either or both of these causes is of fundamental importance to development of dam-safety programs. This book is, therefore, an attempt to present some aspects of earth-dam breach modeling technology. It is hoped that others will be stimulated to write more comprehensive texts on this subject of growing interest and importance. The book is divided into eight chapters. The first chapter is introductory and discusses some aspects of dams and dam failures in the world.

Elliptic Curves, Modular Forms and Cryptography