The Arithmetic of Elliptic Curves

In the preface to the ?rst edition of this book I remarked on the paucity of int- ductory texts devoted to the arithmetic of elliptic curves. That unfortunate state of affairs has long since been remedied with the publication of many volumes, among which may be mentioned books by Cassels [43], Cremona [54], Husemol ¨ ler [118], Knapp [127], McKean et. al [167], Milne [178], and Schmitt et. al [222] that hi- light the arithmetic and modular theory, and books by Blake et. al [22], Cohen et. al [51], Hankerson et. al [107], and Washington [304] that concentrate on the use of elliptic curves in cryptography.However, even among this cornucopia of literature, I hope that this updated version of the original text will continue to be useful. The past two decadeshave witnessed tremendousprogressin the study of elliptic curves. Among the many highlights are the proof by Merel [170] of uniform bou- edness for torsion points on elliptic curves over number?elds, results of Rubin [215] and Kolyvagin [130] on the ?niteness of Shafarevich–Tate groups and on the c- jecture of Birch and Swinnerton-Dyer,the work of Wiles [311] on the modularity of elliptic curves, and the proof by Elkies [77] that there exist in?nitely many supers- gular primes. Although this introductory volume is unable to include proofs of these deep results, it will guide the reader along the beginning of the trail that ultimately leads to these summits.