About this product

Product Identifiers

PublisherSpringer New York

ISBN-100387094938

ISBN-139780387094939

eBay Product ID (ePID)72356938

Product Key Features

Number of PagesXx, 513 Pages

LanguageEnglish

Publication NameArithmetic of Elliptic Curves

SubjectAlgebra / General, Number Theory, Geometry / Algebraic, Arithmetic

Publication Year2009

TypeTextbook

Subject AreaMathematics

AuthorGrigory Mikhalkin, Joseph H. Silverman

SeriesGraduate Texts in Mathematics Ser.

FormatHardcover

Dimensions

Item Weight34 Oz

Item Length9.3 in

Item Width6.1 in

Additional Product Features

Edition Number2

Intended AudienceScholarly & Professional

LCCN2009-926474

ReviewsFrom the reviews of the second edition: "This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS "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. ... 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. ... 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." (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010) "For the second edition of his masterly book, the author considerably updated and improved several results and proofs. ... book contains a great many exercises, many of which develop or complement the results from the main body of the book. ... The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. ... Summarizing, this is an excellent book ... . useful both for experienced mathematicians and for graduate students." (Vasil' I. Andrichuk, Mathematical Reviews, Issue 2010 i) "This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves ... . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. ... The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them." (Ch. Baxa, Monatshefte fr Mathematik, Vol. 164 (3), November, 2011) "The book is written for graduate students ... and for researchers interested in standard facts about elliptic curves. ... A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. ... I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet." (Book Inspections Blog, 2012), From the reviews of the second edition: "This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS "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. ... 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. ... 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." (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010) "For the second edition of his masterly book, the author considerably updated and improved several results and proofs. ... book contains a great many exercises, many of which develop or complement the results from the main body of the book. ... The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. ... Summarizing, this is an excellent book ... . useful both for experienced mathematicians and for graduate students." (Vasil' I. Andrichuk, Mathematical Reviews, Issue 2010 i) "This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves ... . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. ... The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them." (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011) "The book is written for graduate students ... and for researchers interested in standard facts about elliptic curves. ... A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. ... I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet." (Book Inspections Blog, 2012), From the reviews of the second edition: "This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS "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. ... 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. ... 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." (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010) "For the second edition of his masterly book, the author considerably updated and improved several results and proofs. ... book contains a great many exercises, many of which develop or complement the results from the main body of the book. ... The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. ... Summarizing, this is an excellent book ... . useful both for experienced mathematicians and for graduate students." (Vasil' I. Andriichuk, Mathematical Reviews, Issue 2010 i) "This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves ... . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory textand a standard reference. ... The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them." (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011) "The book is written for graduate students ... and for researchers interested in standard facts about elliptic curves. ... A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. ... I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet." (Book Inspections Blog, 2012), From the reviews of the second edition: "This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS "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. ... 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. ... 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." (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010) "For the second edition of his masterly book, the author considerably updated and improved several results and proofs. ... book contains a great many exercises, many of which develop or complement the results from the main body of the book. ... The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. ... Summarizing, this is an excellent book ... . useful both for experienced mathematicians and for graduate students." (Vasil' I. Andriichuk, Mathematical Reviews, Issue 2010 i) "This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves ... . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. ... The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them." (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011) "The book is written for graduate students ... and for researchers interested in standard facts about elliptic curves. ... A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. ... I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet." (Book Inspections Blog, 2012), From the reviews of the second edition: "This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS "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. ... 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. ... 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." (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010) "For the second edition of his masterly book, the author considerably updated and improved several results and proofs. ... book contains a great many exercises, many of which develop or complement the results from the main body of the book. ... The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. ... Summarizing, this is an excellent book ... . useful both for experienced mathematicians and for graduate students." (Vasil' I. Andriichuk, Mathematical Reviews, Issue 2010 i) "This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves ... . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. ... The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them." (Ch. Baxa, Monatshefte fr Mathematik, Vol. 164 (3), November, 2011) "The book is written for graduate students ... and for researchers interested in standard facts about elliptic curves. ... A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. ... I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet." (Book Inspections Blog, 2012), From the reviews of the second edition:"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWSThe book under review is the second, revised, enlarged, and updated edition of J. Silverman's meanwhile classical primer of the arithmetic of elliptic curves. â€¦ 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. â€¦ 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. (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010)For the second edition of his masterly book, the author considerably updated and improved several results and proofs. â€¦ book contains a great many exercises, many of which develop or complement the results from the main body of the book. â€¦ The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. â€¦ Summarizing, this is an excellent book â€¦ . useful both for experienced mathematicians and for graduate students. (Vasil' I. Andrichuk, Mathematical Reviews, Issue 2010 i)This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves â€¦ . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. â€¦ The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them. (Ch. Baxa, Monatshefte fr Mathematik, Vol. 164 (3), November, 2011)

TitleLeadingThe

Dewey Edition22

Series Volume Number106

Number of Volumes1 vol.

IllustratedYes

Dewey Decimal516.352

Table Of ContentAlgebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell Weil Group.- Algorithmic Aspects of Elliptic Curves.

SynopsisThe theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics., The theory of elliptic curves is distinguished by the diversity of the methods used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry., The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, elliptic curves over finite fields, the complex numbers, local fields, and global fields. The last two chapters deal with integral and rational points, including Siegel's theorem and explicit computations for the curve Y 2 = X 3 + DX., 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.

LC Classification NumberQA564-609