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**The arithmetic of elliptic curves.
2nd ed.**
*(English)*
Zbl 1194.11005

Graduate Texts in Mathematics 106. New York, NY: Springer (ISBN 978-0-387-09493-9/hbk; 978-0-387-09494-6/ebook). xx, 513 p. (2009).

Joseph H. Silverman’s popular textbook “The Arithmetic of Elliptic Curves” was first published in 1986. Back then, almost 25 years ago, the author provided the first systematic, comprehensive, and modern textbook on the arithmetic of elliptic curves from the viewpoint of algebraic geometry and algebraic number theory, thereby reflecting the vast amount of research currently being done in this fascinating area of contemporary mathematics. In contrast to its few predecessors, which concentrated mostly on the analytic and modular aspects of the theory, Silverman’s approach relied somewhat more on methods and techniques from algebraic geometry, in particular on the (elementary) geometry of protective plane curves, the basics of which were developed in the first two chapters of the book. Apart from its topicality, this algebraic approach to elliptic curves had the great advantage of both a novel unity of exposition and a much wider range of arithmetic applications. Due to these outstanding features, the first edition of Silverman’s textbook (Zbl 0585.14026) quickly became a standard reference in the field. In the course of the past two decades, there has been a tremendous progress in the study of elliptic curves and their applications, with many spectacular results in various directions, and the former paucity of introductory texts devoted to the subject has been remedied with the publication of quite a large number of related volumes by various authors.

The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. Leaving the well-tried structure of the text basically intact, including its distinguished pedagogical aims, the author has reworked the material under the following guiding principles:

(1) Update and expand several results and references with a view toward recent developments.

(2) Add a new chapter devoted to the modern algorithmic aspects of elliptic curves, with an emphasis on their features that are used in cryptography.

(3) Add an introductory section on Szpiro’s conjecture and the famous ABC conjecture to the chapter on elliptic curves over global fields.

(4) Guide the reader along the beginning of the trail that leads to some of the more recent results of L. Merel (1996), K. Rubin (1987), V. A. Kolyvagin (1988), A. Wiles (1995), N. B. Elkies (1987), and others.

(5) Correct, clarify, and simplify the proofs of some results.

(6) Correct numerous typographical errors and minor mathematical inaccuracies in the original text.

(7) Significantly expand the selection of exercises at the end of each chapter.

In fact, this rather extensive program has been carried out with the greatest care and mastery. The current second edition of the book has grown in size by more than a quarter, that is, by more than 100 pages, due to the numerous updating replenishments indicated above. The material is now organized in eleven chapters and three appendices. For most of the contents, we may refer to the review of the original edition (Zbl 0585.14026), as the major novelties occur in the new Chapter XI, in §11 of Chapter VIII, and in Appendix C. The table of contents of the present second edition of the book-actually reads as follows:

Chapter I: Algebraic varieties; Chapter II: Algebraic curves; Chapter III: The geometry of elliptic curves; Chapter IV: The formal group of an elliptic curve; Chapter V: Elliptic curves over finite fields; Chapter VI: Elliptic curves over local fields; Chapter VII: Elliptic curves over local fields; Chapter VIII: Elliptic curves over global fields, with a new section on Szpiro’s conjecture and the ABC conjecture; Chapter IX: Integral points on elliptic curves; Chapter X: Computing the Mordell-Weil group; Chapter XI (new): Algorithmic aspects of elliptic curves, with an emphasis on algorithms over finite fields such as Lenstra’s factorization, algorithm, Schoof’s point counting algorithm, Miller1s algorithm to compute the Tate and Weil pairings, and their significance in elliptic curve cryptography;

Appendix A: Elliptic curves in characteristics 2 and 3; Appendix B: Group cohomology \((H^0\) and \(H^1)\); Appendix C: Further topics: an overview, with a new section on the variation of the trace of the Frobenius map, the Sato-Tate conjecture (1963), and the very recent progress in this context achieved by L. Clozel, M. Harris, N. Shepherd-Barron, and R. Taylor.

The updated rich bibliography contains the huge number of 317 references, even including several forthcoming research articles, and both the guiding notes on exercises and the extensive index have also been complemented. The carefully compiled list of notation represents another valuable enhancement of this outstanding textbook.

All together, this enlarged and updated version of J. Silverman’s classic “The Arithmetic of Elliptic Curves” significantly increases the unchallenged value of this modern primer as a standard textbook in the field. Despite its advanced character, the book is largely self-contained, and the prerequisites for reading it are still fairly modest. This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.

The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. Leaving the well-tried structure of the text basically intact, including its distinguished pedagogical aims, the author has reworked the material under the following guiding principles:

(1) Update and expand several results and references with a view toward recent developments.

(2) Add a new chapter devoted to the modern algorithmic aspects of elliptic curves, with an emphasis on their features that are used in cryptography.

(3) Add an introductory section on Szpiro’s conjecture and the famous ABC conjecture to the chapter on elliptic curves over global fields.

(4) Guide the reader along the beginning of the trail that leads to some of the more recent results of L. Merel (1996), K. Rubin (1987), V. A. Kolyvagin (1988), A. Wiles (1995), N. B. Elkies (1987), and others.

(5) Correct, clarify, and simplify the proofs of some results.

(6) Correct numerous typographical errors and minor mathematical inaccuracies in the original text.

(7) Significantly expand the selection of exercises at the end of each chapter.

In fact, this rather extensive program has been carried out with the greatest care and mastery. The current second edition of the book has grown in size by more than a quarter, that is, by more than 100 pages, due to the numerous updating replenishments indicated above. The material is now organized in eleven chapters and three appendices. For most of the contents, we may refer to the review of the original edition (Zbl 0585.14026), as the major novelties occur in the new Chapter XI, in §11 of Chapter VIII, and in Appendix C. The table of contents of the present second edition of the book-actually reads as follows:

Chapter I: Algebraic varieties; Chapter II: Algebraic curves; Chapter III: The geometry of elliptic curves; Chapter IV: The formal group of an elliptic curve; Chapter V: Elliptic curves over finite fields; Chapter VI: Elliptic curves over local fields; Chapter VII: Elliptic curves over local fields; Chapter VIII: Elliptic curves over global fields, with a new section on Szpiro’s conjecture and the ABC conjecture; Chapter IX: Integral points on elliptic curves; Chapter X: Computing the Mordell-Weil group; Chapter XI (new): Algorithmic aspects of elliptic curves, with an emphasis on algorithms over finite fields such as Lenstra’s factorization, algorithm, Schoof’s point counting algorithm, Miller1s algorithm to compute the Tate and Weil pairings, and their significance in elliptic curve cryptography;

Appendix A: Elliptic curves in characteristics 2 and 3; Appendix B: Group cohomology \((H^0\) and \(H^1)\); Appendix C: Further topics: an overview, with a new section on the variation of the trace of the Frobenius map, the Sato-Tate conjecture (1963), and the very recent progress in this context achieved by L. Clozel, M. Harris, N. Shepherd-Barron, and R. Taylor.

The updated rich bibliography contains the huge number of 317 references, even including several forthcoming research articles, and both the guiding notes on exercises and the extensive index have also been complemented. The carefully compiled list of notation represents another valuable enhancement of this outstanding textbook.

All together, this enlarged and updated version of J. Silverman’s classic “The Arithmetic of Elliptic Curves” significantly increases the unchallenged value of this modern primer as a standard textbook in the field. Despite its advanced character, the book is largely self-contained, and the prerequisites for reading it are still fairly modest. This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.

Reviewer: Werner Kleinert (Berlin)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11G05 | Elliptic curves over global fields |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

11G07 | Elliptic curves over local fields |

11G10 | Abelian varieties of dimension \(> 1\) |

11G20 | Curves over finite and local fields |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14H52 | Elliptic curves |

14G05 | Rational points |

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |

14H25 | Arithmetic ground fields for curves |

11Y16 | Number-theoretic algorithms; complexity |

20J06 | Cohomology of groups |